Analysis seminar

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For Zoom URL please contact Laura Keller

Spring Semester 2024

Date / Time

Speaker

Title

Location

19 March 2024
15:15-16:15

Prof. Dr. Melanie Rupflin
University of Oxford

Event Details

Analysis Seminar

Title	Sharp quantitative results for maps from $S^2$ to $S^2$ of general degree
Speaker, Affiliation	Prof. Dr. Melanie Rupflin, University of Oxford
Date, Time	19 March 2024, 15:15-16:15
Location	HG G 43
Abstract	As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4\pi vert deg(v)\vert$ with equality if and only if $v$ is a rational map it is natural to ask whether maps with small energy defect $\de_v=E(v)-4\pi \abs{\deg(v)}$ are necessarily close to a rational map. While such a rigidity statement turns out to be false for maps of general degree, we will prove that any map $v$ with small energy defect is essentially given by a collection of rational maps that describe the behaviour of $v$ at very different scales and that the corresponding distance is controlled by a quantitative estimate of the form $\text{dist}^2\leq C \delta_v(1+\abs{\log\delta_v})$ which is indeed sharp.

Sharp quantitative results for maps from $S^2$ to $S^2$ of general degreeread_more

HG G 43

26 March 2024
15:15-16:15

Prof. Dr. Anuj Kumar
UC Berkeley

Event Details

Analysis Seminar

Title	Nonunique solutions of the transport equation for Sobolev vector fields
Speaker, Affiliation	Prof. Dr. Anuj Kumar, UC Berkeley
Date, Time	26 March 2024, 15:15-16:15
Location	HG G 43
Abstract	We construct nonunique solutions of the transport equation in the class $L^\infty$ in time and $L^r$ in space for divergence free Sobolev vector fields $W^{1, p}$. We achieve this by introducing two novel ideas: (1) In the construction, we interweave the scaled copies of the vector field itself. (2) Asynchronous translation of cubes, which makes the construction heterogeneous in space. These new ideas allow us to prove nonuniqueness in the range of exponents beyond what is available using the method of convex integration and sharply matchwith the range of uniqueness of solutions from Bruè, Colombo, De Lellis ’21.

Nonunique solutions of the transport equation for Sobolev vector fieldsread_more

HG G 43

9 April 2024
15:15-16:15

Dr. Vikramaditya Giri
ETH Zurich, Switzerland

Event Details

Analysis Seminar

Title	Spectral gaps for coclosed 1-forms on hyperbolic 3-manifolds
Speaker, Affiliation	Dr. Vikramaditya Giri, ETH Zurich, Switzerland
Date, Time	9 April 2024, 15:15-16:15
Location	HG G 43
Abstract	For hyperbolic 3-manifolds we study two quantifications of the property of having vanishing first Betti number, one analytic and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. First we construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap, which answers a question asked by Lin-Lipnowski in connection with the existence of irreducible solutions to the Seiberg-Witten equations. Interestingly, the proof uses a topological property of the sequence in order to deduce a uniform gap. We'll then explore the connection between a sequence of 3-manifolds having such a uniform gap and exponential in volume torsion homology growth. In the context of towers of arithmetic 3-manifolds, such a connection was suggested by the work of Bergeron-Şengün-Venkatesh. We show that for any sequence of closed hyperbolic 3-manifolds with uniformly bounded rank, if it has a uniform coclosed 1-form spectral gap, then it must have exponential torsion homology growth. Based on joint work with Amina Abdurrahman, Anshul Adve, Ben Lowe, and Jonathan Zung.

Spectral gaps for coclosed 1-forms on hyperbolic 3-manifoldsread_more

HG G 43

16 April 2024
15:15-16:15

Mario Gauvrit
Université Paris Cité

Event Details

Analysis Seminar

Title	Morse index stability for Yang-Mills connections
Speaker, Affiliation	Mario Gauvrit , Université Paris Cité
Date, Time	16 April 2024, 15:15-16:15
Location	HG G 43
Abstract	We investigate Morse index stability for a sequence of Yang-Mills connections on closed 4-manifolds under bubble-tree convergence. As critical points of a conformally invariant energy, Yang-Mills connections share close ties with harmonic maps in various aspects. Simultaneously, their analysis is simpler as long as we work in an appropriate gauge, the so-called Coulomb gauge. Motivated by applications in constructing non-stable solutions of the Yang-Mills equations, this work extends recent methods developed by Da Lio-Gianocca-Rivière for index stability to the Yang-Mills framework, using sharp decay estimates to prove that the neck regions account for a positive contribution to the second variation.

Morse index stability for Yang-Mills connectionsread_more

HG G 43

30 April 2024
15:15-16:15

Prof. Dr. Yi Zhang
Academy of Mathematics and System Science, Beijing

Event Details

Analysis Seminar

Title	John domains in variational problems
Speaker, Affiliation	Prof. Dr. Yi Zhang, Academy of Mathematics and System Science, Beijing
Date, Time	30 April 2024, 15:15-16:15
Location	HG G 43
Abstract	The notion of a John domain was initially introduced in 1961 by Fritz John, and later named after him by Martio and Sarvas. Typically, its study is motivated by its connections to the properties of quasiconformal and quasisymmetric mappings. Moreover, John domains find extensive applications in the theory of Sobolev functions in metric measure spaces and functional analysis, as they represent essentially the sole class of domains that uphold the Sobolev-Poincaré inequality. In this presentation, I will introduce several recent applications of John domains in the theory of the calculus of variations.

John domains in variational problemsread_more

HG G 43

14 May 2024
15:15-16:15

Prof. Dr. Elia Bruè
Università Bocconi Milano

Event Details

Analysis Seminar

Title	Title T.B.A.
Speaker, Affiliation	Prof. Dr. Elia Bruè, Università Bocconi Milano
Date, Time	14 May 2024, 15:15-16:15
Location	HG G 43

Title T.B.A.

HG G 43

14 May 2024
16:30-17:30

Dr. Riccardo Tione
MPI Leipzig

Event Details

Analysis Seminar

Title	Rigidity of critical points of degenerate polyconvex energies
Speaker, Affiliation	Dr. Riccardo Tione, MPI Leipzig
Date, Time	14 May 2024, 16:30-17:30
Location	HG G 43
Abstract	This talk concerns critical points $u$ of polyconvex energies of the form $f(X) = g(det(X))$, where $g$ is (uniformly) convex. It is not hard to see that, if $u$ is smooth, then $\det(Du)$ is constant. I will show that the same result holds for Lipschitz critical points $u$ in the plane. I will also discuss how to obtain rigidity for approximate solutions. This is a joint work with A. Guerra.