Events for 10/20/2021 from all calendars

Mathematical Physics and Harmonic Analysis Seminar

Time: 10:00AM - 11:00AM

Location: Zoom

Speaker: Noema Nicolussi, Ecole Polytechnique

Title: Asymptotics of Green functions: Riemann surfaces and Graphs

Abstract: There are many interesting parallels between the analysis and geometry of Riemann surfaces and graphs. Both settings admit a canonical measure/metric (the Arakelov--Bergman and Zhang measures) and the associated canonical Green function reflects crucial geometric information.

Motivated by the question of describing the limit of the Green function on degenerating Riemann surfaces, we introduce new and higher rank versions of metric graphs and their Laplace operators. We discuss how these limit objects describe the asymptotic of solutions to the Poisson equation and, in particular, the Green function on metric graphs and Riemann surfaces close to the boundary of their respective moduli spaces.

Based on joint work with Omid Amini (Ecole Polytechnique).

Noncommutative Geometry Seminar

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Simone Cecchini, University of Gottingen

Title: Distance estimates in the spin setting and the positive mass theorem

Abstract: The positive mass theorem states that a complete asymptotically Euclidean manifold of nonnegative scalar curvature has nonnegative ADM mass. It relates quantities that are defined using geometric information localized in the Euclidean ends (the ADM mass) with global geometric information on the ambient manifold (the nonnegativity of the scalar curvature). It is natural to ask whether the positive mass theorem can be ``localized’’, that is, whether the nonnegativity of the ADM mass of a single asymptotically Euclidean end can be deduced by the nonnegativity of the scalar curvature in a suitable neighborhood of E.

I will present the following localized version of the positive mass theorem in the spin setting. Let E be an asymptotically Euclidean end in a connected Riemannian spin manifold (M,g). If E has negative ADM-mass, then there exists a constant R > 0, depending only on the geometry of E, such that M must either become incomplete or have a point of negative scalar curvature in the R-neighborhood around E in M. This gives a quantitative answer, for spin manifolds, to Schoen and Yau's question on the positive mass theorem with arbitrary ends. Similar results have recently been obtained by Lesourd, Unger and Yau without the spin condition in dimensions <8 assuming Schwarzschild asymptotics on the end E. I will also present explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end E. The bounds obtained are reminiscent of Gromov's metric inequalities with scalar curvature. This is joint work with Rudolf Zeidler.

Groups and Dynamics Seminar

Time: 3:00PM - 4:00PM

Location: online

Speaker: Heejoung Kim , Ohio State University

Title: Algorithms detecting stability and Morseness for finitely generated groups

Abstract: In geometric group theory, finding algorithms for detection and decidability of various properties of groups is a fundamental question. For a finitely generated group G, we can study not only algorithmic problems for G itself but also algorithms related to a particular class of subgroups. For a word-hyperbolic group G, quasiconvex subgroups have been studied widely and there are algorithmic results. For example, Kapovich provided a partial algorithm which, for a finite set S of G, only halts if S generates a quasiconvex subgroup of G. However, beyond word-hyperbolic groups, the notion of quasiconveixty is not as useful. For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, a ``stable'' subgroup and a ``Morse'' subgroup. In this talk, we will discuss various detection and decidability algorithms for stability and Morseness of a finitely generated subgroup of mapping class groups, right-angled Artin groups, and toral relatively hyperbolic groups.

Topology Seminar

Time: 4:00PM - 5:00PM

Location: Zoom

Speaker: Chris Gerig, Simons Center for Geometry and Physics, Stony Brook University

Title: Studying 4-spheres using near-symplectic geometry and ECH

Abstract: I will introduce certain 2-forms that are "nearly symplectic" and use ECH (embedded contact homology) to probe homotopy 4-spheres. The “invariants” built for homotopy 4-spheres will count pseudoholomorphic curves in the complement of circles, but at the moment they are not sensitive enough to distinguish/detect 4-spheres: I will discuss ideas to possibly refine these “invariants”.