Klaus Widmayer (Universität Zürich)
Global axisymmetric Euler flows with rotation abstract
Abstract:
We discuss the construction of a class of global, dynamical solutions to the 3d incompressible Euler equations near the stationary state given by uniform "rigid body" rotation. At the heart of this result is a dispersive effect due to rotation, which we present with some context.This is based on joint work with Y. Guo and B. Pausader (Brown University).
10:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG , Room G 19.1
Prof. Dr. Thierry Gallay (University Grenobles Alpes )
Interaction and Stabilizaion of Viscous Planar Vortices abstract
Abstract:
As a model of the viscous interaction of planar vortices,we consider the solution of the two-dimensional Navier-Stokesequations with singular initial data corresponding to a finitecollection of point vortices. In the high Reynolds number regime, theevolution of the vortex centers is described, at least for smalltimes, by the classical Helmholtz-Kirchhoff system, and the vortexcores are slightly deformed due to the mutual interactions. Longertimescales can be reached in simple situations corresponding torelative equilibria of the point vortex system. For example, in thecase of a vortex dipole with opposite circulations, an acccurateapproximation of the solution can be constructed, which takes intoaccount the leading order correction to the translation speeddue to finite size effects. The evolution is similar for ill-preparedinitial data, corresponding to smooth and radially symmetric vortices,except that the solution first undergoes a short transient period duringwhich each vortex adapts its shape to the exterior strain.This talk relies on joint works with Martin Donati, Michele Dolce,and Vladimir Sverak.
11:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG , Room G 19.1
Dr. Anshul Adve (Princeton University)
Conformal bootstrap for hyperbolic surfaces and subconvexity abstract
Abstract:
<p>Given a compact hyperbolic surface of fixed topology, we consider its Laplace eigenvalues together with the structure constants for multiplication with respect to a suitable orthonormal basis of automorphic forms. These numbers obey algebraic constraints analogous to the conformal bootstrap equations in physics. In this talk I will present two results. The first is a converse theorem for these constraints: any collection of numbers satisfying the constraints must come from a hyperbolic surface. The second is an application of the constraints to subconvexity for triple product L-functions. This second result is joint with James Bonifacio, Petr Kravchuk, Dalimil Mazac, Sridip Pal, Alex Radcliffe, and Gordon Rogelberg. No knowledge of physics or L-functions will be assumed.</p>
13:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Aitor Iribar López (ETH Zürich)
Cohomology of the Satake compactification IV
13:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43
Elia Brue (Bocconi University)
Instability and Nonuniqueness in Incompressible Fluid Models abstract
Abstract:
The incompressible Navier–Stokes and Euler equations are central to mathematical fluid dynamics, yet their well-posedness theory remains one of the great open problems in PDEs. In recent years, remarkable progress has been made in constructing and understanding non-unique solutions to these equations, often revealing unexpected instability mechanisms. In this talk I will survey some of these developments and highlight connections with the pioneering contributions of Vladimir Sverák.
14:30 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG , Room G 19.1
Markus Tempelmayr
A priori bounds for stochastic porous media equations abstract
Abstract:
We prove a priori bounds for solutions of stochastic porous media equations in their natural $L^1$- based regularity class, in a regime where the equation is both singular and degenerate. For simplicity we consider only a mildly singular regime, which allows to prove modelledness of the solution with respect to the solution of the corresponding linear stochastic heat equation. The proof relies on the kinetic formulation of the equation and a renormalized energy inequality. A careful analysis allows to balance the degeneracy of the diffusion coefficient against sufficiently strong damping of the multiplicative noise for small values of the solution.
15:00 • EPF Lausanne, Bernoulli Center
Dr. Rati Ludhani (Inria Saclay Centre)
Minimal Codewords of Projective Reed-Muller Codes of Order 2 abstract
Abstract:
<p>Minimal codewords of a linear code reveal its important structural properties and are required, for instance, in secret-sharing schemes and certain decoding algorithms. A nonzero codeword is said to be minimal if its support does not properly contain the support of any other nonzero codeword. Determining minimal codewords of a general linear code is NP-hard, so one typically exploits the specific structure of a given code. Here, we consider this problem for projective Reed Muller (PRM) codes of order 2.</p> <p>PRM codes of order 2 are evaluation codes obtained by evaluating quadratic forms over a finite field \\(F\\) at the \\(F\\)-rational points of the corresponding projective space. To characterize their minimal codewords, we reduce the problem to the following geometric question: given two quadrics such that the \\(F\\)-rational points of one are contained in the other, can they differ? Our main result is that for absolutely irreducible quadrics, this almost never happens. In this talk, we present a complete answer to this question, thereby classifying the minimal codewords of PRM codes of order 2.</p> <p>This is joint work with Alain Couvreur.</p>
15:15 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Vladimir Sverak (University of Minnesota)
Three-vortex Dynamics and the Hopf Fibration abstract
Abstract:
It it well-known that the dynamics of three point-vortices in the plane is integrable. The results go back to W. Groebli (1877), H. Poincare (1893), and others. Over the years, many perspectives on the topic have been developed. In this talk I will discuss a recent work with Thierry Gallay that approaches the computations from a geometric viewpoint based on the Hopf fibration. Our main motivation has been to understand the possibilities for regularizing three-vortex collisions.
