Events

Marcelo Viana has conjectured that a smooth diffeomorphism admits a physical measure if the Lyapunov exponents of its orbits in a full volume set do not vanish. I will explain how a technique controlling the continuity of Lyapunov exponents allows to prove this conjecture in the case of smooth surface diffeomorphisms. This is a joint work with Jérôme Buzzi and Omri Sarig.

I will discuss the following new phenomenon in Floer theory: in situations where continuation maps are not homotopy equivalences, so-called secondary continuation maps defined by interpolation carry nontrivial topological information. In a first example, which deals with Rabinowitz Floer homology, the secondary continuation map realizes the Poincaré duality isomorphism and intertwines graded Frobenius algebra structures. In a second example, motivated by string topology, the secondary continuation map controls extensions of the loop coproduct which, together with the loop product, fit into the structure of a unital infinitesimal bialgebra. The talk will be based on joint work with Kai Cieliebak and Nancy Hingston.

Chan, Har-Peled, and Jones [2020] recently developed locality-sensitive ordering (LSO), a new tool that allows one to reduce problems in the Euclidean space R^d to the 1-dimensional line. They used LSO's to solve a host of problems. Later, Buchin, Har-Peled, and Oláh [2019,2020] used the LSO of Chan et al. to construct very sparse reliable spanners for the Euclidean space. A highly desirable feature of a reliable spanner is its ability to withstand a massive failure: the network remains functioning even if 90% of the nodes fail. We develop the theory of LSO's in non-Euclidean metrics by introducing new types of LSO's suitable for general and topologically structured metrics. We then construct such LSO's, as well as constructing considerably improved LSO's for doubling metrics. Afterwards, we use our new LSO's to construct reliable spanners with improved stretch and sparsity parameters. Time permitting, we will also discuss several additional applications of LSO's.

More information: https://math.ethz.ch/news-and-events/events/research-seminars/zurich-colloquium-in-mathematics.html?s=hs22call_made

In this work in progress we follow Monavari's approach (see his talk above)and further determine the fixed locus of the PT moduli space of an arbitrary local curve - we find that it's reduced of pure dimension and its virtual cycle agrees with its fundamental cycle. From this one obtains surprising results about PT of C^2 x P^1 relative to 0 and infinity. This is perhaps the most important 3-fold for PT theory since it governs all the other ones via degeneration. On this basis we conjecture explicit formulas for part of its PT theory with the expectiation that there exist similar formulas for the rest as well - however it's not clear how easily one can guess and prove them. We also elaborate on several consequences that such formulas would have.

The goal of this talk will be to present some puzzling properties of the (two-component) lattice Coulomb gas on the d-dimensional lattice. The connection of this model with integer-valued fields and compact-valued spin systems will be emphasised through the talk. This is a joint work with Avelio Sepúlveda.

I will consider families of quantum lattice systems that have attracted much interest amongst people studying topological phases of matter. Their Hamiltonians are perturbations, by interactions of short range, of a Hamiltonian consisting of strictly local terms and with a (strictly positive) energy gap above its ground-state energy. I will review the main ideas of a novel method based on local Lie-Schwinger conjugations of the Hamiltonians associated with connected subsets of the lattice. By this method fermions and bosons are treated on the same footing, and our technique does not face a large field problem, even though bosons are involved.

This talk aims at being a gentle introduction to Hodge theory and its application to the cohomology of real and complex closed manifolds. If time permits, I will explain some vanishing theorems and introduce the idea of L²-estimates. Basic notions of differential and complex geometry will be recalled.

In a recent effort to push modern tools from machine learning into several areas of science and engineering, deep learning based methods have emerged as a promising alternative to classical numerical schemes for solving problems in the computational sciences -- example applications include fluid dynamics, computational finance, or computational chemistry. This talk seeks to illuminate the limitations and opportunities of this approach, both on a mathematical and an empirical level. In a first part we present computational hardness results for deep learning based algorithms and find that the computational hardness of a deep learning problem highly depends on the specific norm in which the error is measured. In a second part we present a deep learning based numerical algorithm that outperforms the previous state of the art in solving the multi electron Schrödinger equation -- one of the key challenges in computational chemistry.

More information: https://eth-its.ethz.ch/activities/its-science-colloquium.htmlcall_made

We are interested in the study of the mean field optimal stopping problem, that is the optimal stopping of a McKean-Vlasov diffusion, when the criterion to optimize is a function of the distribution of the stopped process. This problem models the situation where a central planner controls a continuous infinity of interacting agents by assigning a stopping time to each of them, in order to maximize some criterion which depends on the distribution of the system. We study this problem via a dynamic programming approach, which allows to characterize its value function by a partial differential equation on the space of probability measures, that we call obstacle problem (or equation) on Wasserstein space by analogy with the classical obstacle problem, which arises in particular in standard optimal stopping. We introduce a notion a viscosity solution for this equation and show that, under appropriate assumptions, the value function of the mean field optimal stopping problem is the unique viscosity solution of the obstacle problem on Wasserstein space. We also study the corresponding finite population problem and prove its convergence to the mean field problem.

In this talk, we will investigate the emergence of geometric patterns in well-trained deep learning models by making use of the layer-peeled model and the law of equi-separation. The former is a nonconvex optimization program that models the last-layer features and weights. We use the model to shed light on the neural collapse phenomenon of Papyan, Han, and Donoho, and to predict a hitherto-unknown phenomenon that we term minority collapse in imbalanced training. This is based on joint work with Cong Fang, Hangfeng He, and Qi Long (arXiv:2101.12699). In the second part, we study how real-world deep neural networks process data in the interior layers. Our finding is a simple and quantitative law that governs how deep neural networks separate data according to class membership throughout all layers for classification. This law shows that each layer improves data separation at a constant geometric rate, and its emergence is observed in an authoritative collection of network architectures and datasets during training. This law offers practical guidelines for designing architectures, improving model robustness and out-of-sample performance, as well as interpreting the predictions. This is based on joint work with Hangfeng He (arXiv:2210.17020).