Seminario di Algebra e Geometria

Sofia Lambropoulou (National Technical University of Athens)

24

2026

The theory of doubly periodic tangles

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Doubly periodic tangles (DP tangles) are entanglements of curves embedded in the thickened plane that are periodically repeated in two transversal directions. They are useful in many scientific fields for the study of physical systems. A better understanding of their topology, often associated to some physical properties, could allow the prediction of functions of the system. In the first part we will establish the topological background of the theory of DP tangles. DP tangles arise as universal covers of knots and links in the thickened torus. We study their DP isotopies via an equivalence relation between their generating (flat) motifs. In the second part we present some isotopy invariants of DP tangles, used for distinguishing their properties.

A spin on Gromov-Witten theory

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

We study linear systems on a smooth complex curve C ⊂ P3 cut out by cones of fixed degree. We develop a systematic study of the families of these systems, considering their limits, their infinitesimal behaviour and some associated geometric structures. As an application, we prove the existence of a pencil of quartics with only one base point and all whose members are irreducible. The work is in collaboration with R. Moschetti and L. Stoppino

Equiaffine immersions and pseudo-Riemannian space forms

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In the first part of the seminar, we will explore, through examples, three different areas of geometry that exhibit a still largely unexplored connection: affine differential geometry, pseudo-Euclidean geometry of neutral signature, and para-complex geometry. The explanation of this connection will form the basis of the second part of the seminar, where we will go into greater detail and show how it can be effectively used to address and solve certain problems that arise naturally in these settings.

Dicembre

09

2025

Gerard Freixas Montplet

On the Deligne-Riemann-Roch isomorphism

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In a letter to Quillen from 1985, Deligne raised the question of lifting the degree-one part of the Grothendieck-Riemann-Roch formula to a canonical, functorial isomorphism of line bundles. This conjectural isomorphism would provide a characteristic class-type expression for the determinant of the cohomology of Knudsen-Mumford. Deligne addressed this question for families of curves, relying on previous work by Mumford and Deligne-Mumford on moduli spaces of curves. Deligne’s program remained essentially open since then. In the recent years, Dennis Eriksson and myself have been developing a functorial relative intersection theory valued in line bundles, relying on some partial developments due to Elkik in the late 80s. As a result, we have established a Grothendieck-Riemann-Roch isomorphism as conjectured by Deligne. In this talk, I will motivate and explain the statement of our result.

On smooth fillings of Calabi-Yau manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

For a family of projective manifolds over a punctured disc, it is not always possible to extend it smoothly. One obstruction to these extensions is given by monodromy. Famous good reductions theorems state that for abelian varieties trivial monodromy implies the possibility to fill smoothly. For general Calabi-Yau families there are however examples of families of trivial monodromy with no smooth fillings. Motivated by our work in mirror symmetry we introduce an invariant that give obstructions to the existence of smooth fillings in these cases, even after finite base extensions and birational equivalence. In this talk I will overview known results in the area and explain how the new invariant generalizes previously known cases. This is joint work with Gerard Freixas.

Relative Brauer groups for finitely generated extensions: a geometric approach

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

I am going to report on joint work in progress with Giulio Bresciani and Zinovy Reichstein. Brauer groups are basic invariants of fields, of great importance in non-commutative algebra, number theory and algebraic geometry; the Brauer group Br(K) of K is a torsion abelian group that can be defined in three different ways: cohomological, algebraic, using central simple algebras, and geometric, using Brauer-Severi varieties, which are varieties over K that become isomorphic to projective spaces extending the scalars to the algebraic closure of K. If L/K is a field extension, the relative Brauer group Br(L/K) is the subgroup of Brauer classes in Br(K) which vanish in Br(L). If L is finitely generated over K, we can think of L as the field of rational functions on an algebraic variety M over K. We are going to present some results connecting finite p-subgroups of Br(L/K) and the geometry of M. As a consequence we obtain that Br(L/K) is finite if M is smooth, and the Euler characteristic of its structure sheaf is plus or minus 1. In the first part of my talk I will review the basic theory of Brauer groups; in the second part I will state the main result, and give a brief sketch of proof.

On the geometry of the Torelli locus

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

I will report on some results on the geometry of the Torelli locus in A_g. In particular, I will concentrate on the study asymptotic directions in the tangent bundle of the moduli space of curves of genus g, namely those tangent directions that are annihilated by the second fundamental form of the Torelli map. I will show that there exist examples of asymptotic directions for any g >3. I will present one main result obtained in collaboration with E. Colombo and G.P. Pirola, saying that if the rank of a tangent direction at [C] (with respect to the infinitesimal deformation map) is less than the Clifford index of the curve C, then the tangent direction is not asymptotic.

The Morrison Cone Conjecture under Deformation

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Let Y be a Calabi—Yau variety. The Morrison Cone Conjecture is a fundamental conjecture in Algebraic Geometry on the geometry of the nef cone and the movable cone of Y: while these cones are usually not rational polyhedral, the cone conjecture predicts that the action of Aut(Y) on Nef(Y) admits a rational polyhedral fundamental domain, and that the action of Bir(Y) on Mov(Y) admits a rational polyhedral fundamental domain. Even though the conjecture has been settled in special cases, it is still wide open in dimension at least 3. We prove that if the cone conjecture holds for a smooth Calabi-Yau threefold Y, then it also holds for any smooth deformation of Y.

Alternative Compactifications of M_{g,n} via Theta-Semistability and Cluster Algebras

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Irreducibility of Severi varieties on toric surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Higher Gaussian maps for special classes of curves

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Hyper-holomorphic bundles on hyperkaehler manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Hyper-holomorphic bundles on hyperkaehler manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Projective models of hyperkähler manifolds and Coble type hypersurfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Geometric descriptions for locally complete families of projective hyperkähler manifolds are only known in very few cases. In this talk, we first describe the projective geometry of the Hilbert square of a K3 surface of genus 7 or 8, by making use of the Mukai model: in both cases, it can be realized as a degeneracy locus on an ambient homogeneous space. From this, we deduce a geometric description for the two locally complete families of K3^[2]-type (square 4 and square 6 with divisibility 1), in terms of Coble type hypersurfaces. If time permits, I will present the construction of an explicit one-dimensional family admitting a large finite automorphism group, as a baby example towards understanding the whole moduli space. Based on a joint work with Ángel Ríos Ortiz and Andrés Rojas, and an ongoing work also with Benedetta Piroddi.

Spaces and the simplicial coalgebra of chains

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Compactified Jacobians: geometric and combinatorial aspects

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The Jacobian variety of a family of singular curves is not proper in general. Several (mostly equivalent) constructions of modular compactifications have been proposed since the 1950s. These classical results, however, do not give a complete classification. After recalling some equivalent constructions and properties of the Jacobian variety of a smooth projective curve, we show what is lost as soon as we drop the smoothness hypothesis. We introduce the notion of a V-compactified Jacobian, together with its dependence on combinatorial V-stability condition, and show that it strictly generalises the preexisting notion of "classical" compactified Jacobian. In particular, we see that V-compactified Jacobians completely classify smoothable compactified Jacobians in the case of a nodal curve. This is joint work with Nicola Pagani and Filippo Viviani.

Perverse schobers and the McKay correspondence

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Two-dimensional McKay correspondence originated in the observation by John McKay that the representation graph of a finite subgroup G of SL_2(C) coincides with the Coxeter graph of an affine Lie algebra \mathfrak{g} of ADE type. It turned out that the combinatorics of \mathfrak{g} control not only the representation theory of G but also the geometry of the minimal resolution Y of C^2/G. In the first half of the talk I will give a gentle introduction to the subject, illustrated by examples. We will review the finite subgroups of SL_2(C), the McKay quiver Q of G, the geometry of the minimal resolution Y, and its construction as a moduli space of semistable representations of Q. The stability parameter space \Theta with the stratification by the semistable walls coincides with the Cartan algebra \mathfrak{h} of \mathfrak{g} stratified by root hyperplanes. I will show how the reflections in the classes of the exceptional curves on Y define an action of the braid group B_{\mathfrak{g}} on the cohomology, K-theory, and the derived category D(Y) of Y. In the second half of the talk, I will report on the ongoing project to construct a certain categorical structure on an affine hyperplane arrangement on \mathfrak{h} refining that of the root hyperplanes. The braid group action above can be viewed as a categorical local system with the fibre D(Y) on the open stratum of \mathfrak{h}/W, where W is the Weyl group. We aim to extend this to a W-equivariant categorical perverse sheaf, a “perverse schober”, on the whole of the affine hyperplane arrangement. This is joint work with Arman Sarikyan (LIMS).

Boundedness of fibered Calabi-Yau varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

I will explain ideas and techniques behind recent results showing that fibered CY varieties are bounded, starting from the elliptic case and then moving to the case of higher relative dimension.

Birational Geometry of Hypersurfaces in Products of Weighted Projective Spaces

06

2025

Francesco Antonio Denisi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Mori dream spaces form a class of algebraic varieties that play a significant role in birational geometry, as they exhibit ideal behaviour within the minimal model program. In this talk, we discuss the birational geometry of hypersurfaces in products of weighted projective spaces, focusing particularly on cases where they are Mori dream spaces. We generalize the results obtained by J.C. Ottem and, if time permits, we will address the Kawamata-Morrison cone conjecture for certain anticanonical Calabi-Yau hypersurfaces in products of weighted projective spaces.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Stability conditions on noncommutative varieties and the geometry of moduli spaces.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

TBA

Cremona transformations of P3 stabilizing quartic surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

We are interested in Gizatullin’s problem which consists of the following question: Given a smooth quartic surface S ⊂ P3, which automorphisms of S are induced by Cremona transformations of P3? Cremona transformations of P3 can be written as a composition of a finite sequence of elementary maps. This is an algorithmic process called the Sarkisov Program. In this talk, we will solve Gizatullin’s problem when S ⊂ P3 has Picard number two by using the Sarkisov program. The results that will be presented are in collaboration with Ana Quedo, and with Carolina Araujo and Sokratis Zikas.

