Nicola Paddeu (Université de Fribourg)
Abstract:
Abstract In this talk, we investigate the geometry of sub-Riemannian geodesics, starting from the fundamental definitions and guiding ideas that shape the field. After presenting the main notions and the central questions surrounding geodesics, we turn to the problem of regularity, highlighting its key features and discussing a representative result. No prior knowledge is required: the presentation is designed to be accessible, and particular emphasis will be placed on the connections between sub-Riemannian geometry and other areas of mathematics.
12:15 • Universität Basel, Lecture Room 117, Kollegienhaus
Leon Staresinic (Universität Zürich)
The Topological Boshernitzan-Kornfeld Conjecture abstract
Abstract:
<div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;"><span style="color: #000000;">Interval Translations Maps (ITMs) are a natural generalisation of the well-known Interval Exchange Transformations (IETs). They are obtained by dropping the bijectivity assumption for IETs. As such, they are exactly the finite piecewise isometries of the interval. There are two types of ITMs, finite-type and infinite-type ones. They are classified by their non-wandering sets: it is a finite union of intervals for finite-type maps, and contains a Cantor set for infinite-type maps.</span></div> <div style="caret-color: #ffffff; color: #ffffff; font-family: Helvetica; font-size: 12px; font-style: normal; font-variant-caps: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration-line: none; text-decoration-thickness: auto; text-decoration-style: solid;">
<div><span style="color: #000000;">One of the basic questions in the field is: How prevalent is each type of map in the parameter space? In this work, we show that the set of finite-type maps contains an open and dense subset of the parameter space of ITMs with a fixed number of intervals, which resolves in positive the topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.</span></div> <div> </div> <span style="color: #000000;">This is a joint work with Kostiantyn Drach and Sebastian van Strien.</span></div>
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.1
Shuchen Guo (University of Oxford, UK)
14:15 • Universität Basel
Farell Brumley
Joint Linnik Problems abstract
Abstract:
A well-known class of arithmetic equidistribution problems, attributed to Linnik, is concerned with periodic toric orbits on quaternionic varieties. Classical examples include the equidistribution of CM points of large discriminant on the modular surface and projections to the sphere of integer solutions to the sum of three squares. These problems were essentially solved by Duke using techniques in automorphic forms and analytic number theory. One can combine any two Linnik problems using a diagonal action of the torus, which encodes their simultaneous equidistribution (or disjointness). This creates a new set of problems, first put forward by Michel and Venkatesh, of considerably greater difficulty. We present new work with Valentin Blomer and Maksym Radziwiłł which uses an array of automorphic and analytic number theoretic techniques to prove the simultaneous equidistribution of two distinct Linnik problems, under a no-Siegel-zero type hypothesis. The latter assumption encodes the abundance of small split primes in quadratic field extensions, a property which interacts directly with competing approaches emanating from ergodic theory.
14:15 • EPF Lausanne, CM 1 517
William Mance (Poznan University)
Normal Numbers abstract
Abstract:
Informally, a real number is normal in base 10 if each of the digits 0, 1, . . . , 9 shows up with frequency 1/10 in its decimal expansion, each pair of digits 00, 01, . . . , 99 shows up with frequency 1/100 in its decimal expansion and so on. There are many easily stated (but very difficult!) open questions revolving around normality. For example, even determining the normality of $\\pi$ in any base appears to be far out of reach of modern mathematics. Depending on the interest of the audience, we may explore connections between normal numbers and other areas of math such as ergodic theory, descriptive set theory, computability theory, probability theory, and others.
14:15 • EPF Lausanne, GR A3 32
Jose A. Carrillo (University of Oxford, UK)
Abstract:
The Stein Variational Gradient Descent method is a variational inference method in statistics that has recently received a lot of attention. The method provides a deterministic approximation of the target distribution, by introducing a nonlocal interaction with a kernel. Despite the significant interest, the exponential rate of convergence for the continuous method has remained an open problem, due to the difficulty of establishing the related so-called Stein-log-Sobolev inequality. Here, we prove that the inequality is satisfied for each space dimension and every kernel whose Fourier transform has a quadratic decay at infinity and is locally bounded away from zero and infinity. Moreover, we construct weak solutions to the related PDE satisfying exponential rate of decay towards the equilibrium. The main novelty in our approach is to interpret the Stein-Fisher information, also called the squared Stein discrepancy, as a duality pairing between H?¹(Rn) and H¹(Rn), which allows us to employ the Fourier transform. We also provide several examples of kernels for which the Stein-log-Sobolev inequality fails, partially showing the necessity of our assumptions.
15:00 • Universität Basel
María Dolores Gómez Olvera (Universidad Rey Juan Carlos)
Cryptographic Protocols Over Noncommutative Group Rings abstract
Abstract:
<p>Group rings represent a compelling alternative for the development of cryptographic protocols. They offer a versatile foundation for building primitives with potential post-quantum resilience.</p> <p>In this talk, we will explore some recent advances in the design of secure protocols over noncommutative group rings. We will dicuss their fundamental algebraic properties and provide a security analysis against current cryptanalytic threats.</p>
15:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Prof. Dr. Helmut Harbrecht (Universität Basel)
Shape optimization under uncertainty abstract
Abstract:
This presentation aims to provide an overview of shape optimization in case of random input parameters. Besides the minimization of the expected objective, we also consider the minimization of failure probabilities. We show that in some situations the objective and its gradient are deterministic despite the random input. We can thus derive cheap, deterministic algorithms to minimize the objective. Several applications and numerical computations are given.
16:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 19.2
Sergi Sánchez Aragón (Universitat Autònoma de Barcelona)
Simplex and MacDonald Codes Over Finite Chain Rings and Their Gray Map Images abstract
Abstract:
<p>A linear code of length \\(n\\) over a finite chain ring \\(R\\) with residue field \\(\\mathbb{F}_q\\) is a \\(R\\)-submodule of \\(R^n\\). A \\(R\\)-linear code is a code over \\(\\mathbb{F}_q\\) (not necessarily linear) which is the generalized Gray map image of a linear code over \\(R\\).</p> <p>In this talk, we present some of the basic properties of linear codes over finite chain rings, as well as the tools used to study linearity of \\(R\\)-linear codes.</p> <p>Finally, we introduce the construction of linear simplex and MacDonald codes of type \\(\\alpha\\) and \\(\\beta\\) over a finite chain ring, showing their properties and those of their Gray map images.</p>
16:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Dr. Yujin Kim (California Institute of Technology)
The low temperature SOS model above a wall and 1:2:3 scaling abstract
Abstract:
Random surfaces play a central role in probability and mathematical physics for decades. Physically, they often arise as models of interfaces: boundaries between distinct regions of space. The Solid-on-Solid (SOS) model is a canonical discrete model for interfaces separating stable (equilibrium) coexisting phases in three dimensions, such as the boundary of a solid that has crystallized in a liquid solution. In this talk, we present the fascinating geometry of the SOS model at "low temperature", conditioned to be non-negative ("above a wall": think of substration on a hard surface). In this setting, the SOS model resembles a wedding cake, being comprised of a sequence of shrinking, stacked layers whose boundaries form a collection of nested loops. Our work sheds light on the fluctuations of these loops away from their Wulff shape scaling limits, and in particular suggests a scaling limit for these fluctuations. Based on joint works with Patrizio Caddeo, Milind Hegde, Eyal Lubetzky, and Christian Serio.
17:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H12