Seminari del Dipartimento di Matematica

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

In this talk, we discuss the \mathcal{G}_0 dichotomy, an important result in descriptive combinatorics, and present negative results in the context of generalized descriptive set theory. This is joint work in progress with Philipp Schlicht.

Settembre

08

2025

Mateusz Rzepecki

TBA

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

logica

Uberhomology, combinatorics and the Mayer-Vietoris spectral sequence

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

SEMINARIO INTERDISCIPLINARE

In this talk we will introduce uberhomology, a combinatorially defined homology theory for simplicial complexes. After proving some of its properties, we’ll show how this invariant is related to dominating sets and to the Mayer-Vietoris spectral sequence. We will conclude with some open questions and problems. All results are joint work with L. Caputi and D. Celoria.

$\aleph_1$-free abelian non-Archimedean Polish groups

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

An uncountable $\aleph_1$-free group can not admit a Polish group topology but an uncountable $\aleph_1$-free abelian group can, as witnessed e.g. by the Baer-Specker group $\mathbb{Z}^\omega$, in fact, more strongly, $\mathbb{Z}^\omega$ is separable. In this paper we investigate $\aleph_1$-free abelian non-Archimedean Polish groups. We prove two main results. The first is that there are continuum many separable (and so torsionless, and so $\aleph_1$-free) abelian non-Archimedean Polish groups which are not topologically isomorphic to product groups and are pairwise not continuous homomorphic images of each other. The second is that the following four properties are complete co-analytic subsets of the space of closed abelian subgroups of $S_\infty$: separability, torsionlessness, $\aleph_1$-freenees and $\mathbb{Z}$-homogeneity.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Categories for probability theory

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Category theory was initially developed to address some structural questions in algebraic topology. Shortly after it was extended to algebraic geometry, logic, universal algebra, and more recently, theoretical computer science. Each of these subjects was heavily influenced by category theory, and in turn, the development of category theory was prominently shaped by the strucures and problems arising in these fields. In the past few years there has been a growing interest in applying categorical techniques to fields such as probability, statistics and information theory, to study their structures and to find patterns in their techniques. Perhaps surprisingly, it turns out that these fields present a rich and principled structure when addressed categorically, with functors and universal properties arising everywhere. However, most of the time, new category theory is needed to study these subjects, as they are quite far from the algebra and geometry for which category theory was initially developed. Two of the current most prolific environments to study probability categorically are Markov categories and dagger categories. In this talk we will give an introduction to both, show their similarities, differences and connections, and use them to prove some core theorems of probability.

Derivations on C*-Algebras: A Revisit

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

analisi matematica

The most basic class of derivations on C*-algebras consists of the inner derivations—those expressible as commutators with elements from the multiplier algebra. A fundamental question in the theory of C*-algebras is to determine which algebras admit only inner derivations. Landmark results by Sakai, Kadison, and Sproston established this property for all von Neumann algebras, simple C*-algebras, and homogeneous C*-algebras. In the separable setting, the problem was completely resolved in 1979 by Akemann, Elliott, Pedersen, and Tomiyama, who showed that a separable C*-algebra has only inner derivations if and only if it is a direct sum of a C*-algebra with continuous trace and a C*-algebra with discrete primitive spectrum. However, the non-separable case remains largely unsettled—even for 2-subhomogeneous algebras. In 1978, Pedersen posed a unifying question, inspired by the work of Sakai and Kadison: given a C*-algebra, does its local multiplier algebra—defined as the C*-direct limit of the multiplier algebras of its essential closed ideals—admit only inner derivations? In this talk, we revisit the classical innerness problem for derivations on C*-algebras, highlighting both recent developments and emerging perspectives.