Homology of configuration spaces of surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

This is joint work with Andreas Stavrou. For a compact orientable surface S of genus g with one boundary component and for an odd prime number p, we study the homology of the unordered configuration spaces C(S) := coprod_{n>=0} C_n(S) with coefficients in F_p. We describe H_*(C(S); F_p) as a bigraded module over the Pontryagin ring H_*(C(D); F_p), where D is a disc, and give a splitting as direct sum of certain well-behaved quotients of this ring. We also consider the action of the mapping class group Mod(S) on the homology, and identify the kernel of the action with the subgroup of Mod(S) generated by separating Dehn twists and p-th powers of Dehn twists. We compare with some partial results about the Mod(S)-action on ordered configuration spaces, obtained in joint work with Jeremy Miller, Jennifer Wilson, and Andreas Stavrou.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Persistent homology and vector fields

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

We give an introduction to Topological Data Analysis (TDA), focusing on Persistent Homology. This is a technique to extract information about the shape of data and summarize it as a collection of intervals, also called a barcode. If the data is given in the form of a nice enough function on a smooth manifold, there are connections to Morse theory, where the topology of the underlying manifold is related to the critical points of the function. We then move the focus to Morse-Smale vector fields, which are a class of vector fields with good structural properties, and present new approaches to extend the aforementioned methods to these objects. We do so by applying different algebraic methods such as parametrized chain complexes and spectral sequences.

Divisorial elementary Mori contractions of maximal length

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

An elementary Mori contraction from a smooth variety X is a morphism with connected fibres onto a normal variety which contracts a single extremal ray of K_X-negative curves. Thanks to a result by P. Ionescu and J. Wisniewsi, we know that the length of such a contraction (i.e. the minimal degree -K_X can have on contracted rational curves) is bounded from above. In a paper which dates back to 2013, A. Höring and C. Novelli studied elementary Mori contractions of maximal length, that is, elementary Mori contractions for which the upper bound is met. Their main result exhibits the structure of a projective bundle for the locus of positive-dimensional fibres up to a birational modification. In my talk, I will move to the submaximal case, in other words the case where the length equals its upper bound minus one, and focus on the divisorial case.

Birational Geometry and the Moduli Spaces of some Varieties of General Type

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In the first part of the seminar, I will discuss the concept of Birational Geometry, introduce some birational iInvariants and the moduli spaces of varieties of general type, along with a few examples that I consider significant. In the second part, I will present some recent results, obtained in collaboration with S. Coughlan, Y. Hu, and T. Zhang, on certain moduli spaces of three-dimensional varieties of general type.

Symmetry counts: an introduction to equivariant Hilbert and Ehrhart series

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Quillen stratification in equivariant homotopy theory

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Growth of automorphisms of special groups

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Given a (nice) group G, we are interested in how fast the length of a group element can grow when we apply powers of a given outer automorphism of G. If the group G is free or the fundamental group of a closed surface, classical train-track techniques give a complete and precise picture. This can be extended to automorphisms of all negatively curved (a.k.a. Gromov hyperbolic) groups G, using Rips-Sela theory and the canonical JSJ decomposition. Very little seems to be known beyond this setting. We study this problem for a broad class of non-positively curved groups: "special" groups in the Haglund-Wise sense. In this setting, we prove that: (1) the top exponential growth rate of any automorphism is an algebraic integer; (2) if the automorphism is untwisted, then it admits only finitely many growth rates, and each of these is polynomial-times-exponential.

Gennaio

21

2025

Pierpaola Santarsiero

TBA

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Dehn funtions of subgroups of direct products of free groups

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Dicembre

03

2024

Jordi Emanuel Hernandez Gomez

The cubo-cubic transformation of the smooth quadric fourfold is very special

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The classification of special Cremona transformations is a classical problem that is completely understood when the dimension of the base locus is at most three. This result is the culmination of the work of B. Crauder, S. Katz, L. Ein, N. Shepherd-Barron, and G. Staglianò from 1987 to 2019. However, what do we know about special self-birational transformations of varieties different from projective spaces? A 2021 article by M. Bernardara, E. Fatighenti, L. Manivel, and F. Tanturri, titled "Fano Fourfolds of K3 Type," explores 64 families of Fano fourfolds with a K3-type structure. One of these families, labeled K3-33, gives rise to a special cubo-cubic self-birational transformation of the smooth quadric fourfold. The base locus of this transformation is a non-minimal K3 surface of degree 10 with two skew (-1)-lines, and the base locus of the inverse map is also a non-minimal K3 surface of the same type. However, the two associated K3 surfaces turn out to be non-isomorphic Fourier-Mukai partners. My recent work shows that this is the only special self-birational transformation for a smooth quadric fourfold and explores its geometry. This represents a first step toward the classification of special self-birational transformations of smooth quadrics with a base locus of dimension at most three.

Classifying Fano 4-folds with a rational fibration onto a 3-fold

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Wild ramification of p-adic analytic curves

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Wild ramification is commonly regarded as pathological. Contrary to this belief, we will explain how it can be effectively controlled via p-adic analytic geometry. Let f:C->D be a cover of p-adic curves. By studying the p-adic analysis of the pullback map f*:Omega_D -> Omega_C on differential forms, one can obtain a ‘different function’ delta: C^{an} ->\R_{ge 0} that measures ramification of the associated p-adic analytification f^{an}:C^{an}\to D^{an}. The potential theory of the different reveals new `analytic’ Riemann-Hurwitz formula: for a simultaneous skeleton the Laplacian of the different equals the `tropical’ relative canonical divisor. This complements results of Temkin e.a. over an algebraically closed ground field, extending them to discretely valued fields. The p-adic different function provides a new tool to study surfaces fibered over Z_p. We illustrate this by explaining the proof of a conjecture of Lorenzini on surface p-cyclic quotient singularities, which relates the combinatorial complexity of a resolution to the valuation theory of local rings.

Thin subgroups of lattices

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Fusion categories and Coxeter theory

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

teoria delle categorie

TBA

An explicit wall crossing for the moduli space of hyperplane arrangements

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The moduli space of hyperplanes in projective space has a family of geometric and modular compactifications that parametrize stable hyperplane arrangements with respect to a weight vector. Among these, there is a toric compactification that generalizes the Losev-Manin moduli space of points on the line. We study the first natural wall crossing that modifies this compactification into a non-toric one by varying the weights. As an application of our work, we show that any Q-factorialization of the blow up at the identity of the torus of the generalized Losev-Manin space is not a Mori dream space for a sufficiently high number of hyperplanes. Additionally, for lines in the plane, we provide a precise description of the wall crossing. This is joint work with Patricio Gallardo.

Punti razionali su 3-varieta' con classe anticanonica nef su campi finiti

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Un teorema di Esnault afferma che le varietà di Fano lisce su campi finiti hanno punti razionali. Cosa succede se rilassiamo le condizioni relative alle proprietà di positività della classe anti-canonica? In questo seminario, discuterò il caso delle 3-varietà con classe anti-canonica nef. In particolare, dimostriamo che, nel caso di dimensione di Kodaira negativa, punti razionali esistono se la cardinalità del campo è maggiore di 19. Nel caso K-triviale, proviamo un risultato simile, a condizione che il morfismo di Albanese sia non banale. Questo è un lavoro con S. Filipazzi.

Good moduli spaces of A_r-stable curves

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Affine laminations and coaffine representations of surface groups

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Ottobre

15

2024

Alvaro Gutierrez Caceres

The combinatorics of plethysm

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The composition of GL(n) representations is know as the plethysm of the respective representations. Understanding the structural coefficients of plethysm is a notoriously hard problem, featured in Stanley's list of important problems in algebraic combinatorics. In this talk, we explore the relevance of this problem, its connections to P vs NP, and discuss some combinatorial approaches to their study.

Symplectic automorphisms on hyperkaehler varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Linear degenerations of Schubert varieties via quiver Grassmannians

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Quiver Grassmannians are projective varieties parametrising subrepresentations of quiver representations. Their geometry is an interesting object of study, due to the fact that many geometric properties can be studied via the representation theory of quivers. For instance, this method was used to study linear degenerations of flag varieties, obtaining characterizations of flatness, irreducibility and normality via rank tuples. We provide a construction for realising smooth Schubert varieties as quiver Grassmannians and desingularizing non-smooth Schubert varieties. We then exploit this construction to define linear degenerations of Schubert varieties, giving a combinatorial description of the correspondance between their isomorphism classes and the B-orbits of certain quiver representations.

Geometry of independence

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Matroids combinatorially abstract the ubiquitous notion of "independence" in various contexts such as linear algebra and graph theory. Recently, an algebro-geometric perspective known as "combinatorial Hodge theory" led by June Huh produced several breakthroughs in matroid theory. We first give an introduction to matroid theory in this light. Then, we introduce a new geometric model for matroids that unifies, recovers, and extends various results from previous geometric models of matroids. We conclude with a glimpse of new questions that further probe the boundary between combinatorics and algebraic geometry. Joint works with Andrew Berget, Alex Fink, June Huh, Matt Larson, Hunter Spink, and Dennis Tseng.

