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
Giardina, C., & Redig, F. (2026). Duality for Markov processes: a Lie algebraic approach. Springer Nature.
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.
Junné, J., Redig, F., & Versendaal, R. (2024). Hydrodynamic limit of the symmetric exclusion process on complete Riemannian manifolds and principal bundles. arXiv:2410.20167.
Giardinà, C., Redig, F., & van Tol, B. (2024). Intertwining and propagation of mixtures for generalized KMP models and harmonic models. arXiv:2406.01160.
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.
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.
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.
Este mini-curso, dividido em duas sessões de duas horas, tem por objetivo dar uma breve introdução aos números $p$-ádicos. A primeira sessão é dedicada à construção dos números $p$-ádicos e alguns aspetos algébricos, nomeadamente:
A analogia de Hensel: um dicionário aritmético-geométrico. Valuação e valor absoluto $p$-ádico.
O anel $\mathbb{Z}_p$ dos inteiros $p$-ádicos.
O corpo $\mathbb{Q}_p$ dos números $p$-ádicos.
Topologia $p$-ádica.
Breve referência ao teorema de Ostrowski e classificação dos corpos locais.
Lemma de Hensel e aplicações.
Extensões finitas e infinitas de $\mathbb{Q}_p$. O corpo $\mathbb{C}_p$.
Na segunda sessão serão abordados os seguintes tópicos introdutórios de análise $p$-ádica:
Funções contínuas em $\mathbb{Z}_p$.
Séries de Mahler e teorema de Mahler.
Funções localmente constantes.
Derivação $p$-ádica.
Derivação da série de Mahler.
Teorema do valor médio.
Destaco, entre a literatura sobre números $p$-ádicos, três referências bibliográficas, pelo seu caráter mais elementar e por abordarem também aspetos analíticos.
Os livros de Fernando Gouvêa e Svetlana Katok são mais elementares, enquanto que o livro de Alain Robert é mais avançado, contendo uma introdução à análise funcional $p$-ádica.
Fernando Gouvêa, p-adic Numbers: An Introduction, Springer, 2020.
Svetlana Katok, p-adic Analysis Compared with Real, AMS, 2007.
Alain Robert, A Course in p-adic Analysis, Springer, 2000.
The Vaidya spacetime is a spherically symmetric solution of the Einstein equations with a null dust source. This can be used to model the gravitational collapse of a thick shell of radiation: a flat interior region is matched at an inner boundary to the null dust filled region, which is then matched at an outer boundary to Schwarzschild spacetime. A central singularity inevitably forms, and depending on the profile of the energy density of the null dust, this singularity can be globally naked. Motivated by the cosmic censorship hypothesis, we consider perturbations of this configuration. We review previous work, and describe recent work where the perturbation of the inner boundary — the past null cone of the central singularity — is analysed using a framework for studying perturbations of general hypersurfaces. This sets boundary conditions for perturbations at the past null cone, and we then consider the 3+1 evolutionary problem, focussing on the question of the stability of the Cauchy horizon of the naked singularity.
We will concentrate on the problem of characterising the minimal speed of monotone fronts in monostable scalar reaction diffusion (and advection) equations. The emphasis will be on variational principles and the puzzling situation of explicitly solvable equations. If time permits, I will also cover voting models and the connection to renormalisation group theory.
We will concentrate on the problem of characterising the minimal speed of monotone fronts in monostable scalar reaction diffusion (and advection) equations. The emphasis will be on variational principles and the puzzling situation of explicitly solvable equations. If time permits, I will also cover voting models and the connection to renormalisation group theory.
We will concentrate on the problem of characterising the minimal speed of monotone fronts in monostable scalar reaction diffusion (and advection) equations. The emphasis will be on variational principles and the puzzling situation of explicitly solvable equations. If time permits, I will also cover voting models and the connection to renormalisation group theory.
Given any open, bounded set $\Omega$, we consider suitable combinations, via a reference function $\Phi$, of the first $p$-eigenvalue of the Dirichlet Laplacian of partitions of $\Omega$. We give two different formulations of the problem, one geometrical and one functional. We prove relations among the two formulations, existence and regularity of optimal partitions, convergence, and stability with respect to $p$ and to $\Phi$. Based on a joint work with G. Stefani (Padova).
We present a unified perspective on the hydrodynamic limits of three interacting particle systems in contact with slow boundary reservoirs: the Simple Symmetric Exclusion Process (SSEP), the Porous Medium Model (PMM), and the Symmetric Zero-Range Process (ZR).