Roelcke percompact groups and their automorphism groups.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Teorema di compattezza per la logica L_{infty,infty}

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

SEMINARIO INTERDISCIPLINARE

TBA

DIFFERENTIAL AUTOMATA THEORY

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

In this talk I will study generalized automata (in the sense of Adámek-Trnková) in Joyal’s category of combinatorial species; as an important preliminary step, I will provide examples of coalgebras for the "derivative" endofunctor ∂ and for the ‘Euler homogeneity operator’ L∂ arising from the adjunction L⊣∂⊣R. The theory is connected with, and in fact provides nontrivial examples of, differential 2-rigs—a concept I recently introduced by treating combinatorial species in the same way that a generic (differential) semiring (R,d) relates to the (differential) semiring N[[X]] of power series with natural coefficients. Joyal himself has long regarded species as categorified formal power series. This perspective aligns with a fundamental category-theoretic insight: free objects in the category of rings naturally acquire a canonical differential structure. At the heart of this phenomenon lies the representability of the prestack of derivations by an object of Kähler differentials. These ideas categorify elegantly within the 2-category of differential 2-rigs, revealing that species possess a universal property as differential 2-rigs. The desire to study categories of ‘state machines’ valued in an ambient monoidal category (K,⊗) gives a pretext to further develop the abstract theory of differential 2-rigs, proving lifting theorems of a differential 2-rig structure from (R,∂) to the category of ∂-algebras on objects of R, and to categories of Mealy automata valued in (R,⊗), as well as various constructions inspired by differential algebra such as jet spaces and modules of differential operators. This talk covers the content of the paper Automata and Coalgebras in Categories of Species (Proceedings of CMCS24, Luxembourg), as well as parts of an ongoing project with Todd Trimble.

Hochschild cohomology of differential graded categories and spectral sequences

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded center of its derived category: the characteristic homomorphism. We interpret this map as an edge homomorphism in a spectral sequence, which allows to study the characteristic homomorphism systematically in many interesting examples from algebra, geometry, topology and physics. To illustrate this, we will discuss several concrete examples related to coherent sheaves on algebraic curves and cochains of classifying spaces of Lie groups. If time permits, I will also indicate a new extension of this framework to $A_\infty$-categories. Some of this is joint work with M. Szymik (Sheffield) other with A. Phimister (Leicester).

A study of Kock's fat delta

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

The simplicial category \Delta plays an important role in category theory. One of the reasons is that there is a fully faithful nerve functor from the category Cat to the category of simplicial sets (that is functors from the opposite of \Delta to Set). Its essential image consists of simplicial sets satisfying additional conditions that the Segal maps are isomorphisms. This allows to think of a small category as a type of simplicial set, and this idea has been carried on in higher dimensions in defining appropriate notions of higher categories. This talk is about a modification of \Delta, introduced by J. Kock, called the fat delta. After explaining the motivation for our interest in fat delta, both from higher category theory and from type theory, we present a study of fat delta in terms of monad with arities. This leads to a nerve theorem for relative semicategories, as well as a description of fat delta as a hypermoment categories in the sense of Berger. This is joint work with Tom de Jong, Nicolai Kraus and Stiephen Pradal, arXiv.2503.10963v1.

On the use of Group Equivariant Non-Expansive Operators for Topological Data Analysis and Machine Learning

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

(Ordered) compact Hausdorff spaces, axiomatically

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Expansions of Hahn fields by algebras of generalized power series

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Dynamics of non-Archimedean Polish groups and dissociation

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

sistemi dinamici

My talk aim to present our joint work with Matthieu Joseph and Remi Barritault on the actions of non-Archimedean Polish groups. I will first present the notion of dissociation for probability measure-preserving actions, which allows us generalize so-called de Finetti theorems. Then, I will outline the model-theoretical conditions that we know imply dissociation and discuss how they may be improved.

Rigidity for metric reduced products under forcing axioms

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

logica

Mild (consequences of) forcing axioms imply a general lifting theorem which allows to characterise precisely those homomorphisms between reduced products of countable structures which have a lift that factorises into homomorphisms between the factor structures. We will discuss recent extensions of such lifting theorems to the metric setting and ensuing rigidity phenomena for particular metric reduced product structures of interest. This is based on joint work with Ilijas Farah and Alessandro Vignati, with Alessandro Vignati, and with Andreas Thom.

Maggio

05

2025

Tullio Ceccherini-Silberstein

Geometric group theory, dynamics, ring theory, and logic in interaction

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

In this completely self-contained talk, I'll discuss various versions of the Garden of Eden theorem, the Gottschalk surjunctivity conjecture, and Kaplansky's conjecture on stable finiteness of group rings. This will include a quick review of the notions of amenability and soficity for groups, of the theory of cellular automata, and entropies of dynamical systems.

