Veranstaltungen

We will develop a new stochastic gradient descent algorithm. By adaptively controlling the variance in the noise term based on the objective function value we can prove global algebraic convergence rate. Earlier results only gave a logarithmic rate. The focus will mainly be on algorithms where the stochastic component is added for global convergence rather than when sampling is used for efficient approximation of the objective or loss function. We will also see that this methodology extends to a gradient free setting.

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

Numerical simulations of merging black holes and neutron stars are essential for the emerging era of gravitational-wave astronomy, but computationally very challenging. Discontinuous Galerkin (DG) methods and a task-based approach to parallelism help us scale these simulations to supercomputers. In this seminar I present our discontinuous Galerkin scheme for the elliptic Einstein constraint equations of general relativity, and applications to problems involving black holes. Our numerical scheme accommodates curved manifolds, nonlinear boundary conditions, and hp-nonconforming meshes. Our generalized internal-penalty numerical flux and our Schur-complement strategy of eliminating auxiliary degrees of freedom make the scheme compact without requiring equation-specific modifications. I also outline our strategy for solving the DG-discretized elliptic problems effectively on supercomputers.

Verdier duality is a key feature of derived categories of constructible sheaves on well-behaved stratified spaces. In this talk we will explain how to extend the duality theorem to constructible sheaves on conically smooth stratified spaces and with values in a general stable bicomplete infinity-category. Our proof relies on two main ingredients, one categorical and one geometric. The first one is an equivalence between sheaves and cosheaves proven by Lurie in Higher Algebra. Lurie's theorem will appear in our discussion both as a fundamental building block for the six functor formalism in a very general setting and as a factor of the duality functor on constructible sheaves. The second is the unzip construction introduced by Ayala, Francis and Tanaka, which provides a functorial resolution of singularities to smooth manifolds with corners. This will be used to prove that the exit path infinity-category of any compact conically smooth stratified space is finite.

A key problem in topology is to relate the volume of a manifold to some other notions of complexity. On one hand, we focus on the simplicial volume of the manifold, a homotopy invariant encoding non-trivial information about the Riemmanian volume. On the other, to describe the complexity of a space, we use integer-valued invariants associated with covers, the so-called categorical invariants. In particular, since amenable groups are small in the context of large-scale geometry, we will see how amenable open covers are the right approximating tool for simplicial volume.

We present a framework of optimal transport aimed at probability measures arising from structural causal models, i.e., models that contain an information structure dictated by a directed graph. This builds on the recent string of literature (called causal or adapted optimal transport) studying this concept for time-series distributions, which has found many applications in financial contexts. We recover the adapted optimal transport problem for a particular temporal graph structure, and additionally show that other special cases like the standard optimal transport problem (fully connected graph) or problems related to Gromov-Wasserstein (graph without edges) are also included in the setting. The concept of coupling for a particular graph structure is characterized in various ways, giving rise to different interpretations and numerical possibilities. Further, we show that the resulting concept of Wasserstein distance can be used to control the difference between average treatment effects under different distributions, and is geometrically suitable to interpolate between different structural causal models.

The Birch-Swinnerton-Dyer conjecture gives a formula for the rank of an elliptic curve or an abelian variety in terms of its L-function. The parity conjecture is the corresponding simpler prediction for the parity of the rank. As I will explain, it is much easier to use in practice and there is a wealth of explicit arithmetic phenomena that it predicts. I will end by discussing some of the recent results on the conjecture in the context of abelian surfaces and Jacobians of curves.