The Hermitian geometry of the Chern connection

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

We survey some analytic problems concerning the geometry of the Chern con- nection of Hermitian manifolds: the existence of metrics with constant Chern-scalar curvature; the generalizations of the K¨ahler-Einstein condition to the non-K¨ahler setting; the convergence of the normalized Chern-Ricci flow on compact complex surfaces; the asymptotic behaviour of Monge-Amp`ere volumes of Hermitian metrics in the ddc-class. The talk is based on joint collaborations with Simone Calamai, Mauricio Corrˆea, Vincent Guedj, Chinh Lu, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti.

Special Hermitian metrics on complex manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Kähler geometry lies in the intersection of complex geometry, Riemannian geometry and symplectic geometry. From the complex geometric point of view, it is therefore natural to consider special non-Kähler Hermitian metrics on complex manifolds. Among them, pluriclosed metrics deserve particular attention. These metrics always exist on compact complex surfaces but the situation in higher dimension is very different. We will discuss several properties concerning these metrics also in relation with the Bismut connection having Kähler-like curvature. This is joint work with G. Barbaro and F. Pediconi.

On Mirror Symmetry for (singular) del Pezzo surfaces out of the known constructions

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Mirror symmetry predicts a correspondence between the complex geometry (the B side) and the symplectic geometry (the A side) of certain pairs of objects. In this talk I will discuss some aspects of mirror symmetry for Fano varieties, focusing on certain orbifold del Pezzo surfaces falling out of the standard mirror constructions. In particular, motivated by Homological Mirror Symmetry, I will describe the derived category of the surfaces (their B side), and mention some results on the category encoding the A side of their only-known Hodge-theoretic mirrors. This is based on work with A. Corti and work in progress with F. Rota.

The importance of being a weighted blow-up

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Blow-ups are fundamental tools in algebraic geometry, and there are several results (e.g the famous Castelnuovo's theorem) that can be used to determine when a variety is obtained as a blow-up of a smooth variety along a smooth center. Weighted blow-ups play a similar role for stacks. In this talk I will present a criterion for finding out if a smooth DM stack is a weighted blow-up. I will apply this result for showing that certain alternative compactifications of moduli of marked elliptic curves are obtained via weighted blow-ups (and blow-downs). This in turn will prove to be useful in order to compute certain invariants, like Chow rings or Brauer groups. First part of this talk is a joint work with Arena, Inchiostro, Mathur, Obinna and Pernice; the second part of this talk is a joint work with L. Battistella.

On derived categories and motivic classes of projective varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Tba

A new vanishing criterion for bounded cohomology

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Promoted by the seminal work of Gromov, bounded cohomology is a powerful and by now well-established tool to study properties of groups and spaces. However, it is often difficult to compute and, beyond the case of amenable groups, many questions remain widely open. In joint work with Francesco Fournier-Facio, Yash Lodha and Marco Moraschini, we present a new algebraic condition that implies the vanishing of the bounded cohomology of a given group for a big family of coefficient modules. This condition is satisfied by many non-amenable groups of topological, geometric or algebraic origin, and even, surprisingly, by the generic countable group (in a precise sense). In the first part we will present the context and examples, and in the second part we will give some ideas beyond the proof of this result.

On finitely Levi nondegenerate closed homogeneous CR manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

A complex flag manifold F= G /Q decomposes into finitely many real orbits under the action of a real form of G. Their embeddings into F define CR manifold structures on them. We give a complete classification of all closed simple homogeneous CR manifolds that have finitely nondegenerate Levi forms.

Involuzioni su schemi di Hilbert su K3

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Torsors for geometrically admissible actions of monoidal categories

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

fisica matematica

In Tannakian formalism, groups and generalizations like groupoids and Hopf algebras can be reconstructed from the fiber functor which is a forgetful strict monoidal functor from its category of modules to the category of vector spaces. Can we do something similar for the actions of groups, their properties and the generalizations ? If a Hopf algebra H coacts on an algebra A by a Hopf action (that is, A becomes an H-comodule algebra) then the category of H-modules acts on the category of A-modules, this action strictly lifts the trivial action of the category of vector spaces on A-modules and also H lifts to a comonoid in H-modules inducing a comonad on the category of A-modules. The comodules over this comonad are the analogues of H-equivariant sheaves and a Galois condition can be stated in terms of affinity in the sense of Rosenberg. We propose taking these properties as defining for a general framework allowing for the definition of Galois condition/principal bundles/torsors in a number of geometric situations beyond the cases of Hopf algebras coacting on algebras. We also sketch how many other examples like coalgebra-Galois extensions and locally trivial nonaffine noncommutative torsors fit into this framework.

Locality for quantum principal bundles

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

fisica matematica

In topology, principal bundles are often assumed to be locally trivial. While the original Cartan definition of a principal bundle is inherently global, and has been successfully promoted to noncommutative geometry in the compact/unital setting, the standard concept of local triviality is hard to implement for quantum principal bundles. The goal of this talk is to review the state of the art of the concept of locality for quantum principal bundles. We begin with a guided tour concerning the topology of compact Hausdorff principal bundles. In particular, the difference between the piecewise and local triviality shall be explained. In the main part of the talk, the local triviality of compact quantum principal bundles will be compared with the piecewise triviality of principal comodule algebras. (This is a review talk based on joint works with many collaborators.)

Voisin's Conjecture and Voisin's Map

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The opening segment will explore the conjectural relationships between Hodge structures and Chow groups. The Bloch-Beilinson conjecture suggests a functorial filtration on the Chow groups of smooth projective varieties, underpinned by natural axioms. We anticipate refined structures of Bloch-Beilinson filtrations, particularly within projective hyper-Kähler and Calabi-Yau manifolds, as proposed by Beauville and Voisin. Linking these to the generalized Hodge conjecture allows the formation of explicit conjectures. Verifying these for specific Calabi-Yau manifolds or projective hyper-Kähler manifolds could substantiate both the Bloch-Beilinson and generalized Hodge conjectures. **Part 2 title:** *Voisin's Conjecture and Voisin's Map* Voisin's work, which crafts a series of K-trivial varieties from cubic hype-resurfaces and self-rational maps on them, called the Voisin's map will be the focus here. Notable among these is the Fano variety of lines of a cubic fourfold, a dimension 4 hyper-Kähler manifold. The Voisin's map in this case has been extensively studied. We'll examine higher-dimensional examples, which are all Calabi-Yau manifolds. This session aims to study the geometry of these manifolds and apply their structural insights to the conjectures on algebraic cycles discussed in Part 1, utilizing Voisin's self-rational map as a pivotal analytical tool.

Derived categories of G/P

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

teoria delle categorie

Starting from the pioneering works of Beilinson and Kapranov, derived categories of coherent sheaves on homogeneous varieties G/P have attracted a lot of attention over the last decades. We’ll begin by an introduction into this area and then discuss more recent developments related to Lefschetz exceptional collections, quantum cohomology and homological mirror symmetry.

Curve counting on the Enriques surface and Borcherds automorphic form

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. Klemm and Marino conjectured a formula expressing the Gromov–Witten invariants of the local Enriques surface in terms of automorphic forms. In particular, the generating series of elliptic curve counts on the Enriques should be the Fourier expansion of (a certain power of) Borcherds famous automorphic form on the moduli space of Enriques surfaces. In this talk I will explain a proof of this conjecture.

Geometry of Hyperkaehler Manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

TBA

Mapping class group, triangolazioni e cicli interi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Il volume simpliciale è un invariante per varietà compatte introdotto da Gromov nel 1982. Pur essendo definito solo utilizzando l'omologia singolare, è strettamente correlato alle strutture geometriche che una varietà può supportare: ad esempio, si annulla su varietà che ammettano metriche con curvatura di Ricci non negativa, ed è positivo per varietà di curvatura negativa. In questo seminario confronteremo il volume simpliciale con alcuni invarianti ad esso correlati, come il minimo numero di simplessi in una triangolazione, o il minimo numero di simplessi singolari in un rappresentante della classe fondamentale a coefficienti interi. A tale scopo, introdurremo un nuovo invariante, chiamato "Filling volume", definito sul mapping class group di varietà. Lavoro in collaborazione con Federica Bertolotti.

On semiample vector bundles

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In the first part we will review some notions of positivity and base loci for line bundles and how to generalise them to the case of higher rank vector bundles. In the second part we will discuss some geometric interpretations of semiampleness of the cotangent bundle, and some characterizations of abelian verieties and compact complex parallelizable manifolds.

Character stacks and varieties for Riemann surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Given a Riemann surface X, character stacks and varieties are geometric objects which parametrize certain representations of the fundamental group of X or, equivalently, local systems with prescribed local monodromies. These objects have a rich geometry and are related, for instance, to the moduli spaces of Higgs bundles through non abelian Hodge correspondence. The cohomology of character stacks and varieties is almost completely understood in the case of a generic choice of monodromies, thanks to the work of Hausel, Letellier and Rodriguez-Villegas and Mellit. In the non-generic case, the geometry of these objects becomes considerably more complicated and their cohomology has not been studied much until recently. In the first part of the talk, I will introduce and define character stacks and varieties and review the known results about the generic case. In the second part, I will focus on the non-generic case and give a sketch of the proof of a formula for the E-series of non-generic character stacks, which is the main result of my PhD thesis.

