Weekly Bulletin

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

For a given genus $g\geq1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over $\mathbb{F}_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $psilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over $\mathbb{F}_q$ with at least $1+q+(2g-psilon)\sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71\sqrt{q}$ valid for $g\geq3$ and odd $q\geq11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4\sqrt{q}-32$ valid for all $g\geq2$ and for all $q$. This talk is based on a joint work with joint work with Jonas Bergström, Everett W. Howe, and Christophe Ritzenthaler. <BR> <BR> (**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)

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.

Let $X$ be an algebraic variety over a field $k$. In arithmetic geometry we are interested in the set $X(k)$ of $k$-rational points on $X$. For example, is $X(k)$ empty? And if not, how `large' is $X(k)$? If $k$ is infinite, is $X(k)$ dense in $X$ with respect to the Zariski topology? Del Pezzo surfaces are surfaces classified by their degree~$d$, which is an integer between 1 and 9; for $d\geq3$, these are the smooth surfaces of degree $d$ in~$\mathbb{P}^d$. They contain a fixed number of `lines' (exceptional curves), depending on their degree. The lower the degree, the more complicated these surfaces become, and the more open questions there are. I will talk about the density of the set of rational points and the configurations of lines on del Pezzo surfaces of degree 1. I will show why these two topics are interesting to study, what they have to do with each other, and talk about results in both directions. <BR> <BR> (**This eSeminar will also be live-streamed on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact simran.tinani@math.uzh.ch **)

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.

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.