Filter off: No filter for categories.
This paper, based on joint work with Badia, Behounek, and Cintula, concerns a generalisation of the Tarskian definition of consequence relation to accommodate relevant logics. In particular, these are non-monotone multiset consequence relations. In this course of the talk, I present some motivations for the central definitions (building on a picture of relevance as it concerns derivations in an axiomatic system), consider a range of examples, and study the properties of these consequence relations. The upshot is to capture the idea that relevant implication expresses the real consequence relation of these logics, and to study what these consequence relations must look like when studied from a more abstract perspective.
In this talk, I will start by introducing spin systems on the lattice Z^d. I will then focus mainly on spin systems whose underlying symmetry is continuous (as opposed to the celebrated Ising model whose symmetry sigma->-sigma is discrete). The goal of this talk will be to explain a surprising link which appeared recently between these models in statistical physics and questions in Bayesian statistics / statistical reconstruction. I will introduce a new way to identify long-range-order in these spins systems with continuous symmetry (also called "symmetry breaking") which is based on the concept of "group synchronization" and relies in particular on a recent work by Abbe, Massoulié, Montanari, Sly and Srivastava (2018).
The talk will not require any background in statistical physics. This is a joint work with Thomas Spencer (IAS, Princeton).
In this talk, we compute the thermodynamic limit of the loop-HYL model, an approximation to the Feynman representation of the hard-core Bose gas. We show that the excess density concentrates on the random interlacements, with a discontinuity at the critical point. Joint work with Matthew Dickson, LMU.
After a short presentation by Andreas Wiese, there will be an open discussion where everyone is invited to participate and give their own input, opinions and ideas on the topic.
We study dynamic programming, duality and optimality conditions in general convex stochastic optimization problems introduced by Rockafellar and Wets in the 70s. We give a general formulation of the dynamic programming recursion and derive an explicit dual problem in terms of two dual variables, one of which is the shadow price of information while the other one gives the marginal cost of a perturbation much like in classical Lagrangian duality. Existence of primal solutions and the absence of duality gap are obtained without compactness or boundedness assumptions. In the context of financial mathematics, the relaxed assumptions are satisfied under the well-known no-arbitrage condition and the reasonable asymptotic elasticity condition of the utility function. We extend classical portfolio optimization duality theory to problems of optimal semi-static hedging. Besides financial mathematics, we obtain several new results in stochastic programming and stochastic optimal control.
https://www.fm.mathematik.uni-muenchen.de/teaching/teaching_winter_term_2022_2023/seminars/christmas_workshop/index.html
In this talk we will talk about a class of particle system defined on top of random graphs that has the stochastic Kuramoto model as a particular example. We are interested on limit theorems for the empirical measure of the particles. In other words, we investigate the behavior of a typical particle proving law of large numbers and large deviations results. The graphs we consider include the Erdös-Rényi graph with different levels of sparsity.
In this talk we will talk about a class of particle system defined on top of random graphs that has the stochastic Kuramoto model as a particular example. We are interested on limit theorems for the empirical measure of the particles. In other words, we investigate the behavior of a typical particle proving law of large numbers and large deviations results. The graphs we consider include the Erdös-Rényi graph with different levels of sparsity.
The discovery of solitons and completely integrable partial differential equations (PDEs) provides a paradigm in mathematics and modern physics. Its impact on the development of PDEs in physics (both classical and quantum), pure analysis, and differential geometry can hardly be overrated. In this colloquium talk, I will give an introduction to a class of newly discovered completely integrable PDEs, which exhibit turbulent behavior, i.e., the degree of smoothness of solutions cannot be generally controlled by an infinite hierarchy of conservation laws and thus singularities can form. Indeed, these systems can be seen as infinite-dimensional continuum versions of classical so-called Calogero-Moser systems introduced by Francesco Calogero and Jürgen Moser in 1970s. Part of my talk is based on joint work with Patrick Gérard (Paris-Sud).
