Departement Mathematik

Analysis seminar

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Frühjahrssemester 2024

Datum / Zeit Referent:in Titel Ort
19. März 2024
15:15-16:15 		Prof. Dr. Melanie Rupflin
University of Oxford
Details

Analysis Seminar

Titel Sharp quantitative results for maps from $S^2$ to $S^2$ of general degree
Referent:in, Affiliation Prof. Dr. Melanie Rupflin, University of Oxford
Datum, Zeit 19. März 2024, 15:15-16:15
Ort 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. März 2024
15:15-16:15 		Prof. Dr. Anuj Kumar
UC Berkeley
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Analysis Seminar

Titel Nonunique solutions of the transport equation for Sobolev vector fields
Referent:in, Affiliation Prof. Dr. Anuj Kumar, UC Berkeley
Datum, Zeit 26. März 2024, 15:15-16:15
Ort 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
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Analysis Seminar

Titel Spectral gaps for coclosed 1-forms on hyperbolic 3-manifolds
Referent:in, Affiliation Dr. Vikramaditya Giri, ETH Zurich, Switzerland
Datum, Zeit 9. April 2024, 15:15-16:15
Ort 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
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Analysis Seminar

Titel Morse index stability for Yang-Mills connections
Referent:in, Affiliation Mario Gauvrit , Université Paris Cité
Datum, Zeit 16. April 2024, 15:15-16:15
Ort 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
Details

Analysis Seminar

Titel John domains in variational problems
Referent:in, Affiliation Prof. Dr. Yi Zhang, Academy of Mathematics and System Science, Beijing
Datum, Zeit 30. April 2024, 15:15-16:15
Ort 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
7. Mai 2024
15:15-16:15 		Dr. Mateus Sousa
BCAM
Details

Analysis Seminar

Titel Local and global extremizers for Fourier restriction estimates
Referent:in, Affiliation Dr. Mateus Sousa, BCAM
Datum, Zeit 7. Mai 2024, 15:15-16:15
Ort HG G 43
Abstract In this talk we will see a brief history of sharp Fourier restriction theory and some recent developments related to Fourier restriction estimates on spheres. We will discuss the problem of finding sharp constants for such inequalities, as well as the questions of existence and classification of extremizers of these estimates.
Local and global extremizers for Fourier restriction estimatesread_more
HG G 43
14. Mai 2024
15:15-16:15 		Prof. Dr. Elia Bruè
Università Bocconi Milano
Details

Analysis Seminar

Titel Non-Uniqueness in Two-Dimensional Euler Equations
Referent:in, Affiliation Prof. Dr. Elia Bruè, Università Bocconi Milano
Datum, Zeit 14. Mai 2024, 15:15-16:15
Ort HG G 43
Abstract In 1962, Yudovich established the well-posedness of the two-dimensional incompressible Euler equations within the class of solutions with bounded vorticity. Since then, a central unresolved problem has been the question of uniqueness within the broader class of solutions with L^p-vorticities. Recent years have witnessed significant progress in this investigation. In my talk, I aim to provide an overview of these developments and highlight recent results obtained thanks to the convex integration method.
Non-Uniqueness in Two-Dimensional Euler Equationsread_more
HG G 43
14. Mai 2024
16:30-17:30 		Dr. Riccardo Tione
MPI Leipzig
Details

Analysis Seminar

Titel Rigidity of critical points of degenerate polyconvex energies
Referent:in, Affiliation Dr. Riccardo Tione, MPI Leipzig
Datum, Zeit 14. Mai 2024, 16:30-17:30
Ort 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.
Rigidity of critical points of degenerate polyconvex energiesread_more
HG G 43
21. Mai 2024
15:15-16:15 		Prof. Dr. Daniele Valtorta
Università degli Studi di Milano
Details

Analysis Seminar

Titel Energy Identity for Stationary Harmonic Maps
Referent:in, Affiliation Prof. Dr. Daniele Valtorta, Università degli Studi di Milano
Datum, Zeit 21. Mai 2024, 15:15-16:15
Ort HG G 43
Abstract We present the proof for Energy Identity for stationary harmonic maps. In particular, given a sequence of stationary harmonic maps weakly converging to a limit with a defect measure for the energy, then m-2 almost everywhere on the support of this measure the density is the sum of energy of bubbles. This is equivalent to saying that annular regions (or neck regions) do not contribute to the energy of the limit. This result is obtained via a quantitative analysis of the energy in annular regions for a fixed stationary harmonic map. The proof is technically involved, but it will be presented in simplified cases to try and convey the main ideas behind it.
Energy Identity for Stationary Harmonic Mapsread_more
HG G 43
28. Mai 2024
15:15-16:15 		Prof. Dr. Antoine Gloria
Sorbonne Université
Details

Analysis Seminar

Titel What is large-scale regularity?
Referent:in, Affiliation Prof. Dr. Antoine Gloria, Sorbonne Université
Datum, Zeit 28. Mai 2024, 15:15-16:15
Ort HG G 43
Abstract The aim of this talk is to investigate what survives of the standard estimates valid for operators with constant coefficients in the case of variable coefficients. The general strategy is based on quantifying how far the (inverse) operator with variable coefficients is from an (inverse) operator with constant coefficients, and obtain the desired estimates by perturbation. Whereas this is classically done at the level of the coefficients themselves (Meyers’ estimates, Schauder theory e.g.), in this talk I will use closeness in the sense of homogenization. As an illustration, I will discuss large-scale Meyers’ estimates, large-scale Lipschitz estimates, and conclude with large-scale dispersive estimates.
What is large-scale regularity?read_more
HG G 43
28. Mai 2024
16:30-17:30 		Dr. André Guerra
ETH Zurich, Switzerland
Details

Analysis Seminar

Titel Harmonic maps and the vectorial obstacle problem: singularities vs free boundaries
Referent:in, Affiliation Dr. André Guerra, ETH Zurich, Switzerland
Datum, Zeit 28. Mai 2024, 16:30-17:30
Ort HG G 43
Abstract I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain O in R^m. These minimizers can be thought of as solutions of a vectorial obstacle problem, or as harmonic maps into the manifold-with-boundary given by the complement of O. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.
Harmonic maps and the vectorial obstacle problem: singularities vs free boundariesread_more
HG G 43
4. Juni 2024
15:15-16:15 		Dr. Severin Schraven
TU Munich
Details

Analysis Seminar

Titel Two-sided Lieb-Thirring bounds
Referent:in, Affiliation Dr. Severin Schraven, TU Munich
Datum, Zeit 4. Juni 2024, 15:15-16:15
Ort HG G 43
Abstract We discuss upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. For atomic Hamiltonians with Kato potentials one can strengthen the result to obtain two-sided estimates for the sum of the negative eigenvalues. Instead of being in terms of the potential itself, as in the usual Lieb-Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of $(-\Delta + V +M)u_M =1$ in $\mathbb{R}^d$; here $M\in\mathbb{R}$ is chosen so that the operator is positive. This talk is based on the preprint \href{https://arxiv.org/abs/2403.19023}{arXiv:2403.19023} which is joint work with S. Bachmann and R. Froese.
Two-sided Lieb-Thirring boundsread_more
HG G 43

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