Holomorphic dynamics in C^2: trascendental Hénon maps

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

analisi matematica

While the dynamical behaviour of the iteration of holomorphic functions in one variable is well known, the situation is drastically different in several variables. This should not be a surprise. After all, even from the geometrical point of view the two situations are drastically different: in several variables there is no theorem similar to the Riemann uniformization theorem, and even simple domains as the ball and the polydisk are not biholomorphically equivalent; a holomorphic function of several variables is not determined if known on a set with an accumulation point; there are open domains which are not the maximal natural domain of any holomorphic function (Hartogs' phenomenon). Thus, understanding the dynamical behaviour of the iterations of holomorphic maps, even of automorphisms of C^2, is quite difficult. There are some classes of functions, which can be thought of as being of dimension 1.5, for which it is easier to find results, using theorems of the 1-dimensional theory together with some tools of geometrical flavour. Among these, are the Hénon maps: F(z,w)=(f(z)-\delta w , z) where f is a one-dimensional entire function, and \delta is a complex number. If f is a polynomial, they are a valid playground to understand the behaviour of all polynomial automorphisms of C^2. If f is trascendental, they are not enough to grasp all the possible dynamical behaviours of automorphisms of C^2, but nevertheless they are a starting point. In the first part of the seminar I will present the state of the art of holomorphic dynamic in C^2, while in the second part I will talk about recent results on trascendental Hénon maps, in collaboration with Anna Miriam Benini, Veronica Beltrami and Michela Zedda.

On the (non-)uniqueness of minimal surfaces in hyperbolic three-space

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The Asymptotic Plateau Problem in the hyperbolic space is the problem of existence of minimal surfaces with a prescribed Jordan curve as a boundary “at infinity”. Since the work of Anderson in the 1980s, it is known to have a solution, which is in general not unique. In this talk, I will present an example of a Jordan curve bounding uncountably many minimal discs. I will also present some criteria for uniqueness. This is joint work with Zheng Huang and Ben Lowe.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Delle tre geometrie a curvatura costante K, quella iperbolica (corrispondente a K = -1) è di gran lunga la più ricca. In questo seminario parleremo di varietà iperboliche, cioè varietà Riemanniane di curvatura costante -1 compatte (o più generalmente complete e di volume finito). Vedremo che tali oggetti esistono in ogni dimensione, e cercheremo di capirne la topologia - in particolare esaminando quali fra queste varietà possono avere una struttura di fibrato (da un lavoro in collaborazione con Italiano e Migliorini).

The 3D index and Dehn surgery

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

After giving a general introduction to Dimofte-Gaiotto-Gukov's 3D index for cusped hyperbolic 3-manifolds, we'll dive into some of its relations with basic hypergeometric series. Then we'll describe an ongoing effort to prove how the 3D index changes under Dehn surgery. This is work in progress with Profs C. Hodgson and H. Rubinstein.

On finite generation in magnitude (co)homology, and its torsion

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

interdisciplinare

logica

teoria delle categorie

Magnitude homology, as introduced by Hepworth and Willerton, is a bigraded homology theory of metric spaces that categorifies Leinster's notion of magnitude. We will show that, when restricting to graphs of bounded genus, magnitude homology is a finitely generated functor. As a consequence, we will prove that the ranks of magnitude homology, in each homological degree, grow at most polynomially in the number of vertices, and that its torsion is bounded. We will use the categorical framework of Groebner categories developed by Sam and Snowden, in the spirit of Ramos, Miyata and Proudfoot. This is joint work with C. Collari.

Alexander polynomials and symplectic curves in CP^2

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Libgober defined Alexander polynomials of (complex) plane projective curves and showed that it detects Zariski pairs of curves: these are curves with the same singularities but with non-homeomorphic complements. He also proved that the Alexander polynomial of a curve divides the Alexander polynomial of its link at infinity and the product of Alexander polynomials of the links of its singularities. We extend Libgober's definition to the symplectic case and prove that the divisibility relations also hold in this context. This is work in progress with Hanine Awada.

Deformation theory via DGLA and application to deformations of sheaf.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In this talk we first introduce differential graded Lie algebra (DGLA) and their role in deformation theory. Then, we apply this techniques to analyse some deformation problems, such as deformations of pair (variety,sheaf) or deformations of locally free sheaves and subspace of sections.

Semiregularity maps and deformations of modules over Lie algebroids

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Let A\subset L be a flat inclusion of Lie algebroids, i.e., a Lie pair, on a smooth separated scheme over a field of characteristic 0. For every locally free A-module M we define its semiregularity maps and prove that, under some additional assumptions, they annihilate obstructions to deformations of M. In case A=0 and L=tangent bundle, this construction gives to the usual Buchweitz-Flenner's semiregularity maps for coherent sheaves.

A Feigin-Frenkel theorem with n singularities

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

TBA

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

TBA

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

TBA

Birational equivalence and deformation equivalence for HK manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Hyperkähler manifolds are one of the building blocks of compact complex Kähler manifolds with trivial first Chern class. Huybrechts proved that if X and Y are birational HK then they have to be deformation equivalent, but this implication is far from being an equivalence. It is then natural to ask which additional assumptions are needed to get that two HK manifolds of the same deformation type are birational. In this talk I will give a gentle introduction on HK manifolds and lattice theory for HK manifolds, then I will provide a lattice-theoretic criterion to determine when two HK manifolds of OG10 type are birational (joint work with C.Felisetti and F.Giovenzana), and I will show an application to the Li-Pertusi-Zhao variety. If time permits I will discuss another related topic about understanding sufficient numerical conditions to determine the deformation type of a given HK manifolds of a fixed dimension. This is a first result about a joint work in progress with P.Beri.

An algorithmic method to compute plat-like Markov moves in genus two 3-manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

This dissertation will deal with the equivalence of links in 3-manifolds of Heegaard genus two. We construct an algorithm (implemented in C++) which, starting from a description of such a manifold introduced by Casali and Grasselli that uses 6-tuples of integers and determines a Heegaard decomposition of the manifold, allows to find the words in B_2,2n, the braid group on 2n strands of a surface of genus two, that realize the plat-equivalence for links in that manifold. In this way we extend the result obtained by Cattabriga and Gabrovšek for 3-manifolds of Heegaard genus one to the case of genus two. We describe explicitly the words for a notable family of 3-manifolds.

The combinatorics and algebra of the Chow ring of a matroid

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Matroids encode in a combinatorial way the notion of linear independence and can be seen as a generalization of matrices, graphs and hyperplane arrangements. The main protagonist of this talk is an invariant called the Chow ring of a matroid, whose definition is given in analogy with the one arising from Algebraic Geometry. Long-standing combinatorial conjectures were solved by the introduction of this and other related geometric tools, which in turn have remarkable combinatorial features; for example, their Hilbert series seem to be real-rooted. After a friendly introduction to Matroid Theory, the plan of the talk is to answer the following questions. 1) How can we study the Hilbert series without actually building the whole graded vector space?While trying to answer this question, different algebraic and combinatorial objects will arise along the way, like the Kazhdan-Lusztig-Stanley polynomials. Help will come both from Poset Theory and Polytope Theory. 2) After obtaining these combinatorial answers, which tools can be lifted back to the higher categorical level we started from?In particular, we are concerned with questions regarding properties of some functors in a new category of matroids. Time permitting, we will also transform all these invariant into graded representations of the group of symmetries of the matroid. This is based on a joint work with Luis Ferroni, Jacob Matherne, and Matthew Stevens and an ongoing project with Ben Elias, Dane Miyata, and Nicholas Proudfoot.

Aprile

20

2023

Kieran Gregory O'Grady

Stable modular sheaves with moduli

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Prima parte: Introduzione ai moduli di fasci stabili su varieta' complesse proiettive lisce, in particolare superfici K3. Seconda parte: Esporro' la costruzione di componenti irriducibili di spazi di moduli di fasci (semi)stabili su varieta' HK polarizzate di tipo K3^{[n]} che sono deformazioni di spazi di moduli di dimensione arbitrariamente alta di fasci (semi)stabili su superfici K3. La costruzione e' stata motivata da esempi di Enrico Fatighenti.

Foliazioni sulle varietà razionali omogenee.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Una foliazione di codimension 1 su una varietà proiettiva X puo' essere vista come un sotto fascio (saturato) F del fibrato tangente TX, stabile per il bracket di Lie. Una volta fissato il determinante di F, tali foliazioni formano un sottoinsieme localmente chiuso dello spazio delle 1-forme sur X a valori in un fibrato in rette L. Studieremo lo spazio di queste foliazioni quando X è una varietà razionale omogenea di numero di Picard 1, per le scelte più semplici possibili di L, in particolare quando X è una grassmanniana o più generalmente una varietà cominuscola. Lavoro in collaborazione con V. Benedetti e A. Muniz.

Loop braid groups and generalisations of Hecke algebras

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

TBA

Shi arrangement and low element

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In this talk, I will survey the notion of Garside families in relation with the word problem for Artin-Tits groups. In particular, Dehornoy, Dyer and I proved that there exist a finite Garside family in any Artin-Tits group G. Then I will discuss how this family is (suprisingly) related to a well-known finite hyperplane sub-arrangements, the Shi arrangement, of the reflection arrangement of the Coxeter group associated to G.

The monodromy of moduli spaces of sheaves on K3 surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

There are very few known deformation classes of irreducible symplectic manifolds, i. e. compact, connected Kahler manifolds that are simply connected and carry a holomorphic symplectic form spanning the space of closed holomorphic 2-forms. Among the tools that are used to study the birational geometry of these manifolds, the monodromy group is of the utmost important, and has been calculated for all the known deformations classes by Markman, Mongardi, Rapagnetta and Onorati. As soon as we allows singularities, we get into the world of irreducible symplectic varieties, for which we know many more deformations classes, and the monodromy group is still defined and plays a major role in the study of their birational geometry. In a joint work with Onorati and Rapagnetta we calculate the monodromy group of the examples of irreducible symplectic varieties given by moduli spaces of semistable sheaves on K3 surfaces.