15:45 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG , Room G 19.1
Amit Rajaraman (MIT, US)
Eigenvalue Bounds for Random Matrices via Zerofreeness abstract
Abstract:
We introduce a new technique to prove bounds for the spectral radius of a random matrix, based on using Jensen\'s formula to establish the zerofreeness of the associated characteristic polynomial in a region of the complex plane. Our techniques are entirely non-asymptotic, and we instantiate it in three settings:(i) The spectral radius of non-asymptotic Girko matrices---these are asymmetric matrices M ∈ ℂ^{n × n} whose entries are independent and satisfy 𝔼 Mᵢⱼ = 0 and 𝔼 |Mᵢⱼ|² ≤ 1/n.(ii) The spectral radius of non-asymptotic Wigner matrices---these are symmetric matrices M ∈ ℂ^{n × n} whose entries above the diagonal are independent and satisfy 𝔼 Mᵢⱼ = 0, 𝔼 |Mᵢⱼ|² ≤ 1/n, and 𝔼 |Mᵢⱼ|⁴ ≤ 1/n.(iii) The second eigenvalue of the adjacency matrix of a random d-regular graph on n vertices, as drawn from the configuration model.In all three settings, we obtain constant-probability eigenvalue bounds that are tight up to a constant. Applied to specific random matrix ensembles, we recover classic bounds for Wigner matrices, as well as results of Bordenave--Chafaï--García-Zelada, Bordenave--Lelarge--Massoulié, and Friedman, up to constants.
16:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room D 1.1
Prof. Dr. Alejandro José Giangreco Maidana (Faculty of Engineering, National University of Asunción)
Cyclicity of the Group of Rational Points of Abelian Varieties Over Finite Fields and Totally Real Algebraic Integers abstract
Abstract:
<div dir="auto">Vladuts characterized in 1999 the set of finite fields k such that all elliptic curves defined over k have a cyclic group of rational points. Under the conjecture of infinitely many Mersenne primes, this set is infinite. </div> <div dir="auto">In this talk, I study the question in higher dimension. Precisely, I prove that there is no such fields. This is related with the existence of some totally real algebraic integers having some arithmetic properties.</div> <div dir="auto">I am going to present a result about cyclicity of maximal abelian varieties as well, and how this is related to some totally positive algebraic integers.</div> <div dir="auto">In both problems, open questions about totally real algebraic integers arise, some of which are addressed from an algorithm point of view.</div>
16:30 • UZH Irchel, Winterthurerstrasse 190, Zürich, Building Y27, Room H 28
Dr. Catherine Cawley Wolfram (ETH Zürich, Switzerland)
Epstein curves and holography of the Schwarzian action abstract
Abstract:
The circle can be seen as the boundary at infinity of the hyperbolic plane. We give a 1-to-2 dimensional holographic interpretation of the Schwarzian action, by showing that the Schwarzian action (which is a function of a diffeomorphism of the circle) is equal to the hyperbolic area enclosed by an "Epstein curve" in the disk. A dimension higher, the Epstein construction was used to relate the Loewner energy (a function of a Jordan curve related to SLE and Brownian loop measures) to renormalized volume in hyperbolic 3-space.In this talk I will explain how to construct the Epstein curve, how the bi-local observables of Schwarzian field theory can be interpreted as a renormalized hyperbolic length using the same Epstein construction, and (time permitting) discuss a bit what we know so far about the relationship between the Schwarzian action and the Loewner energy. This is based on joint work with Franco Vargas Pallete and Yilin Wang.
17:15 • ETH Zentrum, Rämistrasse 101, Zürich, Building HG, Room G 43