UC San Diego Geometry Seminar 2022-2023

Unless otherwise noted, all seminars are on Thursday 1:00 - 2:00 in Zoom setting or in person at APM 5829. Zoom ID:93127945240

Fall 2023 Schedule

October 5, 2023

Mat Langford (ANU), Cancelled

October 12 , 2023


October 19 , 2023

Yevgeny Liokumovich, Toronto
Title: Interplay of ideas in scalar curvature and macroscopic scalar curvature
Abstract:The work of Gromov, Lawson, Schoen, Yau on scalar curvature inspired progress in metric geometry. In turn, this progress inspired new results about geometry and topology of manifolds with positive scalar curvature. In my talk I will give some examples of how ideas travel between these two worlds. The talk will be based on joint works with Lishak-Nabutovsky-Rotman, Maximo, Chodosh-Li, Wang.

October 26, 2023


November 2, 2023

Aaron Naber, Colloquium at 4:00pm, APM 6402
Title: Ricci Curvature, Fundamental Group and the Milnor Conjecture
Abstract:It was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated. In this talk we will discuss a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated. There are several new points behind the result. The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake. The ability to build such a fractal structure will rely on a very twisted gluing mechanism. Thus the other new point is a careful analysis of the mapping class group \pi_0Diff(S^3\times S^3) and its relationship to Ricci curvature. In particular, a key point will be to show that the action of \pi_0Diff(S^3\times S^3) on the standard metric g_{S^3\times S^3} lives in a path connected component of the space of metrics with Ric>0.

November 9, 2023

Paul Minter, IAS
Title:Singularities in minimal submanifolds
Abstract:Over the last few years there have been significant developments in the techniques used to understand singularities within minimal submanifolds. I will discuss this circle of ideas and explain how they enable us to reconnect the study of these geometric singularities with more classical PDE techniques, such as those used in unique continuation.

November 16, 2023

Chi Fai Chau, UCI (Postponed)

November 30, 2023

Nienhaus, Jan, UCLA
Title: Einstein metrics on (even-dimensional) spheres
Abstract:The first non-round Einstein metrics on spheres were described in 1973 by Jensen in dimensions 4n+3 (n > 0). For the next 25 years it remained an open problem whether the same could be done in even dimensions. This question was settled in 1998 when C. Böhm constructed infinite families of Einstein metrics on all Spheres of dimension between 5 and 9, in particular on $S^6$ and $S^8$. In the 25 years since then, all spheres of odd dimension (at least 5) have been shown to admit non-round Einstein metrics. However, there have been no new developments in even dimensions above 8, leaving open to speculation the question of whether, if the dimension is even, non-uniqueness of the round metric is a low-dimensional phenomenon or to be expected everywhere. I will give an overview of the methods used to construct such Einstein metrics, which we recently used to construct the first examples of non-round Einstein metrics on $S^{10}$. This is joint work with Matthias Wink.

December, 7, 2023

Guozhen Lu, Univ. Connecticut
Title: Helgason-Fouruer analysis on hyperbolic spaces and applications to sharp geometric inequalities
Abstract: Sharp geometric and functional inequalities play an important role in analysis, PDEs and differential geometry. In this talk, we will describe our works in recent years on sharp higher order Poincare-Sobolev and Hardy-Sobolev-Maz'ya inequalities on real and complex hyperbolic spaces and noncompact symmetric spaces of rank one. The approach we have developed crucially relies on the Helgason-Fourier analysis on hyperbolic spaces and establishing such inequalities for the GJMS operators. Best constants for such inequalities will be compared with the classical higher order Sobolev inequalities in Euclidean spaces. The borderline case of such inequalities, such as the Moser-Trudinger and Adams inequalities will be also considered.

Spring 2022 Schedule

Questions: lni@math.ucsd.edu