Non-degeneracy of Enriques surfaces

21

2023

Davide Cesare Veniani

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Non-degeneracy of Enriques surfaces Enriques' original construction of Enriques surfaces dates back to 1896. It involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron. The question whether all Enriques surfaces arise through Enriques' construction has remained open for more than a century. In two joint works with G. Martin and G. Mezzedimi, we have now settled this question in all characteristics by studying particular configurations of genus one fibrations, and two invariants called maximal and minimal non-degeneracy. The proof involves so-called `triangle graphs' and the distinction between special and non-special 3-sequences of half-fibers. In this talk, I will present the classification of Enriques surfaces of low non-degeneracy and explain how this classification solves this long-standing problem.

Divisible convex sets with properly embedded cones

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

An open subset of real projective space is said to be properly convex if it is contained in an affine chart where it is convex and bounded. Together with their natural Hilbert metric and group of projective symmetries, properly convex sets are a rich source of geometry, dynamics, and group theory. Of particular interest are those that are divisible, that is, they admit a compact quotient by a discrete group of symmetries. In general, it is a challenging problem to construct such examples. After reviewing the global classification scheme, I will describe a new class of examples of divisible convex sets with special geometric properties (non-symmetric and non-strictly convex) that completes a missing part of the picture. This is joint work with Pierre-Louis Blayac.

Explicit measurable cocycles for actions at infinity of some semisimple Lie groups

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Since the formulation of Dupont's conjecture, it has been evident the importance to understand the boundedness of characteristic classes appearing in the cohomology ring of a semisimple Lie group. This problem is deeply related to Monod's conjecture, which relates the continuous bounded cohomology of a semisimple Lie group with its continuous variant. An important step towards a possible proof of those conjectures was the isometric realization of the continuous bounded cohomology of a semisimple Lie group G as the cohomology of the complex of essentially bounded functions on the Furstenberg-Poisson boundary (and more generally for any regular amenable G-space). Surprisingly, Monod has recently proved that the complex of measurable unbounded functions on the same boundary does not compute the continuous cohomology of G unless the rank of the group is not one, but an additional term appears. Nevertheless, there is a way to characterize explicitly the defect in terms of the invariant cohomology of a maximal split torus. In this seminar we will exhibit two main examples of such phenomenon: the product of isometry groups of real hyperbolic spaces and the group SL3. The first part of the seminar will be devoted to an overview about the state of art. Then we will move to examples and we will give a characterization of Monod's Kernel in low degree. Finally we will show that Monod's conjecture is true in those cases. In the second part of the seminar we will discuss in details the main results and the techniques we used, such as the explicit computation on Bloch-Monod spectral sequence. If time allows we will show how we can implement all this stuff using a software like Sagemath.

Algebre di vertice di Poisson, W-algebre e triple integrabili

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Nel seminario rispolvereremo alcune nozioni base di geometria di Poisson e di algebre di vertice di Poisson e il loro legame con le gerarchie integrabili di ODE/PDE. Parleremo in seguito di triple integrabili per le algebre di Lie semplici e della loro classificazione. La classificazione è usata per dimostrare che (quasi) tutte le W-algebre classiche affini W(g,f), una vasta famiglia di algebre di vertice di Poisson associate a un'algebra di Lie semplice g e un suo elemento nilpotente f, possiedono gerarchie integrabili di PDE bi-Hamiltoniane. Queste gerarchie generalizzano le gerarchie di Drinfeld-Sokolov che sono ottenute quando f è la somma dei vettori radice corrispondenti alle radici semplici.

Geometry of some infinite-type hyperbolic 3-manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In this talk I will give some purely topological construction for hyperbolic 3-manifolds with infinitely generated fundamental group, this will let us construct many infinite-type hyperbolic 3-manifolds. Then, I will say something about the set of hyperbolic structures that they can admit.

Infinitary combinatorics in condensed mathematics

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The rapid development, in the few short years since their introduction, of the closely related condensedand pyknotic mathematics frameworks of Clausen and Scholze, and Barwick and Haine, respectively, forms the immediate background for this talk. In the standard heuristic, these are frameworks for ``doing algebra with objects carrying a topology''; more concretely (but still imprecisely), they are frameworks for coordinated approaches to a wide variety of mathematical objects via their (pre)sheaf representations over the category of profinite sets (i.e., of totally disconnected compact Hausdorff spaces). In our talk's first half, we will introduce and survey these frameworks. In its second half, we will discuss some of the questions in infinitary combinatorics and set theory which these frameworks pose (with a particular focus on the ``condensed image'' of the Whitehead Problem), as well as some answers. This work is joint with Chris Lambie-Hanson.

Parametrizing subvarieties of a given algebraic variety

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Given an algebraic variety $X$ in a projective space, one can try to describe the families of curves (lines, conics...), surfaces (planes, quadrics...), and other subvarieties contained in $X$. These families are themselves algebraic varieties that can help understand the geometry of $X$. I will review the classical case when $X$ is a cubic hypersurface and then continue with the case where $X$ is a so-called Gushel-Mukai variety. This is work in collaboration with Alexander Kuznetsov.

A gentle introduction to measurable cocycles, rigidity and recent advances

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Measurable cocycles arise in different fields of mathematics, and cocycle rigidity is a self standing research topic that goes back to Zimmer's work. In the first part of this talk I will give a gentle introduction to cocycles, describing some explicit examples coming out in different situations such as orbit equivalences and measure equivalences. In the the second part I will describe a technique to investigate rigidity that involves bounded cohomology. A comparison with groupoids will be carried on and, time permitting, in the last part I will show how the above machinery might be described in the category of measured groupoids. This is the starting point of a joint work in progress with A. Savini about a theory of bounded cohomology for groupoids.

Sixfolds of generalized Kummer type and K3 surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

I will present a construction which associates to any sixfold K of generalized Kummer type a hyper-Kähler manifold Y deformation equivalent to a Hilbert scheme of lenght 3 subscheme on a K3 surface, relating the most well studied deformation types of hyper-Kähler manifolds in dimension 6. Our construction is reminiscent of the classical construction of Kummer K3 surfaces, in the sense that Y is obtained as resolution of the quotient of K by a group of symplectic automorphisms. As a consequence we are able to show that any projective sixfold K as above determines a well-defined K3 surface. We use this construction to prove that the Kuga-Satake correspondence is algebraic for infinitely many new families of K3 surfaces of general Picard rank 16.

Hyperelliptic Odd Coverings

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Hyperelliptic Odd Coverings" are a class of odd coverings of C -> P^1, where C is a hyperelliptic curve. They are characterized by the behavior of the hyperelliptic involution of C with respect to an involution of P^1. I will talk about some ways for studying this type of coverings: by fixing an effective theta characteristic, they correspond to the solutions of a certain type of differential equations. Considering them as elements in a suitable Hurwitz space, they can be characterized using monodromy and then studied from the point of view of deformations. When C is general in H_g, the number of possible Hyperelliptic Odd Coverings C -> P^1 of minimum degree is finite. The main result will be how to compute this number. This is a work in collaboration with Gian Pietro Pirola. In the first part of the talk, I will introduce the Hyperelliptic Odd Coverings from a geometric point of view, contextualizing them in the panorama of other enumerative works (in collaboration with Farkas, Naranjo, Pirola, Lian). In the second part of the talk he will dedicate myself to deepening some demonstrations and some open problems subject to future analysis.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Discs and other surfaces in 4-manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Symplectic rigidity of O'Grady tenfolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In the first half of the talk I will talk about automorphism groups of varieties, focusing in a second moment on irreducible symplectic varieties. In the second part of the talk I will present a result obtained jointly with L. Giovenzana, A. Grossi and D. Veniani on the (non-)existence of symplectic automorphisms on irreducible symplectic varieties obtained as symplectic desingularisation of moduli spaces of sheaves on K3 surfaces.

The Hodge structure of O'Grady's singular symplectic varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Starting from a question of Bakker and C. Lehn about the singularities type of O’Grady’s singular moduli spaces, we investigated their Hodge structure. As consequence of our (partial) computation of it, we deduced that O’Grady’s moduli spaces do not have finite quotient singularities. In the first part of this talk I will introduce O'Grady's moduli spaces, placing them in the setting of singular symplectic varieties. I will state our main result on their Hodge structure, and I will underline the relevant properties of it giving consequences on the singularities type of these spaces. In the second part I will go into the techniques we used for the (partial) computation of the Hodge structure, and I will present a strategy to complete the computation. This is a joint project with Franco Giovenzana

Delta and Theta operators expansions

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Delta and Theta operators are two families of operators on symmetric functions that show remarkable combinatorial properties. Delta operators generalise the famous nabla operator by Bergeron and Garsia, and have been used to state the Delta conjecture, an extension of the famous shuffle theorem proved by Carlsson and Mellit. Theta operators have been introduced in order to state a compositional version of the Delta conjecture, with the idea, later proved successful, that this would have led to a proof via the Carlsson-Mellit Dyck path algebra. We are going to give an explicit expansion of certain instances of Delta and Theta operators when t=1 in terms of what we call gamma Dyck paths, generalising several results including the Delta conjecture itself, using interesting combinatorial properties of the forgotten basis of the symmetric functions.

On Pfaffian representations of cubic hypersurfaces.