The totally asymmetric simple exclusion process is a conservative particle system that has been studies though various mathematical lenses. Results for this particle system include hydrodynamic limits, invariant distributions, fluctuations and large deviations. It has connections to the celebrated KPZ class via a coupling with the corner growth model and last passage percolation; it is considered one of the exactly solvable models of the KPZ class.
In this talk we will discuss a (non-exactly solvable) generalisation of TASEP in which the rates that govern the particle jumps depend on the location of the particle and the time that we are observing the process. The rates come from a background function that can be discontinuous in space and time. We will discuss the hydrodynamic limit of this version of TASEP (for particle current and density), which will be the solution to certain discontinuous PDEs.
We consider augmented training of ODE based neural networks, such as ResNets, to increase robustness with respect to adversarial attacks. This is done by adding a first order sensitivity term to the loss function, derived from the corresponding robust optimal control problem. To reduce memory cost in the training process, we take an optimize then discretize approach and compute the gradients via solving the adjoint sensitivity equation, which is of second order due to the modified loss function. The presented work is a collaboration with Enrique Zuazua.
In this talk, I am going to explain several approaches to explain the dynamics of neural networks. First, I will argue, why neural networks should always be viewed within the framework of dynamical systems. Then I am going to prove, how we may be able to derive mean-field differential equations for neural networks even in the intermediate/sparse coupling recurrent neural network regime. This construction involves network dynamics on graphops, which is a framework useful far beyond neural nets. Finally, I am going to indicate how to employ rigorous validated computation to prove dynamics of neural networks rigorously when pencil-and-paper methods fail.
Clustering is one of the fundamental tasks in data analytics and machine learning. In many situations, different clusterings of the same data set become relevant. For example, different algorithms for the same clustering task may return dramatically different solutions. We are interested in applications in which one clustering has to be transformed into another; such a scenario arises, for example, when a gradual transition from an old solution to a new one is required. Based on linear programming and network theory, we develop methods for the construction of a sequence of so-called elementary moves that accomplishes such a transition. Specifically, we discuss two types of transitions: short transitions with a low number of steps and transitions that retain separation of clusters throughout.
The main focus of the talk will be on a new, probabilistic, concept of solution to singular free boundary problems, in which boundary points may move at infinite speed. I will discuss this new concept in the context of Stefan problems from mathematical physics that describe melting/solidification of a solid/liquid (e.g., ice/water) in the presence of supercooling. In particular, I will present new global existence, regularity and uniqueness results for the two geometrically simplest settings: flat and radial. Based on joint works with Sergey Nadtochiy, Francois Delarue and Yucheng Guo.
In her talk „Molecules and Light“ Caroline Lasser illuminates the mathematical background on how a molecule aligns when exposed to laser light: Compared to us, molecules are small, and we describe them in the language of quantum mechanics with wave functions and Schrödinger operators. The Christmas lecture strolls into this exciting interdisciplinary field of research, highlighting some of the basic mathematics to be found there.
We prove a duality between the asymmetric simple exclusion process (ASEP) with non-conservative open boundary conditions and an asymmetric exclusion process with particle-dependent hopping rates and conservative reflecting boundaries. This is a reverse duality in the sense that the duality function relates the measures of the dual processes rather than expectations. Specifically, for a certain parameter manifold of the boundary parameters of the open ASEP this duality expresses the time evolution of a family of shock product measures with N microscopic shocks in terms of the time evolution of N particles in the dual process. The reverse duality also elucidates some so far poorly understood properties of the stationary matrix product measures of the open ASEP given by finite-dimensional matrices.
Calculating averages with respect to probability measures on submanifolds (level-sets of a certain function) is crucial in computational statistical mechanics. Inspired by the idea of encoding constraints via stiff drift terms (so called soft constraints), in this talk I will present sampling schemes on submanifolds using various diffusion processes. This talk is based on joint work with Lara Neureither and Wei Zhang.