Analysis seminar

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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 Title T.B.A.
Speaker, Affiliation Mario Gauvrit , Université Paris Cité
Date, Time 16 April 2024, 15:15-16:15
Location HG G 43
Title T.B.A.
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 Title T.B.A.
Speaker, Affiliation Dr. Riccardo Tione, MPI Leipzig
Date, Time 14 May 2024, 16:30-17:30
Location HG G 43
Title T.B.A.
HG G 43

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