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Can a given polynomial (or some power thereof) be written as the determinant of a matrix with linear polynomials as entries? The answer to this question very much depends on the number of variables, the degree, and the generality of the polynomial. As it turns out, a positive answer is related to the existence of special vector bundles on the hypersurface cut out by the polynomial. In this talk I will try to explain this link and I will give an overview of the known results. In the second part, I will focus on two particular cases, cubic surfaces and cubic threefolds, and I will show how to explicitly realize their defining equation as the Pfaffian of a matrix with linear polynomials as entries.

Positivity on irreducible holomorphic symplectic manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The purpose of this talk is to discuss some positivity questions in hyper-Kähler geometry. The first part will be a short introduction to positivity in algebraic geometry, with an emphasis on the themes I will focus on during the second part. To be more precise, given an irreducible holomorphic symplectic (IHS) manifold, we study the volume function, the structure of the big cone and of the pseudo-effective cone and, time permitting, some convex bodies associated with any big divisor.

Zeta Functions in Arithmetic and Geometry

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In this talk we briefly introduce several definitions of Zeta functions that are used in the context of arithmetic varieties. These functions are in a vague sense related to each other and, at least the version defined by Hasse and Weil, arise from the Riemann zeta function. We will focus on the motivic zeta function attached to a Calabi-Yau variety X defined over a field K endowed with an ultrametric absolute value. We will explain what it means for a formal series with coefficients in the Grothendieck ring of varieties to be rational and how poles are defined. We will finally discuss the monodromy conjecture that relates those poles with the action of the absolute Galois group of K on the (étale) cohomology of X. We discuss how the Hilbert schemes of points on a surface behave with respect to the monodromy conjecture.

Symmetries of supergeometries related to nonholonomic superdistributions

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

We extend the Tanaka theory to the context of supergeometry and obtain an upper bound on the supersymmetry dimension of geometric structures related to strongly regular bracket-generating distributions on supermanifolds and their structure related to strongly regular bracket-generating distributions on supermanifolds and their structure reductions. Several examples will be demonstrated, including distributions with at most simple Lie superalgerbas as maximum symmetry. The talk is based on joint works with Andrea Santi, Dennis The and Andreu Llabres.

Special cubulation of strict hyperbolization

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

Hyperbolization procedures are constructions that turn a simplicial complex into a metric space of non-positive curvature. They were first introduced by Gromov, and later refined by Charney and Davis to produce spaces of strictly negative curvature. In the first part of the talk I will describe some notions of curvature, some hyperbolization procedures, and then showcase some applications in the theory of manifolds. In the second part of the talk I will present joint work with J. Lafont, in which we show that, while the spaces obtained via hyperbolization are often topologically exotic, their fundamental groups are as nice as possible. Namely, these groups admit nice actions on cubical complexes, and are therefore linear over the integers. As an application, we obtain new examples of hyperbolic groups that algebraically fiber.

From configurations on graphs to high dimensional cohomology of moduli spaces M_{2,n}

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The moduli spaces of algebraic curves with marked points have hugely complicated and interesting cohomology. While in low dimensions the cohomology exhibits a form of (representation-) stability, near the top dimension very little is known. Tropical geometry gives access to some of this high dimensional cohomology, namely its top weight quotient. In joint work with Bibby, Chan, Yun and Hainaut we relate the top weight cohomology of the moduli space of genus 2 curves with marked points with that of a configuration space on a graph, opening the door to extensive new calculations and qualitative analyses.

Giugno

21

Francesco Sala (Università di Pisa)

Representations of cohomological Hall algebras of surfaces via torsion pairs

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The first part of the talk is devoted to a gentle introduction to the theory of cohomological Hall algebras (COHAs for brevity) of surfaces and their categorification. Moreover, their “quantum” nature will be discussed. In addition, the Dolbeaut cohomological Hall algebras of curves will be introduced. During the second part of the talk, I will address the problem of constructing pairs consisting of a COHA and a representation of it canonically associated to torsion pairs of the abelian category of coherent sheaves on a smooth projective complex surface S.

Giugno

14

Francesco Esposito (Università di Padova)

Geometric interpretation of approximations of Nichols algebras

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Nichols algebras are graded Hopf algebras playing a prominent role in the study of pointed Hopf algebras and comprising symmetric algebras, exterior algebras and the positive part of quantized universal enveloping algebras. Kapranov and Schechtman have given a geometric interpretation of primitive bialgebras in terms of factorizable perverse sheaves. Through this equivalence, Nichols algebras correspond to IC complexes. In ongoing joint work in progress with Giovanna Carnovale and Lleonard Rubio y Degrassi, we give a geometric interpretation as well of approximations of Nichols algebras. This allows to reformulate geometrically various open problems on Nichols algebras.

Giugno

07

Marco Trevisiol (Sapienza Università di Roma)

Normality of Closures of Orthogonal Nilpotent Symmetric Orbits

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Kraft and Procesi showed that the Zariski closure of the conjugacy classes of type A are all normal and, in type B, C and D, they have described which ones are normal. In their work the Lie group acts on its Lie algebra by the adjoint action. In types B, C, D, a similar question can be asked for the action of the Lie group on the odd part of the general linear Lie algebra; that is the orthogonal group acting on the symmetric matrices and the symplectic group acting on the symmetric-symplectic matrices. Ohta showed that in the latter case every orbit has normal closures while this conclusion is not valid in the former case. In this talk I will present the main result of my Ph.D. thesis which gives a combinatorial description of the orbit whose closures are normal in the orthogonal case.

Moduli of rational elliptic surfaces of index two

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Elliptic surfaces are ubiquitous in Mathematics. Examples include Enriques surfaces, Dolgachev surfaces, all surfaces of Kodaira dimension one, and many rational surfaces. In this talk we will focus on the latter. It is a classical result that all rational elliptic surfaces can be realized as a nine-fold blow-up of the plane, where the nine points (possibly infinitely near) are the base points of a pencil of plane curves of degree 3m, each of multiplicity m. The number m is called the index of the fibration. In joint work with Rick Miranda we have constructed a moduli space for rational elliptic surfaces of index two as a toric GIT quotient. The goal of my talk will be to explain our construction.

Surface braid groups and the splitting problem of the mixed braid groups of the projective plane

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The aim of this talk is to give an introduction to the surface braid groups and to present both the splitting problem of surface braid groups and certain results about this problem, concerning the mixed braid groups of the real projective plane. Surface braid groups are a generalisation, to any connected surface, of both the fundamental group of a surface and the braid groups of the plane, which are known as Artin braid groups and were defined by Artin in 1925. Surface braid groups were initially introduced by Zariski and then, during the 1960’s, Fox gave an equivalent definition from a topological point of view. In the first part of the talk, we will define the surface braid groups from both a geometric and a topological point of view and we will present their close relation to the symmetric groups. Moreover, we will present an important family of surface braid groups, the so-called mixed braid groups. Finally, we will describe the splitting problem of surface braid groups, which we will see in detail in the second part of the talk. In the second part of the talk, we will focus on the splitting problem, which, during the 1960’s, the period of the development of the theory of surface braid groups, was studied by many mathematicians; notably by Fadell, Neuwirth, Van Buskirk and Birman, and more recently by Gonçalves–Guaschi and Chen–Salter. In particular, we will focus on the case of the projective plane: we will present its braid groups as well as certain results that we obtained concerning the splitting problem of its mixed braid groups.

31

Irreducibility of Severi varieties on K3 surfaces

Margherita Lelli Chiesa

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Let (S,L) be a general K3 surface of genus g. I will prove that the closure in |L| of the Severi variety parametrizing curves in |L| of geometric genus h is connected for h>=1 and irreducible for h>=4, as predicted by a well known conjecture. This is joint work with Andrea Bruno.

A differential geometric description of exceptional projective irreducible (discrete) representations of the Hamiltonian Lie algebra of Cartan type

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Some exact complexes of irreducible discrete representations of the unique irreducible central extension of H_n have only been constructed by hand. I will provide a differential geometric description by using base change in the pseudoalgebra language.

24

Leone Slavich (Università di Pavia)

Totally geodesic immersions of hyperbolic manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The study of totally geodesic immersions between (complete, finite-volume) hyperbolic manifolds is a classical problem in the field of hyperbolic geometry. There are two main approaches to this problem which often interplay with each other: 1) Given a hyperbolic manifold N, determine the hyperbolic manifolds in which N can be immersed geodesically; 2) Given a hyperbolic manifold, determine its totally geodesic immersed submanifolds. We will show how it is possible to build totally geodesic immersed submanifolds in a hyperbolic manifold M using finite subgroups in the commensurator of M. We will then focus on the class of arithmetic manifolds i.e. those whose fundamental groups is commensurable with the integral points of some k-form of Isom(H^n)=PO(n,1,R), for some real algebraic number field k. We will show how to characterise all totally geodesic immersions in this setting through the analysis of Vinberg's commensurability invariants: the adjoint trace field (which is an algebraic number field) and the ambient group (an algebraic group defined over the adjoint trace field). This is joint work with Mikhail Belolipetski, Nikolay Bogachev and Alexander Kolpakov.

Sviluppi recenti sulla congettura di Lang: quozienti di domini limitati

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

La congettura di Lang (1986) caratterizza le varietà complesse proiettive (o, più generalmente, Kähler compatte) iperboliche nel senso di Kobayashi come quelle di tipo generale assieme a tutte le loro sottovarietà. Lungi dall’essere dimostrata al momento, la congettura è però nota in una serie di casi paradigmatici ancorché particolari. Ci concentreremo in particolare su una direzione della congettura, spiegando come sia possibile verificare ad esempio che un quoziente libero e compatto di un dominio limitato dello spazio affine complesso abbia tutte sottovarietà di tipo generale (lavoro in collaborazione con S. Boucksom). Tempo permettendo, descriveremo alcune variazioni sul tema, considerando tipi di quozienti più generali: non più necessariamente lisci, né compatti (lavoro in collaborazione con B. Cadorel e H. Guenancia).

