Seminars from until

Let $L = (L(t))_{t\geq 0}$ be a multivariate Lévy process with Lévy measure $\nu(dy) = \exp(-f(|y|)) dy$ for a smoothly regularly varying function $f$ of index $\alpha>1$. The process $L$ is renormalized as $X^\epsilon(t) = \epsilon L(r_\epsilon t)$, $t\in [0, T]$, for a scaling parameter $r_\epsilon = o(\epsilon^{-1})$, as $\epsilon \to 0$. We study the behavior of the bridge $Y^{\epsilon, x}$ of the renormalized process $X^\epsilon$ conditioned on the event $X^\epsilon(T) = x$ for a given end point $x\neq 0$ and end time $T>0$ in the regime of small $\epsilon$. Our main result is a sample path large deviations principle (LDP) for $Y^{x, \epsilon}$ with a specific speed function $S(\epsilon)$ and an entropy-type rate function $I_{x}$ on the Skorokhod space in the limit $\epsilon \to 0^+$. We show that the asymptotic energy minimizing path of $Y^{\epsilon, x}$ is the linear parametrization of the straight line between $0$ and $x$, while all paths leaving this set are exponentially negligible. Since on these short time scales ($r_\epsilon = o(\epsilon^{-1})$) direct LDP methods cannot be adapted we use an alternative direct approach based on convolution density estimates of the marginals $X^{\epsilon}(t)$, $t\in [0, T]$, for which we solve a specific nonlinear functional equation.

I will review two results pertaining to 3-dimensional Reshetikhin–Turaev TQFTs, defined from modular tensor categories M. These theories were not constructed as “fully local” TQFTs (in the framework of Lurie’s Cobordism Hypothesis): no algebraic structures were assigned to points. (The obstruction was the Witt class of M.) Kevin Walker solved the locality problem in the setting of anomalous theories. A ‘no-go’ theorem (joint with Dan Freed) showed that, if localized as linear theories, these RT theories did not admit local topological boundary conditions, and could therefore not be generated from a point by this method. (The group-like case had been addressed by Kapustin and Saulina.) In recent work with Freed and Claudia Scheimbauer, we displayed a fully local realization of these theories, by objects in a target 3-category which enlarges that of fusion categories. This allowed us to settle some conjectures relating orientations and spherical structures.

In this talk, we study fast diffusion equations (FDEs) in the context of interacting particle systems (IPS). The term fast diffusion refers to the fact that the diffusion coefficient diverges as the density approaches zero. These equations have been extensively studied in the literature and arise in a wide range of physical applications. For instance, they model diffusion in plasmas, appear in the study of cellular automata and interacting particle systems exhibiting self-organized criticality, and describe the evolution of plane curves shrinking along the normal direction at a curvature-dependent speed.

From the perspective of interacting particle systems, the first part of the talk is devoted to deriving an FDE as the scaling limit of a sequence of zero-range processes with symmetric unit rates. To capture the fast diffusion behavior at the microscopic level, we introduce an appropriate rescaling of models featuring a typically large number of particles per site. In the second part, we introduce a family of zero-range processes aimed at establishing a connection between the results of Landim (1996), Morris (2006), and Nagahata (2010). Certain processes within this family are naturally associated with fast diffusion equations, and our main goal is to determine the order of the relaxation time, a key ingredient in the derivation of scaling limits. Starting from a heuristic argument that estimates the relaxation time of a general zero-range process in terms of its partition function, we identify a parametric family of partition functions arising as solutions of a specific ordinary differential equation. By analyzing the asymptotic behavior of the coefficients in their power series expansions, we derive the corresponding family of rate functions. Finally, we present numerical evidence—obtained via deterministic iterative methods and Monte Carlo simulations—supporting the predicted order of the relaxation time for these processes.

This is joint work with Milton Jara (IMPA) and Freddy Hernández (UFF / Universidad Nacional de Colombia).

Existing machine learning frameworks operate over the field of real numbers ($\mathbb{R}$) and learn representations in real (Euclidean or Hilbert) vector spaces (e.g., $\mathbb{R}^d$). Their underlying geometric properties align well with intuitive concepts such as linear separability, minimum enclosing balls, and subspace projection; and basic calculus provides a toolbox for learning through gradient-based optimization.

But is this the only possible choice? In this seminar, we study the suitability of a radically different field as an alternative to $\mathbb{R}$ — the ultrametric and non-archimedean space of $p$-adic numbers, $\mathbb{Q}_p$. The hierarchical structure of the $p$-adics and their interpretation as infinite strings make them an appealing tool for code theory and hierarchical representation learning. Our exploratory theoretical work establishes the building blocks for classification, regression, and representation learning with the $p$-adics, providing learning models and algorithms. We illustrate how simple Quillian semantic networks can be represented as a compact $p$-adic linear network, a construction which is not possible with the field of reals. We finish by discussing open problems and opportunities for future research enabled by this new framework.

