Oct 21, 2024
Monday
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09:15 AM - 09:30 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
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- Supplements
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09:30 AM - 10:30 AM
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Rigidity theorems for the area widths of Riemannian manifolds
Fernando Marques (Princeton University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In this lecture we discuss rigidity results for the variationally defined widths of the area functional. This is based on joint work with Lucas Ambrozio and Andre Neves.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Scalar curvature and the length of a shortest closed geodesic
Regina Rotman (University of Toronto)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Let M be a closed Riemannian 3-manifold with scalar curvature bounded below by some positive constant k. We will prove that there exists a closed non-trivial geodesic on M of length at most $\frac{c}{\sqrt{k}}$. (Joint with Y. Liokumovich, D. Maximo.)
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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The geometry of pp-waves
Sven Hirsch (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In Special Relativity, the vanishing of mass occurs exclusively in the cases of vacuum and radiation. In this talk, we demonstrate that an analogous result holds within the framework of General Relativity. The proof combines ideas from spin geometry and spacetime harmonic functions. This is based upon joint work with Yiyue Zhang.
- Supplements
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03:00 PM - 03:00 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Smoothing $L^\infty$ Riemannian metrics with nonnegative scalar curvature outside of a singular set
Paula Burkhardt-Guim (Stony Brook University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We show that any $L^\infty$ Riemannian metric $g$ on $\R^n$ that is smooth with nonnegative scalar curvature away from a singular set of finite $(n-\alpha)$-dimensional Minkowski content, for some $\alpha>2$, admits an approximation by smooth Riemannian metrics with nonnegative scalar curvature, provided that $g$ is sufficiently close in $L^\infty$ to the Euclidean metric. The approximation is given by time slices of the Ricci-DeTurck flow, which converge locally in $C^\infty$ to $g$ away from the singular set. We also identify conditions under which a smooth Ricci-DeTurck flow starting from a $L^\infty$ metric that is uniformly bilipschitz to Euclidean space and smooth with nonnegative scalar curvature away from a finite set of points must have nonnegative scalar curvature for positive times.
- Supplements
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Oct 22, 2024
Tuesday
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09:30 AM - 10:30 AM
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Uniqueness of Semigraphical Translators
Mariel Saez Trumper (Pontificia Universidad Católica de Chile)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
In this talk we prove the uniqueness of pitchfork and helicoid translators of the mean curvature flow in $\mathbb{R}^3$. This solves a conjecture by Hoffman, White and Martin.
The proof is based on an arc-counting argument motivated by Morse-Rad\'o theory for translators and a rotational maximum principle.
This is joint work with F. Martin and R. Tsiamis.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Einstein manifolds with boundary
Zhongshan An (University of Michigan)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We will talk about existence of Einstein metrics on manifolds with boundary, while prescribing the induced conformal metric and mean curvature of the boundary. In dimension 3, this becomes the existence of conformal embeddings of surfaces into constant sectional curvature space forms, with prescribed mean curvature. We will show existence of such conformal emebeddings near generic Einstein background. We will also discuss the existence question in higher dimensions, where things become more subtle and a non-degenerate boundary condition is used to construct metrics with nonpositive Einstein constant.
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Nonnegative Ricci curvature, nilpotency, and asymptotic geometry
Jiayin Pan (University of California, Santa Cruz)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
The interplay between geometry and topology is always one of the central topics in Riemannian geometry. For open (noncompact and complete) manifolds with nonnegative Ricci curvature, it is known that the fundamental groups could be torsion-free nilpotent. This is distinct from open manifolds with nonnegative sectional curvature, whose fundamental groups are virtually abelian. This talk will cover how the virtual nilpotency/abelianness of the fundamental group is related to various geometric quantities and equivariant asymptotic geometry.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Minimal surfaces in spheres and random permutations
Antoine Song (California Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Minimal surfaces in spheres, which are invariant by a group of symmetries, are closely related to group theory, hyperbolic geometry and geometric topology. In this talk, I will discuss a new connection with random matrices. As we will explain, from two permutations, one can associate a 2d minimal surface in a Euclidean sphere. The main property is a probabilistic rigidity phenomenon: if the two permutations are chosen uniformly at random, then with high probability, the minimal surface has a geometry close to that of the hyperbolic plane.
- Supplements
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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Oct 23, 2024
Wednesday
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09:30 AM - 10:30 AM
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Harmonic maps into Euclidean Buildings
Christine Breiner (Brown University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We describe a regularity result for equivariant harmonic maps from the universal cover of a Riemannian manifold into a (not necessarily locally finite) Euclidean building. As an application we prove non-Archimedean superrigidity for rank 1 symmetric spaces. This result extends the work of Gromov-Schoen, who proved p-adic superrigidity by considering locally finite targets. This work is joint with B. Dees and C. Mese.