Moltiplicazione, numeri primi ed errori fortunati

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Parlerò di come affrontare due problemi familiari (moltiplicare numeri interi e stabilire se un numero sia primo) in modo insolito e di quali ricadute questo abbia al di fuori della matematica.

Resolution of Ideals Associated to Subspace Arrangements

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The ideal of definition of a linear subspace in a projective space has a very simple structure and hence a very simple free resolution, i.e. a Koszul complex. What can we say for the ideal that defines a finite collection of linear subspaces, subspace arrangements, in a projective space? Here we can take the intersection of the ideals defining the individual subspaces or their product. For the intersection, the structure of the resolution remains largely mysterious. For the product instead the resolution can be described and it turns out that it is supported on a polymatroid associated with the subspace arrangement. Joint work with Manolis Tsakiris (Chinese Academy of Sciences). arXiv:1910.01955v2

03

Bifurcation currents for families of group representations in higher rank

Florestan Martin-Baillon

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Finitely generated groups acting on projective spaces are examples of holomorphic dynamical systems which exhibit a variety of different behaviours. We introduce the notion of proximal stability which measures a form of dynamical stability for the action of a holomorphic family of group representations and we will explain how this property can be detected using a bifurcation current on the parameter space of the family. This bifurcation current measure the pluriharmonicity of the top Lyapunov exponent of the family of representation, defined using a random walk on the group.

Symmetric quivers and symmetric varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In this talk I will report on ongoing joint work with Ryan Kinser and Jenna Rajchgot, on varieties of symmetric quiver representations. These varieties are acted upon by a reductive group via change of basis, and it is natural to ask for a parametrisation of the orbits, for the closure inclusion relation among them, for information about the singularities arising in orbit closures. Since the Eigthies, same (and further) questions about representation varieties for type A quivers have been attached by relating such varieties to Schubert varieties in type A flag varieties (Zelevinsky, Bobinski-Zwara, ...). I will explain that in the symmetric setting it is possible to interpret the above questions in terms of certain symmetric varieties. For example, we show that singularities of an orbit closure of a symmetric quiver representation variety are smoothly equivalent to singularities of an appropriate Borel orbit closure on a symmetric variety. As a consequence, we obtain an infinite class of symmetric quiver loci that are normal and Cohen-Macaulay.

Hyperbolic 4-Manifolds with perfect circle-valued Morse functions

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In dimension 3, combining the study of the geometry and the topology of manifolds led to interesting and surprising results. The generalization of such a connection in dimension 4 seems to be a promising approach to better understand this complicated world. An intriguing 3-dimensional phenomenon is the existence of hyperbolic manifolds which fiber over the circle. Such manifolds cannot exist in dimension 4, due to a constraint given by Euler Characteristic and the Gauss - Bonnet formula. We will introduce the notion of "perfect circle-valued Morse function", which appears to be the natural generalization of "fibration over S^1", and we will introduce some tools to build a hyperbolic 4-manifold that admits such a function. To do this we will elaborate on a paper of Jankiewicz - Norin - Wise that makes use of Bestvina - Brady theory. Joint work with Bruno Martelli.

Aprile

05

On the Behrend function and the blowup of some fat points

Michele Graffeo (SISSA)

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Not much is known about the geometric properties of the punctual Hilbert scheme of fat points of length n supported at the origin of the affine plane A^2. In order to investigate them, a huge number of invariants, for fat points, has been introduced (e.g. multiplicity, order, type, blowup tree...). I will focus on the Behrend number v_Z of a fat point Z in A^2. Such invariant can be defined in terms of the blowup of the affine plane with center the subscheme Z. I will discuss the problem of computing the Behrend number of a monomial fat point following a joint work with Andrea T. Ricolfi. In particular, I will explain, first in the normal setting, how toric geometry methods apply in the construction of the blowup and in the computation of v_Z. Then, I will move to the non-normal setting, and I will show some examples of computation. Finally, if time permits, I will show some difficulties that arise in higher dimension.

Aprile

05

Allen Knutson (Cornell University)

Schubert calculus and quiver varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Since the 1970s we have known that the structure constants for intersection theory on compact complex homogeneous spaces (such as Grassmannians) are nonnegative, but our only formulae for these constants (outside special cases) are essentially as alternating sums. The most effective tool to date for giving manifestly positive formulae are the "puzzles" that Terry Tao and I introduced, but the connection to quantum integrable systems observed by Paul Zinn-Justin made it clear that the puzzles should be solving a richer problem. This turns out to involve Nakajima quiver varieties, and has shed light even on the original problem of intersecting three cells in a Grassmannian. This work is joint with Paul Zinn-Justin.

29

Javier Aramayona (Instituto de Ciencias Matemáticas)

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing braided Thompson groups, we introduce the asymptotic mapping class group of C, whose elements are proper isotopy classes of self-diffeomorphisms of C that are ”eventually trivial.” This group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C. B acts on a contractible cube complex X of infinite dimension. We use the action to determine finiteness properties of B: in well-behaved cases, X is CAT(0) and B is of type F∞. More concretely, the methods apply when Y is a 2-dimensional torus, S2 × S1, or Sn × Sn for n at least 3. In these cases, the group B contains the mapping class groups of every compact surface with boundary, the automorphism groups of every finitely generated free group, or an infinite familiy of arithmetic symplectic or orthogonal groups. In particular, the cases where Y is a sphere or a torus in dimension 2 yields a positive answer to a question of Funar-Kapoudjian-Sergiescu. In addition, we find cases where the homology of B coincides with the stable homology of the relevant mapping class groups. (Joint work with Kai-Uwe Bux, Jonas Flechsig, Nansen Petrosyan, and Xiaolei Wu.)

08

Jerzy Weyman (Jagiellonian University)

On the structure of Gorenstein ideals of codimension 4

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

I will discuss the new approach to Gorenstein ideals of codimension 4, with n generators. This allows us to construct (by a single calculation, starting from scratch) the examples of such ideals with 4<= n<= 8 generators. For n=4,5,6 they give generic models of resolutions of ideals of that type (in the sense that each such resolution comens from the generic model). We conjecture that for n=7,8 these resolutions are also generic models. The main idea is a construction of a certain generic ring which has unexpected symmetry of the type E_n. For n >= 9 such construction is not possible which indicates that the classification is much more difficult.

Severi Varieties on Enriques Surfaces

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Given a (smooth) projective (complex) surface S and a complete linear (or algebraic) system of curves on S, one defines the Severi varieties to be the (possibly empty) subvarieties parametrizing nodal curves in the linear system, for any prescribed number of nodes. These were originally studied by Severi in the case of the projective plane. Afterwards, Severi varieties on other surfaces have been studied, mostly rational surfaces, K3 surfaces and abelian surfaces, often in connection with enumerative formulas computing their degrees. Interesting questions are nonemptiness, dimension, smoothness and irreducibility of Severi varieties. In this talk I will first give a general overview and then present recent results about Severi varieties on Enriques surfaces, obtained with Ciliberto, Dedieu and Galati, and the connection to a conjecture of Pandharipande and Schmitt.

22

Marco Moraschini (Università di Bologna)

Simplicial volume and aspherical manifolds

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Simplicial volume is a homotopy invariant for compact manifolds introduced by Gromov in the early 80s. It measures the complexity of a manifold in terms of singular simplices. Since simplicial volume behaves similarly to the Euler characteristic, a natural problem is to understand the relation between these two invariants. More precisely, a celebrated question by Gromov (~’90) asks whether all oriented closed connected aspherical manifolds with zero simplicial volume also have vanishing Euler characteristic. In this talk, we will introduce the notion of simplicial volume and then we will describe Gromov's question. Then, we will discuss some new possible strategies to approach the problem as well as the relation between Gromov’s question and other classical problems in topology. This is part of a joint work with Clara Löh and George Raptis.

15

Stefano Riolo (Università di Bologna)

La segnatura delle 4-varietà iperboliche con cuspidi

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Le 4-varietà iperboliche (orientate) compatte hanno segnatura nulla per il Teorema della segnatura di Hirzebruch. Cosa si può dire, invece, per 4-varietà non compatte, complete e di volume finito? In questo caso, la segnatura dipende solo dalla topologia delle parti finali (le cuspidi) e si annulla sempre su qualche rivestimento finito della varietà. Inoltre, tutti gli esempi noti fino a poco tempo fa avevano segnatura nulla. Considerazioni naïve potrebbero quindi dare la sensazione che questo sia vero in generale. Vedremo invece che la segnatura può essere qualsiasi numero intero e, tempo permettendo, affronteremo di conseguenza qualche considerazione "geografica". In collaborazione con Sasha Kolpakov e Steve Tschantz.

01

Giovanni Cerulli Irelli (Sapienza Università di Roma)

Symmetric representation theory of quivers and connections to Lie theory

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

The representation theory of quivers and finite dimensional algebras deals with representations of products of general linear groups, and it is hence a ``type A situation''. There are many attempts to drag into the pictures groups of other nature. In this talk I will talk about a new attempt to get actions of groups of type B,C and D on the representation varieties associated to algebras with self-duality based on joint works with Magdalena Boos and partially with Francesco Esposito. For hereditary algebras this reduces to the approach due to Derksen and Weyman in 2002 when they introduced the so-called ``symmetric quivers''. In the first part I will mostly talk about quivers of type A and their symmetric representation theory, and state one of our main result with Lena which states that the symmetric orbit closures are induced by non-symmetric ones for symmetric quivers of finite type. Then I will talk about the connection with 2-nilpotent Borel orbits in classical Lie algebra worked out with Lena and Francesco and give an example that shows that in this context is not true the orbit closures of type D are induced by type A. I will close the talk by stating various conjectures and open problems concerning the problem of when symmetric orbit closures are induced by type A.