THE REALCOMPACTNESS NUMBER IN COUNTABLE (CROWDED) SPACES

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Given a realcompact space X, we denote by Exp(X) the smallest infinite cardinal κ such that X is homeomorphic to a closed subspace of Rκ. In this talk, we analyze the realcompactness number of countable spaces. We will show that, for every cardinal κ, there exists a countable crowded space X such that Exp(X) = κ if and only if p ≤ κ ≤ c. On the other hand, we show that a scattered space of weight κ has pseudocharacter at most κ in any compactification. This will allow us to calculate Exp(X) for an arbitrary (that is, not necessarily crowded) countable space. This is a joint work with Andrea Medini and Lyubomyr Zdomskyy.

Representation theory of non-Archimedean Polish groups and dissociation

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

logica

sistemi dinamici

algebra e geometria

For a countable homogeneous structure M, a natural question in representation theory is the classification of unitary representations of the Polish, non-archimedean group Aut(M). This question was completely answered by Tsankov for ω-categorical structures. In joint work with R. Barritault and C. Jahel, we generalize this result and go beyond ω-categoricity by addressing structures such as the integral/rational Urysohn space and the integral/rational universal diversity. We will explain how the notion of dissociation, a structural property of unitary representation for Polish non-archimedean groups, is used in our work. No background knowledge of representation theory will be assumed for this talk.

Additive Vs monoidal categorification of cluster algebras

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

In this talk, we will show how representations of a (Dynkin) quiver allow to construct cluster variables for the associated Fomin-Zelevinsky cluster algebra. We will start from the basics on quiver representations, notably Gabriel's theorem establishing a bijection between positive roots and indecomposable representations. By combining this bijection with Fomin-Zelevinsky's between positive roots and non-initial cluster variables, we will obtain a map associating a non-initial cluster variable with each indecomposable representation. The starting point of additive categorification is an explicit formula for this map due to Caldero-Chapoton. It involves Euler characteristics of varieties of subrepresentations and is typical of links between cluster algebras and algebraic geometry.

A primer on cluster algebras

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

teoria delle categorie

Since their invention by Fomin-Zelevinsky in 2002, cluster algebras have shown up in an ever growing array of subjects in mathematics (and in physics). In this talk, we will approach their theory starting from elementary examples. More precisely, we will see how the remarkable integrality properties of the Coxeter-Conway friezes and the Somos sequence find a beautiful unification and generalization in Fomin-Zelevinsky's definition of cluster variables and their Laurent phenomenon theorem. Motivated by the periodicity of Coxeter-Conway friezes, we will conclude with a general periodicity theorem (Keller 2013), whose proof is based on the interaction between discrete dynamical systems and quiver representations through the combinatorial framework of cluster algebras.

Categorie quasi-abeliane e loro applicazioni alla geometria algebrica.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

teoria delle categorie

Given an abelian category A its derived category D(A) admits a natural t-structure whose heart is A. Moreover by the Auslander’s Formula A is equivalent to the quotient category of coherent functors by the Serre subcategory of effaceable functors. Given a quasi-abelian category E its derived category D(E) admits two canonical t-structures (left and right) whose hearts L and R are derived equivalent and their intersection in D(E) is E, moreover E is a tilting torsion class (rep. cotilting torsion-free class) in the right heart R (resp. L). We generalise the Auslander’s Formula to quasi-abelian categories and we extend this picture to its higher version introducing n-quasi-abelian categories. Given X a smooth algebraic variety of dimension n the category of locally free O_x-modules of finite rank is n-quasi-abelian.

C*-rigidity: a bridge between coarse geometry and C*-algebras

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

sistemi dinamici

With every proper metric space are associated C*-algebras of special geometric importance (Roe algebras), originally defined for index-theoretic considerations. It is a simple observation that metric spaces that are coarsely equivalent give rise to isomorphic C*-algebras, and recently we could prove that the converse also holds. Namely, a C*-rigidity Theorem shows that spaces with isomorphic Roe algebras must be coarsely equivalent. This establishes a strong connection between the categories of coarse spaces and C*-algebras. In this introductory talk I aim to properly introduce the C*-rigidity question and illustrate the roadmap of the proof of rigidity.