- Supplements
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10:30 AM - 10:40 AM
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Group Photo
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- Location
- SLMath: Front Courtyard
- Video
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- Abstract
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- Supplements
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10:40 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Ricci Curvature, Fundamental Groups, and Milnor Conjecture
Elia Bruè (Università Commerciale ``Luigi Bocconi'')
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
This seminar will examine the structure of fundamental groups on manifolds with Ricci curvature bounded below. A central focus will be Milnor's famous 1968 conjecture and the recent counterexample developed in collaboration with Naber and Semola.
- Supplements
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Oct 24, 2024
Thursday
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09:30 AM - 10:30 AM
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Mass-angular momentum inequalities, singular harmonic maps, and flows of stationary vacuum black holes
Marcus Khuri (Stony Brook University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
We establish the conjectured mass-angular momentum inequality for multiple black holes, modulo the extreme black hole `no hair theorem'. More precisely it is shown that either there is a counterexample to black hole uniqueness, in the form of a regular axisymmetric stationary vacuum spacetime with an asymptotically flat end and multiple degenerate horizons which is `ADM stable', or the following statement holds. Complete, simply connected, maximal initial data sets for the Einstein equations with multiple ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum, under the assumption of nonnegative energy density and axisymmetry. Moreover, equality is achieved in the mass lower bound only for a constant time slice of an extreme Kerr spacetime. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the harmonic map equations and their linearization. This is joint work with Qing Han, Gilbert Weinstein, and Jingang Xiong.
- Supplements
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10:40 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Toral symmetries of collapsed ancient homogeneous Ricci flows
Anusha Mangala Krishnan (Indian Institute of Technology Bombay)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
Zoom Link
Collapsed ancient solutions to the homogeneous Ricci flow on compact manifolds occur only on the total space of principal torus bundles. Under an algebraic assumption that guarantees flowing through diagonal metrics and a tameness assumption on the collapsing directions, we prove that such solutions have additional symmetries, i.e., they are invariant under the right action of their collapsing torus. As a byproduct of these additional torus symmetries, we prove that these solutions converge, backward in time, in the Gromov-Hausdorff topology to an Einstein metric on the base of a torus bundle. (Joint work with F. Pediconi and S. Sbiti.)
- Supplements
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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The Art and Science of Writing About Math
Siobhan Roberts (New York Times)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
Siobhan Roberts will recount her trajectory from history major to math journalist, and give a behind-the-scenes look at writing about mathematics for the New York Times.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Toward a stability theory for higher codimension minimal surfaces
Ailana Fraser (University of British Columbia)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
The stability theory for minimal submanifolds in codimension one has been well studied. In this talk we will survey the state of affairs in higher codimension and discuss recent progress.
- Supplements
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Oct 25, 2024
Friday
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09:30 AM - 10:30 AM
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Singularities in mean curvature flow
Otis Chodosh (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
I will discuss some recent results (joint works with Kyeongsu Choi, Josh Daniels-Holgate, Christos Mantoulidis, Felix Schulze) concerning the singularities that can arise (or not) in mean curvature flow of certain hypersurfaces.
- Supplements
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10:30 AM - 11:00 AM
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Morning Break
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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11:00 AM - 12:00 PM
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Positive scalar curvature metrics and aspherical summands
Shuli Chen (Stanford University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
A closed manifold is called aspherical if it has contractible universal cover. It has been conjectured since the 80s that all closed aspherical manifolds do not admit metrics with positive scalar curvature. In dimensions 3,4,5 this conjecture is solved by works of Schoen—Yau, Gromov—Lawson, Chodosh—Li, and Gromov. We prove for n = 3,4,5 that the connected sum of a closed aspherical n-manifold with an arbitrary non-compact manifold does not admit a complete metric with nonnegative scalar curvature. More generally, for n = 3,4,5, we generalize Chodosh, Li, and Liokumovich's partial classification result of closed PSC n-manifolds to the non-compact case. This is joint work with Jianchun Chu and Jintian Zhu.
- Supplements
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12:00 PM - 12:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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02:00 PM - 03:00 PM
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Introducing various notions of distances between space-times
Anna Sakovich (Uppsala University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
I will introduce the class of causally-null-compactifiable spacetimes that can be canonically converted into compact timed-metric spaces using the cosmological time function of Andersson-Galloway-Howard and the null distance of Sormani-Vega. This class of space-times includes future developments of compact initial data sets and regions exhausting asymptotically flat space-times. I will discuss various intrinsic notions of distance between such space-times and show that some of them are definite in the sense that they are equal to zero if and only if there is a time-oriented Lorentzian isometry between the space-times. These definite distances allow us to define notions of convergence of space-times to limit space-times that are not necessarily smooth. This is joint work with Christina Sormani.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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- Location
- SLMath: Atrium
- Video
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- Abstract
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- Supplements
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03:30 PM - 04:30 PM
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Smoothness of Riemannian submersions in nonnegative curvature
Burkhard Wilking (Westfälische Wilhelms-Universität Münster)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
Zoom Link
- Supplements
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