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Seminari periodici
DIPARTIMENTO DI MATEMATICA

Seminars in Mathematical Physics and Beyond

Organizzato da: Gabriele Sicuro

Seminari passati

Matteo Casarosa

The \mathcal{G}_0 dichotomy in the generalized context

nell'ambito della serie: SEMINARS IN MATHEMATICAL PHYSICS AND BEYOND

SEMINARIO INTERDISCIPLINARE

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.

Godwin Osabutey

Structural Complexity of the Rooted-Tree Factorization of Integers

nell'ambito della serie: SEMINARS IN MATHEMATICAL PHYSICS AND BEYOND

fisica matematica

interdisciplinare

probabilità

sistemi dinamici

We encode natural numbers as planar rooted trees via iterated Euclidean decomposition, yielding a fully deterministic symbolic text of Dyck words governed by intrinsic syntactic rules. Without introducing probabilistic assumptions or generative models, we quantify how information, redundancy, and correlations are distributed within this corpus. Both the dictionary of distinct Dyck words and the entropy grow sublinearly, while the rank–frequency curve exhibits a stable parabolic fractal form indicative of hierarchical self-similarity rather than Zipfian scaling. Correlation measurements based on anomalous diffusion reveal a transition from near-diffusive to quasi-ballistic regimes, demonstrating persistent structural organization across scales. Together, these findings show that the sequence of natural numbers, when expressed through its tree-based Dyck-word representation, forms a highly non-random, statistically structured text in which large-scale order and predictability emerge directly from arithmetic determinism.

Transformer Wave Function for two dimensional frustrated magnets: emergence of a Spin-Liquid Phase in the Shastry-Sutherland Model

nell'ambito della serie: SEMINARS IN MATHEMATICAL PHYSICS AND BEYOND

fisica matematica

Understanding quantum magnetism in two-dimensional systems represents a lively branch in modern condensed-matter physics. In the presence of competing super-exchange couplings, magnetic order is frustrated and can be suppressed down to zero temperature. Still, capturing the correct nature of the exact ground state is a highly complicated task, since energy gaps in the spectrum may be very small and states with different physical properties may have competing energies. Here, we introduce a variational Ansatz for two-dimensional frustrated magnets by leveraging the power of representation learning. The key idea is to use a particular deep neural network with real-valued parameters, a so-called Transformer, to map physical spin configurations into a high-dimensional feature space. Within this abstract space, the determination of the ground-state properties is simplified and requires only a shallow output layer with complex-valued parameters. We illustrate the efficacy of this variational Ansatz by studying the ground-state phase diagram of the Shastry-Sutherland model, which captures the low-temperature behavior of SrCu2(BO3)2 with its intriguing properties. With highly accurate numerical simulations, we provide strong evidence for the stabilization of a spin-liquid between the plaquette and antiferromagnetic phases. In addition, a direct calculation of the triplet excitation at the Γ point provides compelling evidence for a gapless spin liquid. Our findings underscore the potential of Neural-Network Quantum States as a valuable tool for probing uncharted phases of matter, and open up new possibilities for establishing the properties of many-body systems.

Statistical physics of learning with two-layer neural networks

nell'ambito della serie: SEMINARS IN MATHEMATICAL PHYSICS AND BEYOND

fisica matematica

Feature learning - or the capacity of neural networks to adapt to the data during training - is often quoted as one of the fundamental reasons behind their unreasonable effectiveness. Yet, making mathematical sense of this seemingly clear intuition is still a largely open question. In this talk, I will discuss a simple setting where we can precisely characterise how features are learned by a two-layer neural network during the very first few steps of training, and how these features are essential for the network to efficiently generalise under limited availability of data.

Alexander Zlokapa

Slow mixing from quantum bottlenecks

nell'ambito della serie: SEMINARS IN MATHEMATICAL PHYSICS AND BEYOND

nel ciclo di seminari: SEMINARS IN MATHEMATICAL PHYSICS AND BEYOND

fisica matematica

Locandina seminario

In classical algorithms, tools such as the overlap gap property and free energy barrier are used to provide lower bounds for algorithms that are local, stable, or low-degree. In this talk, we review quantum algorithms for Gibbs sampling and show that they face analogous obstructions due to a general quantum bottleneck lemma. When applied to Metropolis-like algorithms and classical Hamiltonians, our result reproduces classical slow mixing arguments. Unlike previous techniques to bound mixing times of quantum Gibbs samplers, however, our bottleneck lemma provides bounds for non-commuting Hamiltonians. We apply it to systems such as random classical CSPs, quantum code Hamiltonians, and the transverse field Ising model. Key to our work are two notions of distance, which we use to measure the locality of quantum samplers and to construct the bottleneck.