A duality for the class of compact T1 -topological spaces

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

SEMINARIO INTERDISCIPLINARE

We present a duality between compact $ T_1 $-spaces and a class of distributive lattices (subfit, compact, and complete), which captures key aspects of both Stone duality and $\Omega$-point duality in particular instances. We then show how this duality extends to a contravariant adjunction between $ T_1 $-spaces and bounded distributive lattices. This adjunction gives rise to a canonical compactification---the \emph{Wallman compactification}---for $ T_1 $ spaces, such that any \textit{strongly continuous} map from a $ T_1 $ space $ X $ into a compact $ T_1 $ space factors uniquely through the Wallman compactification of $ X $. This is joint work with Matteo Viale and Mai Gehrke.

Higher limits for wider systems

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

We discuss a class of pro-groups whose derived limits are relevant to the additivity of strong homology. These are indexed by the set of functions {}^\kappa \omega. The case of width \omega, that is, the one where \kappa = \omega, is linked to the additivity of strong homology on the class of locally compact separable metric spaces. While an equivalence was proved for \lim^1 between certain narrow and wider system, the analogous equivalence for higher limits has long been an open question. Here we present a negative answer to that question and, time permitting, some ideas toward a consistency result on simultaneous vanishing for all cardinals and all \lim^n. This is joint work with Jeffrey Bergfalk.

The exponential law and bounded cohomology for semidirect groupoids.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Given a measured groupoid G, together with Filippo Sarti, we defined a cohomology theory which generalizes the measurable bounded cohomology of a locally compact group. In the particular case of a groupoid associated to a measure preserving action, our cohomology boils down to the usual bounded cohomology of the group with twisted coefficients. We will discuss the possible applications of this result to orbit equivalence.

A topos-theoretic perspective on reconstruction theorems in model theory

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

Non-commutative derived geometry

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

Definable Obstruction Theory

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

The simplicial volume of Davis' manifolds

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

analisi matematica

interdisciplinare

The simplicial volume is a homotopy invariant of manifolds; this talk is about the simplicial volume of a Davis' manifolds, obtained from the so-called reflection group trick, which is a powerful method for constructing aspherical manifolds. I will describe an approach based on the study of triangulations of spheres and simplicial maps between them. This approach also presents connections with the theory of graph minors. No knowledge about simplicial volume or Davis' reflection group trick is expected from the audience.

Gennaio

20

2025

Lorenzo Luperi Baglini

Ramsey Pairs

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

logica

We introduce the concept of Ramsey pairs, and show how they can be prove several infinitary results in combinatorics.

Strong vector spaces and some torsion theories of Ind(Vect^{op})

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Generic properties of countably infinite groups

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Group operations on a fixed countably infinite universe form a Polish space G. Thus we can view group properties as isomorphism-invariant subsets of G, and it makes sense to ask: what properties are generic (in the sense of Baire category)? In my talk, I will address this question and if time permits, I may also say a few words about generic properties of compact groups.

Fragments of Geometric Logic and their behaviour

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

Inspired by a recent characterisation of coherent topoi as a class of Kan injectives, we provide a tentative definition of fragment of geometric logic. We treat them as mathematical objects, and study them from the point of view of Lindstrom-type theorems.

Maximality in first-order theories and the PM_k hierarchy.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

In the classification of first-order theories, many "dividing lines" have been defined in order to understand the complexity and the behavior of some classes of theories. These classes are usually defined by forbidding some specific configurations of definable subsets. Patterns (of consistency and inconsistency) are essentially descriptions of abstract configurations of sets and hence can give a general framework to study dividing lines. In this talk, we describe this general framework and we introduce a notion of maximal complexity, first defined by Shelah, by requesting the presence of all the exhibitable patterns of definable sets. Weakening this notion, we define Positive Maximality and the PM_k hierarchy, and describe some results about them.

Computability, Consistency, and a Theorem on Fractals

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

teoria delle categorie

I will talk about the relationship between computability theory, the complexity of sets of reals, and set-theoretical axioms. Using all these, I will prove a ZFC consistency result in fractal geometry, which shows that the Hausdorff dimension of “simple" sets can behave badly under projections.