Level structures on logarithmic (and tropical) curves

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Level structures are extra data that can be added to some moduli problems in order to rigidify the situation. For example, in the case of curves, they yield smooth Galois covers of the moduli space M_g, and the problem of extending this picture to the boundary was studied by several authors, using in particular admissible covers and twisted curves. I will report on some work in progress with M. Ulirsch and D. Zakharov, in which we consider a tropical notion of level structure on a tropical curve. The moduli space of these is expected to be closely related to the boundary complex of the stack of G-admissible covers. As usual, logarithmic geometry stands in the middle and provides a convenient language to bridge the two worlds.

Special varieties and hyperbolicity

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Campana proposed a series of conjectures relating algebro-geometric and complex-analytic properties of algebraic varieties and their arithmetic. The main ingredient is the definition of the class of special varieties, which conjecturally identity the class of varieties with a potential dense set of rational points (when defined over a number field) and admitting a dense entire curve (when defined over the complex numbers). In the talk we will review the main conjectures and constructions, and we will discuss some recent results that give evidence for some of these conjectures. This is joint work with E. Rousseau and J. Wang

From polytopes to singularities on moduli spaces of Fano varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

A projective algebraic variety is called Fano (after the Italian mathematician Gino Fano) if it has positive curvature. Fano varieties play a prominent rôle in algebraic geometry for many reasons. Recently there has been fundamental work on constructing moduli spaces of (certain) Fano varieties. The aim of my talk is to show how polytopes and combinatorics can help in proving that moduli spaces of Fano varieties are, in general, quite singular. My talk is based on joint work with Anne-Sophie Kaloghiros. The first 45 minutes of my talk do not require any knowledge in algebraic geometry.

Chiusure di classi di Jordan in algebre di Lie, gruppi algebrici ed algebre di Lie Z_n-graduate

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Le classi di Jordan sono state introdotte da Borho e Kraft nel loro studio delle sheet per algebre di Lie semisemplici. Sono le classi di equivalenza di elementi in un'algebra di Lie che hanno stessa decomposizione di Jordan, o, equivalentemente di elementi che hanno stabilizzatori (per l'azione aggiunta) coniugati tra loro. Sono localmente chiuse, irriducibili, lisce, e le loro chiusure danno luogo ad una stratificazione finita. La stessa costruzione può essere adattata per definire le classi di Jordan in gruppi algebrici riduttivi: la stratificazione che ne risulta compare nello studio di Lusztig dei fasci carattere. In collaborazione con Ambrosio ed Esposito abbiamo osservato che localmente le chiusure di classi di Jordan nel gruppo si comportano come chiusure di classi di Jordan in un'opportuna algebra di Lie. Un analogo di classe di Jordan per algebre di Lie Z_2-graduate è stato introdotto da Tauvel e Yu e le chiusure sono state studiate da Bulois ed Hivert: si perdono alcune delle caratteristiche dei casi precedenti ma il quadro complessivo è ancora chiaro. Motivato dallo studio della modalità per azioni di gruppi, Popov ha recentemente introdotto le classi di Jordan anche per algebre di Lie ciclicamente graduate. In collaborazione con Esposito e Santi abbiamo fornito una descrizione geometrica locale delle loro chiusure, mostrando in particolare che anche in questo caso la chiusura delle classi di Jordan è un'unione di classi. Con una serie di esempi mostreremo affinità e divergenze tra i vari contesti e le situazioni nelle quali la partizione in classi di Jordan ha un ruolo importante.

Hodge Theory for Polymatroids

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In the first part, we present classical results about hyperplane arrangements and matroids: starting from the definitions, we present the construction by De Concini and Procesi of wonderful models. We discuss the cohomology of the wonderful model and of the complement of the arrangement. We also sketch the proof by Huh, Adiprasito, Katz of log-concavity of coefficients of the characteristic polynomial. In the second part we introduce subspace arrangements and polymatroids. We provide a generalization of the Goresky, MacPherson formula and we discuss the Hodge package (i.e. Poincaré duality, Hard Lefschetz and Hodge Riemann bilinear relations) for the Chow ring of a polymatroid. This is a joint work with Gian Marco Pezzoli.

Horocycle flows in moduli spaces of abelian differentials

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

There is a fruitful analogy between Lie group actions and the dynamics of SL(2,R) on the moduli space of abelian differentials. Along the lines of this dictionary, the horocycle flow corresponds to a unipotent flow on a homogeneous space, for which Ratner’s theory is available. I will report on recent progress regarding the dynamics of the horocycle flow in the moduli space of abelian differentials.

Localisation in enumerative geometry

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Very few techniques are available for performing calculations in enumerative geometry. One of these is localisation. We will present different examples, flavours and refinements of the localisation formula, including applications to Donaldson-Thomas invariants.

Functoriality of the c-map

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

scarica allegato

This talk will present some of the results obtained in the paper "The c-map as a functor on certain variations of hodge structure" by Mantegazza and Saha. In particular we will see how projective special Kähler manifolds can be interpreted in terms of variations of polarised Hodge structure, and how the latter can be used to give a description of the c-map that is manifestly functorial.

30

Paul-Konstantin Oehlmann (Uppsala University)

Non-flat elliptic fourfold and three-form cohomology (and strongly coupled theories in four dimensions)

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

We consider compact elliptic four-folds whose fiber ceases to be flat over Riemann surfaces of genus g in the base. We show that those contributions generically lead to non-trivial threeform cohomology proportional to g and the number of non-flat fiber components. These non-flat components can be viewed as compactifications of non-flat three-folds where they correspond to superconformal matter theories. Moreover we show, that one can perform conifold transitions that remove those non-flat fibers, corresponding to non-flat fibers in codimension three and second to birational base changes. The former phase is interpreted as a non-perturbative gauge invariant fourpoint coupling and the second one is closer to a classical 4D Coulomb branch.

Santiago Estupinan Salamanca (Universidad de los Andes)

Schur and Power Sum Polytopes

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Aguiar and Ardila defined a Hopf monoidal structure on the collection of generalized permutahedra of all dimensions; and from it, constructed a Hopf algebra on the same polytopes, which is isomorphic to the Hopf algebra of symmetric functions, Sym. This endows each element of Sym with a formal sum of permutahedra, so that we can think of symmetric functions as members of McMullen’s polytope algebra. In this talk, we give geometric models for the Schur and power sum symmetric functions, when regarded as elements of the aforesaid polytope algebra. This is accomplished through a combinatorial rule for the former ones and in the way of an explicit description for the latter ones. We also characterize when the resulting geometric objects correspond to polytopes with missing faces. (Joint work with Carolina Benedetti and Mario Sanchez)

Luis Ferroni Rivetti (Università di Bologna)

EHRHART THEORY OF MATROID POLYTOPES

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Matroids are combinatorial structures that admit several "cryptomorphic" definitions. It is possible to define matroids as a certain kind of polytopes with vertices with 0/1-coordinates. Also, for every lattice polytope, the function counting the number of integer points inside of every integral dilation is known to be a polynomial, named after Eugene Ehrhart. In this seminar we will talk about the Ehrhart polynomials of matroid polytopes. We will discuss some open problems and recent results on the area, using just minimal prerequisites.

09

Aline Zanardini (University of Pennsylvania)

Stability of Halphen pencils of index two

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

In this talk I will present some results about the GIT stability of Halphen pencils of index two under the action of SL(3). These are pencils of plane curves of degree six having nine (possibly infinitely near) base points of multiplicity two. Inspired by the work of Miranda on pencils of plane cubics, I will explain how we can explore the geometry of the associated rational elliptic surfaces.

02

Mariel Supina (University of California, Berkeley)

The universal valuation of Coxeter matroids

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Matroids are combinatorial objects that generalize the notion of independence, and their subdivisions have rich connections to geometry. Thus we are often interested in functions on matroids that behave nicely with respect to subdivisions, which are called valuations. Matroids are naturally linked to the symmetric group; generalizing to other finite reflection groups gives rise to Coxeter matroids. I will give an overview of these ideas and then present some recent work with Chris Eur and Mario Sanchez on constructing the universal valuative invariant of Coxeter matroids. Since matroids and their Coxeter analogues can be understood as families of polytopes with special combinatorial properties, I will present these results from a polytopal perspective.

02

José Bastidas (Cornell University)

The polytope algebra of generalized permutahedra

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

McMullen used the fundamental operation of Minkowski sum to construct the polytope algebra of real vector space. In this talk, I will consider the subalgebra generated by deformations of a fixed zonotope and endow it with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. In the particular case of Coxeter arrangements of type A and B, we find striking relations between the corresponding module structure and certain statistics on permutations and signed permutations, respectively. I will explain how these statistics give information on families of polytopes that generate all (type B) generalized permutahedra as signed Minkowski sums.

Thomas Lam (University of Michigan)

Positroid varieties

nell'ambito della serie: SEMINARIO DI ALGEBRA E GEOMETRIA

algebra e geometria

Positroid varieties are intersections of cyclically rotated Schubert varieties in the Grassmannian, introduced in my work with Knutson and Speyer. I will discuss some aspects of these very nice spaces, including a recent computation of the cohomology of open positroid varieties in joint work with Galashin.