Although these systems share the same type of boundary dynamics — particle creation and annihilation at rates of order $N^{-\theta}$ — their bulk dynamics differ substantially: linear exclusion, constrained exclusion with nonlinear mobility, and unbounded occupancy with nonlinear jump rates.
Under diffusive scaling, the empirical density evolves according to a parabolic equation whose form depends on the microscopic interaction. We show how the strength of the reservoirs determines a phase transition in the macroscopic boundary conditions: Dirichlet for $\theta < 1$, Robin for $\theta = 1$, and Neumann for $\theta > 1$.
This comparison highlights how microscopic mechanisms shape macroscopic diffusion, while revealing a universal boundary transition driven by slow reservoirs.
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.
Recent results on black hole interiors suggest a failure of strong cosmic censorship for charged black holes in the presence of a positive cosmological constant. In this talk we show that, in the context of the Einstein-Maxwell-real scalar field system, such violations are non-generic in a larger moduli space of non-smooth (spherically symmetric) initial data.
Dynamical systems are inevitably subject to noise perturbations, making the stability of invariant measures under noise perturbations a fundamental problem. Such a stability is well-known for physical measures in hyperbolic systems, but remains widely open for more general systems. This talk will present some recent results on stochastic stability of physical measures in both conservative and dissipative systems.
Categorical algebra is a fundamental branch of mathematics that lies at the intersection of category theory and algebra. On the one hand, it captures the fruitful properties and structures studied in algebra via category theory. On the other hand, it investigates the global categorical properties that algebraic objects enjoy when collected together. Both these endeavors are essential to extend and transport the fundamental concepts and theorems of algebra to different and broader settings. In this talk, we present an innovative theory that generalizes categorical algebra to the framework of 2-dimensional category theory. This has the notable advantage that the second dimension can be used both to weaken conditions that are too strict in nature and to refine algebraic invariants, obtaining a richer theory which encompasses a broader range of examples. Furthermore, 2-dimensional categorical algebra is essential to effectively compare different algebraic categories with each other. This talk is based on a joint work with Elena Caviglia and Zurab Janelidze.
Abelian categories and triangulated categories provide fundamental frameworks to study homological and cohomological problems across algebraic geometry, topology and representation theory.
In this talk we will explain how we can study the 2-category AbCat of abelian categories and the 2-category Triang of triangulated categories through the lenses of 2-dimensional categorical algebra. Surprisingly, through these lenses, AbCat and Triang look extremely similar.
We will show that the important notions of Serre subcategories and Serre quotients of abelian categories precisely correspond respectively with the 2-dimensional kernels and cokernels in AbCat. In a similar way, thick triangulated subcategories and Verdier localizations of triangulated categories are exactly the 2-kernels and the 2-cokernels inTriang.
Furthermore, even more striking similarities between the two contexts arise when characterizing these 2-kernels and 2-cokernels in terms of categorical properties satisfied by their underlying functors.
These results will allow us to show that both the 2-categories AbCat and Triang are exact, in appropriate 2-dimensional senses. In particular, AbCat is 2-Puppe exact in a 2-dimensional sense, while Triang satisfies the weaker exactness property of a 2-homological category.
This talk is based on a joint work in progress with Zurab Janelidze, Luca Mesiti and Ulo Reimaa.
Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenaria. In particular, alternating patterns between synchronization and desynchronization behaviors are given by studying the asymptotics of the Markov perturbed stationary distributions. This talk is based on joint works with Arno Berger, Wen Huang, Hong Qian, Felix X.-F. Ye, and Yingfei Yi.
Diffusion probabilistic models have become the state-of-the-art tool in generative methods, used to generate high-resolution samples from very high-dimension distributions (e.g. images). Although very effective, they suffer some drawbacks:
as opposed to variational encoders, the dimension of the problem remains high during the generation process and
they can be prone to memorization of the training dataset.
In this talk, we first provide an introduction to generative modeling, with a focus on diffusion models from the point of view of stochastic PDEs. Then, we introduce a kernel-smoothed empirical score and study the bias-variance of this estimator. We find improved bounds on the KL-divergence between a true measure and an approximate measure generated by using the smoothed empirical score. This score estimator leads to less memorization and better generalization. We demonstrate these findings on synthetic and real datasets, combining diffusion models with variational encoders to reduce the dimensionality of the problem.