Dicembre

16

2024

Francesco Paolo Gallinaro

Automorphisms of valued fields: amalgamation and existential closedness

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

A valued difference field is a field equipped with a valuation and an automorphism which preserves the valuation ring setwise. In this talk I will discuss various results on these objects, concerning the existence of a section of the valuation compatible with the automorphism and of extensions of the valuation to difference fields extension, the solvability of amalgamation problems of valued difference fields, and the classification of the theory of valued difference fields in the sense of positive model theory. This is joint work with Jan Dobrowolski and Rosario Mennuni.

Factorization in generalized power series

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

A classical tool in the study of real closed fields are the fields K((G)) of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field K of characteristic 0 and exponents in an ordered abelian group G. We generalize previous results about irreducible elements and unique factorization in the subring K((G≤0)).

Homological algebra for pro-Lie Polish groups

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

Enriching traditional algebraic structures such as abelian groups with a topology runs into some issues from the Homological Algebra point of view, as the categories thus obtained are not, in general, classical abelian categories, but instead only satisfy the weaker notion of being quasi-abelian. The additional homological work required to deal with topological abelian groups has traditionally been carried out for Locally Compact Groups by Moskowitz with extensive use of the Pontryagin duality. However, this work has shown that Locally Compact Groups do not satisfy some useful homological properties. In this talk I will introduce the category of pro-Lie Polish groups, a recently proposed generalization of Polish Locally Compact Groups, and discuss why they are better suited as an object of study of Homological Algebra.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Complete (curved) Lie algebras as models of spaces

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

The rational homotopy type of simply connected spaces is fully captured by its Quillen model, a differential graded Lie algebra constructed from the space. Conversely, any positively graded differential Lie algebra can be "realized" as a topological space, with rational homotopical and homological invariants preserved by these two functors. However, these constructions are inherently limited to connected and simply connected spaces. To remove these constraints, we must move to the category of complete Lie algebras. Within this category, there exists a cosimplicial object that gives rise to a pair of adjoint functors between the categories of complete Lie algebras and topological spaces. In this talk, we will explore the construction of this pair of functors and some important properties. Concretely, we will show that composing both of them results in the Bousfield-Kan $\mathbb{Q}$-completion. Additionally, we will discuss how this framework can be extended to curved Lie algebras, leading to a unpointed theory.

A Hahn-like construction for finitely ramified valued fields

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

Explicit constructions of models of the theory of a valued field are useful tools for understanding its model theory. Since Kaplansky’s work, it has been a topic of interest to characterize value fields in terms of fields of power series. In particular, Kaplansky proved that, under certain assumptions, an equicharacteristic valued field is isomorphic to a Hahn field. In this talk, we show that in the mixed characteristic case, assuming the Continuum Hypothesis, we can provide a characterization, in terms of power series, of pseudo-complete finitely ramified valued fields with a fixed residue field k and valued in a Z-group G, using a Hahn-like construction with coefficients in a finite extension of the Cohen field C(k) of k. In this construction, the elements of the field are “twisted” power series, i.e. powers series whose product is defined by having an extra factor, given by the cross-section and a 2-cocycle determined via the value group. This generalizes a result by Ax and Kochen, who characterize pseudo-complete valued fields elementarily equivalent to the field of p-adic numbers Q_p. If time permits, we will see some consequences of this characterization regarding the problem of lifting automorphisms of the residue field and the value group to automorphisms of the valued field in the mixed characteristic case.

Abelian groups are to abelian categories as Hilbert spaces are to what?

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

teoria delle categorie

I will introduce the new notion of R*-category (arXiv:2312.02883) and explain the ways in which it is a good answer to the question in the title. The main difference between R*-categories and abelian categories is extra structure that generalises adjoints of morphisms: R*-categories come with a functorial and involutive choice of morphisms f*:Y → X for each morphism f: X → Y. Many elementary aspects of the theory of Hilbert spaces generalise to arbitrary R*-categories, including the relationship between positive operators and contractions. Familiarity with only the most basic concepts from category theory (categories, products, equalisers, monomorphisms) will be sufficient to understand most of the talk.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