Based on:
André F. T. Martins, Learning with the $p$-adics

Recently it was shown through simulations studies that Sub-D produces estimates with unbiased and lower variance-covariance estimates than the ANOVA-based estimator, except in case of random “one-way” balanced designs. In this designs the simulations studies suggested they have the same variance-covariance estimates. This paper aims to compare the common ANOVA-based estimator to Sub-D in random “one-way” designs with two groups of treatment and in random “one-way” balanced designs. The comparison will be conducted through theoretical results and corroborated with simulation studies. It will be proved that the ANOVA-base estimator and Sub-D have exactly the same variance-covariance estimates in both above referred designs. The proof will be given firstly for random “one-way” designs with two groups of treatment and then for random “one-way” balanced designs.

The Malmquist-Takenaka (MT) system is a complete orthonormal system in $H^2(T)$ generated by an arbitrary sequence of points in the unit disk that do not approach the boundary very fast. The nth point of the sequence is responsible for multiplying the nth and subsequent terms of the system by a Möbius transform taking the point to 0. One can recover the classical trigonometric system, its perturbations or conformal transformations, as particular examples of the MT system. However, for many interesting choices of the generating sequence, the MT system is less understood. We prove almost everywhere convergence of the MT series for three different classes of generating sequences.

By Alexander's theorem, every link in the 3-sphere can be represented as the closure of a braid. Lorenz links and twisted torus links are two families that have been extensively studied and are well-described in terms of braids. In this talk, we will present a natural generalization of Lorenz links and twisted torus links that produces all links in the 3-sphere. This provides a simpler braid description for all links in the 3-sphere.

The Hopfield Neural Network has played, ever since its introduction in 1982 by John Hopfield, a fundamental role in the inter-disciplinary study of storage and retrieval capabilities of neural networks, further highlighted by the recent 2024 Physics Nobel Prize.

From its strong link with biological pattern retrieval mechanisms to its high-capacity Dense Associative Memory variants and connections to generative models, the Hopfield Neural Network has found relevance both in Neuroscience, as well as the most modern of AI systems.

Much of our theoretical knowledge of these systems however, comes from a surprising and powerful link with Statistical Mechanics, first established and explored in seminal works of Amit, Gutfreund and Sompolinsky in the second half of the 1980s: the interpretation of associative memories as spin-glass systems.

In this talk, we will present this duality, as well as the mathematical techniques from spin-glass systems that allow us to accurately and rigorously predict the behavior of different types of associative memories, capable of undertaking various different tasks.

Here we introduce basic concepts, various models (SIP, SEP, independent random walkers) and how they are linked to each other via the Lie algebraic formalism.

From the Lie algebraic formalism we infer that interacting particle systems with dualities come in "families" characterized by an underlying Lie algebra.

These are SU(2) for SEP, SU(1,1) for SIP, and the Heisenberg algebra for independent particles.

References

1. Giardina, C., & Redig, F. (2026). Duality for Markov processes: a Lie algebraic approach. Springer Nature.
2. Van Ginkel, B., & Redig, F. (2020). Hydrodynamic Limit of the Symmetric Exclusion Process on a Compact Riemannian Manifold: B. van Ginkel et al. Journal of Statistical Physics, 178(1), 75-116.
3. Junné, J., Redig, F., & Versendaal, R. (2024). Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles. arXiv:2410.20167.
4. Giardinà, C., Redig, F., & van Tol, B. (2024). Intertwining and propagation of mixtures for generalized KMP models and harmonic models. arXiv:2406.01160.
5. Schütz, G., & Sandow, S. (1994). Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems. Physical Review E, 49(4), 2726.
6. Giardina, C., Kurchan, J., Redig, F., & Vafayi, K. (2009). Duality and hidden symmetries in interacting particle systems. Journal of Statistical Physics, 135(1), 25-55.
7. Frassek, R., & Giardinà, C. (2022). Exact solution of an integrable non-equilibrium particle system. Journal of Mathematical Physics, 63(10).

Here we use duality to characterize the ergodic invariant measures, and use duality to also look at the stationary state of systems driven by reservoirs at the boundary.

Special attention is given to the harmonic model and propagation of mixed product states.

Here we use duality to characterize hydrodynamic limits and fluctuation fields.

Special attention is given to the hydrodynamic limit of SEP in a geometric setting, i.e., on graphs that approximate a Riemannian manifold.