The notion of ultracategory was introduced by Michael Makkai in a paper in APAL in 1990 for the characterisation of categories of models of pretoposes, an ample extension to (intuitionistic) first order theories of Stone duality for Boolean algebras, providing a kind of Stone duality for first order theories -- aka conceptual completeness. Recently, Jacob Lurie refined that notion in unpublished notes producing another approach to the duality for pretoposes -- the two notions of ultracategory appear to be different, though no separating example has been produced yet. In the talk, we shall give intuitions about Makkai's and Lurie's notions, providing examples and applications. Then we shall introduce an algebraic notion of structured category which subsumes the two kinds of ultracategories mentioned above -- technically, the "ultracompletion" 2-functor on the 2-category of small categories, and extend it to a pseudomonad. Next we show how it relates to the two existing notions. This is joint work with Richard Garner.

Solving equations involving the j-function and its derivatives

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

The j-function is a central player in the study of elliptic curves. It satisfies many interesting algebraic properties, but there many things still unknown. In this talk we will discuss the problem of trying to solve polynomial equations that involve the j-function and its first two derivatives, and we will discuss some important cases that can be solved. This is joint work with Vahagn Aslanyan and Vincenzo Mantova.

Enriched bicategories and string diagrams

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

teoria delle categorie

Two main generalizations of categories are bicategories and enriched categories. The first one allows morphisms one level up, the other one allows morphisms to be much more general objects rather than just sets. This talk will try to explain what happens if we do the two at the same time. In particular, we will explore the main available results and tools with which enriched bicategories can be tamed. Among the results we have strictification theorems, for what concerns the tools, most notably, the extremely powerful language of string diagrams. Time permitting, we will see the combination of the two in action.

Universality properties of graph homomorphism: one construction to prove them all.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

SEMINARIO INTERDISCIPLINARE

We show that the natural operation of connected sum for graphs can be used to prove at once most of the universality results from the literature concerning graph homomorphism. In doing so, we significantly improve many existing theorems and solve some natural open problems. Despite its simplicity, our technique unexpectedly leads to applications in quite diverse areas of mathematics, such as category theory, combinatorics, classical descriptive set theory, generalized descriptive set theory, model theory, and theoretical computer science. (Joint work with S. Scamperti)

Wadge hierarchy on general topological spaces

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

SEMINARIO INTERDISCIPLINARE

The Wadge preorder is a tool to compare the complexity of subsets of topological spaces: if $A,B$ are subsets of the topological spaces $X,Y$, respectively, $A$ \emph{Wadge reduces} to $B$ if there exists a continuous function $f:A\to B$ such that $A=f^{-1}(B)$. While most of the earlier work on the Wadge preorder concerned zero-dimensional Polish spaces, recent investigations have involved more general kinds of spaces. This talk surveys some of the results and presents a few open problems and perspectives in the field.

Arithmetic Ramsey theory by nonstandard methods.

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

logica

In recent years there has been a growing interest in Ramsey theory and related problems in combinatorics of numbers. Historically, the earliest results in this field are Schur's Theorem ("In every finite coloring of the naturals there exists a monochromatic triple a, b, a+b") and van der Waerden's Theorem ("In every finite coloring of the naturals there exist monochromatic arithmetic progressions of arbitrary length"). A peculiar aspect of this area of research is the wide variety of methods used: in addition to the tools of elementary combinatorics, also methods of discrete Fourier analysis, ergodic theory, and ultrafilter space algebra have been successfully applied. Recently, a further line of research has been undertaken, in which combinatorial properties of sets of integers are studied by methods of nonstandard analysis. In this seminar I will discuss these methods and present some examples of their applications.

Higher Theories and Monads

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

teoria delle categorie

Monads and algebraic theories are two categorical approach to universal algebra. In his book Higher Algebra, Jacob Lurie established a relatively comprehensive theory of monads on infinity categories. However, his approach can be difficult in practice to use due to its highly technical nature. In this talk, we will describe a version of generalized algebraic theories in the $\infty$-categorical setting, and show that it is recovers Lurie's theory for nice monads. As an application, we will prove several structural results about monads in the $\infty$-categorical setting. We will also use our result to describe the algebraic theories of E_1, E_2, and E_\infty algebras.

HILBERT GEOMETRIES, DIVISIBLE DOMAINS AND HYPERBOLICITY

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

Hilbert geometries have been introduced as a generalization of hyperbolic geometry, and provide a family of metric spaces where the Euclidean straight lines are geodesics. A Hilbert geometry is said to be divisible if it admits a group of isometries that acts cocompactly on the space. The aim of this talk is to introduce the class of divisible Hilbert geometries and to look at a characterization of hyperbolicity in this class.

Febbraio

07

2024

Aristotelis Panagiotopoulos

The class and dynamics of α-balanced Polish groups

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

analisi matematica

logica

sistemi dinamici

A Polish group is TSI if it admits a two-side invariant metric. It is CLI if it admits complete and left-invariant metric. The class of CLI groups contains every TSI group but there are many CLI groups that fail to be TSI. In this talk we will introduce the class of α-balanced Polish groups where α ranges over all countable ordinals. We will show that these classes completely stratify the space between TSI and CLI. We will also introduce "generic α-unbalancedness": a turbulence-like obstruction to classification by actions of α-balanced Polish groups. Finally, for each α we will provide an action of an α-balanced Polish group whose orbit equivalence relation is not classifiable by actions of any β-balanced Polish group with β<α. This is joint work with Shaun Allison.

Gennaio

26

2024

Antongiulio Fornasiero

Hilbert polynomials for finitary matroids

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

Let d be a finite tuple of commuting derivations on a field K. A classical result allows us to associate a numerical polynomial to d (the Kolchin polynomial), measuring the "growth rate" of d. We show that we can abstract from the setting of fields with derivations, and consider instead a matroid with a tuple d of commuting (quasi)-endomorphisms. In this setting too there exists a polynomial measuring the growth rate of d. Joint work with E. Kaplan

Bounded cohomology of groups and amenability

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

teoria delle categorie

Bounded cohomology of groups is a variant of group cohomology that, given a group and a Banach coefficient module over such group, gives graded semi-normed vector spaces. A major role in the theory of bounded cohomology is played by amenable groups and amenable actions as they provide vanishing conditions for bounded cohomology. The goal of this talk is to introduce bounded cohomology of groups and look into its realtion with amenability.

Recent developments in the geometry of topoi

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

teoria delle categorie

Mackey 2-functors and Mackey 2-motives

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

teoria delle categorie

In this talk we will give a brief introduction to the theory of bicategories, in order to present a bicategorical gadget, called Mackey 2-functor, axiomatizing an ubiquitous phenomena in finite equivariant mathematics, especially in representation theory of finite groups and equivariant topology. The key point of this theory so far is the existence of a universal bicategory (the Mackey 2-motives) encoding the properties of such 2-functors, and allowing furthers constructions.

Some applications of forcing

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

analisi matematica

interdisciplinare

logica

In this talk, after reviewing the main concepts related to forcing and giving some examples of frequently used forcing notions (i.e. partially ordered sets deployed for this technique), we discuss some "concrete" mathematical statements that can be shown to be undecidable. In particular, we will show that the existence of a Suslin Line is independent of ZFC.

Condensation and left-orderable groups

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

In this talk we shall discuss condensed points in the Polish space of left-orderings of a fixed left-orderable groups. We will describe new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. We discuss how these methods apply to a large class of left-orderable groups, and they shed light on spaces of left-orderings with low Borel complexity. This is joint work with Adam Clay.

A gentle introduction to forcing for mathematicians

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

analisi matematica

interdisciplinare

logica

In this talk, we introduce some set-theoretic tools to prove consistency results. More precisely, the presentation will cover Goedel's Constructible Universe as well as Cohen's method of forcing. No previous knowledge on this subject will be assumed.

Elliott classification of AF-algebras, logic and computation

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

analisi matematica

logica

teoria delle categorie

Rational discrete cohomology for totally disconnected locally compact groups

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

logica

With the solution of Hilbert’s fifth problem, our understanding of connected locally compact groups has significantly increased. Therefore, the contemporary structure problem on locally compact groups concerns the class of totally disconnected locally compact (= t.d.l.c.) groups. The investigation of the class of t.d.l.c. groups can be made more manageable by dividing the infinity of objects under investigation into classes of types with “similar structure”. To this end we introduce the rational discrete cohomology for t.d.l.c. groups and discuss some of the invariants that it produces. For example, the rational discrete cohomological dimension, the number of ends, finiteness properties FP_n and F_n, and the Euler-Poincaré characteristic.

The computational core: its reduction and intersection type discipline

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

SEMINARIO INTERDISCIPLINARE

Moving from the abstract definition of monads, we introduce a version of the call-by-value computational λ-calculus based on Wadler’s variant. We call the calculus computational core and study its reduction, prove it confluent, and study its operational properties on two crucial properties: returning a value and having a normal form. The cornerstone of our analysis is factorization results. In the second part, we study a Curry-style type assignment system for the computational core. We introduce an intersection type system inspired by Barendregt, Coppo, and Dezani system for ordinary untyped λ-calculus, establishing type invariance under conversion. Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterizes convergent terms via their types. For greater accessibility, the presentation will begin with a brief introduction to lambda calculus, monads, and intersection types.

Derived Smooth Manifolds, Part II

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

teoria delle categorie

A well-known shortcoming of the category of smooth manifolds is its lack of arbitrary pullbacks. A pullback of manifolds, and in particular an intersection of submanifolds, exists only along maps which are transversal. This problem can be overcome by passing to the larger category of derived smooth manifolds. The construction of this category combines ideas from algebraic geometry, homotopy theory and of course differential topology. We can describe this construction in several steps. Firstly, we consider the relation between manifolds and schemes. Here, we employ the so-called C^\infty-rings, which are algebraic objects encoding the structure of the collection of smooth functions on R^n beyond that of an R-algebra. By the general philosophy of algebraic geometry, their duals give rise to geometric objects, called C^\infty-schemes. These geometric objects are primarily studied as models for synthetic differential geometry. Secondly, we introduce homotopy theory into the picture. This step adapts the ideas of derived algebraic geometry to the setting of C^\infty-schemes. Our approach replaces the algebraic objects involved by their simplicial counterparts. In this context, the main objective is to develop a homotopy theory of presheaves which allows us to work with sheaf axioms 'up to homotopy'. Succinctly, a derived smooth manifold can be described as a homotopical C^\infty scheme of finite type. In my talk, I will highlight some steps of the rather intricate construction described above. Hopefully, this will give the audience a perspective from which to think further about these exciting interactions between algebraic geometry, homotopy theory and differential topology.

Derived Smooth Manifolds, Part I

nell'ambito della serie: LOGIC, CATEGORIES, AND APPLICATIONS SEMINAR

algebra e geometria

interdisciplinare

teoria delle categorie

A well-known shortcoming of the category of smooth manifolds is its lack of arbitrary pullbacks. A pullback of manifolds, and in particular an intersection of submanifolds, exists only along maps which are transversal. This problem can be overcome by passing to the larger category of derived smooth manifolds. The construction of this category combines ideas from algebraic geometry, homotopy theory and of course differential topology. We can describe this construction in several steps. Firstly, we consider the relation between manifolds and schemes. Here, we employ the so-called C^\infty-rings, which are algebraic objects encoding the structure of the collection of smooth functions on R^n beyond that of an R-algebra. By the general philosophy of algebraic geometry, their duals give rise to geometric objects, called C^\infty-schemes. These geometric objects are primarily studied as models for synthetic differential geometry. Secondly, we introduce homotopy theory into the picture. This step adapts the ideas of derived algebraic geometry to the setting of C^\infty-schemes. Our approach replaces the algebraic objects involved by their simplicial counterparts. In this context, the main objective is to develop a homotopy theory of presheaves which allows us to work with sheaf axioms 'up to homotopy'. Succinctly, a derived smooth manifold can be described as a homotopical C^\infty scheme of finite type. In my talk, I will highlight some steps of the rather intricate construction described above. Hopefully, this will give the audience a perspective from which to think further about these exciting interactions between algebraic geometry, homotopy theory and differential topology.

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DIPARTIMENTO DI MATEMATICA

Logic, Categories, and Applications Seminar

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Logic, Categories, and Applications Seminar

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DIPARTIMENTO DI MATEMATICA