Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

Speaker: 

Yongjia Zhang

Institution: 

UC San Diego

Time: 

Tuesday, May 21, 2019 - 4:00pm

Location: 

RH 306

Entropy has been an important topic in the study of Ricci flow
ever since it was invented by Perelman. We consider Perelman's entropy
defined on an ancient solution, and prove a gap theorem for its backward
limit: If Perelman's entropy limits to a number too close to zero as time
approaches negative infinity, then the ancient solution must be the trivial
Euclidean space.

Ancient and translating solutions to mean curvature flow

Speaker: 

Mat Langford

Institution: 

University of Tennessee

Time: 

Tuesday, April 9, 2019 - 4:00pm

Host: 

Location: 

RH 306

A deep result of X.-J. Wang states that a convex ancient solution to mean curvature flow either sweeps out all of space or lies in a stationary slab (the region between two fixed parallel hyperplanes). We will describe recent results on the construction and classification of convex ancient solutions and convex translating solutions to mean curvature flow which lie in slab regions, highlighting the connection between the two. Work is joint with Theodora Bourni and Giuseppe Tinaglia.

One Dimensional Rectifiable Varifolds and Some Applications

Speaker: 

Robert Hardt

Institution: 

Rice University

Time: 

Tuesday, March 12, 2019 - 4:00pm

Host: 

Location: 

RH 306

A k dimensional varifold on R^n is a Radon measure on the Grassmann bundle R^n x G(n,k) of k planes in R^n. Varifolds were originally introduced to describe limiting behavior of minimizing sequences of functions, paths, or surfaces. Stationary one-dimensional rectifiable varifolds have a simple regularity description due to F.Almgren and W.Allard (1976).  Oriented 1d varifolds are useful in describing various optimal transport problems. Also, signed 1d varifolds can be used to model Michell trusses. These are cost minimal 1d balanced structures consisting of beams and cables. Introduced in 1904, they have been treated in the Mechanical Engineering  literature and in interesting mathematics papers by R.Kohn and G. Strang (1983) and by G.Bouchitte, W.Gangbo, and  P.Sepulcher (2008). There are many basic open questions about the location and structure of Michel trusses. The varifold model allows one to consider associated evolution and higher dimensional problems.

Information Geometry and Entropy-Based Inference

Speaker: 

Jun Zhang

Institution: 

University of Michigan

Time: 

Tuesday, March 5, 2019 - 4:00pm

Location: 

RH 306

Information Geometry is the differential geometric study of the manifold of probability models, and promises to be a unifying geometric framework for investigating statistical inference, information theory, machine learning, etc. Instead of using metric for measuring distances on such manifolds, these applications often use “divergence functions” for measuring proximity of two points (that do not impose symmetry and triangular inequality), for instance Kullback-Leibler divergence, Bregman divergence, f-divergence, etc. Divergence functions are tied to generalized entropy (for instance, Tsallis entropy, Renyi entropy, phi-entropy) and cross-entropy functions widely used in machine learning and information sciences. It turns out that divergence functions enjoy pleasant geometric properties – they induce what is called “statistical structure” on a manifold M: a Riemannian metric g together with a pair of torsion-free affine connections D, D*, such that D and D* are both Codazzi coupled to g while being conjugate to each other. Divergence functions also induce a natural symplectic structure on the product manifold MxM for which M with statistical structure is a Lagrange submanifold.  We recently characterize holomorphicity of D, D* in the (para-)Hermitian setting, and show that statistical structures (with torsion-free D, D*) can be enhanced to Kahler or para-Kahler manifolds. The surprisingly rich geometric structures and properties of a statistical manifold open up the intriguing possibility of geometrizing statistical inference, information, and machine learning in string-theoretic languages. 

Nonnegative Ricci curvature, stability at infinity, and structure of fundamental groups

Speaker: 

Jiayin Pan

Institution: 

UCSB

Time: 

Tuesday, February 12, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

We study the fundamental group of an open n-manifold of nonnegative Ricci curvature with some additional condition on the Riemannian universal cover. We show that if the universal cover satisfies certain geometric stability condition at infinity, the \pi_1(M) is finitely generated and contains an abelian subgroup of finite index. This can be applied to the case that the universal cover has a unique tangent cone at infinity as a metric cone or the case that the universal cover has Euclidean volume growth of constant 1-\epsilon(n).

 

Mass, Kaehler Manifolds, and Symplectic Geometry

Speaker: 

Claude LeBrun

Institution: 

Stony Brook University

Time: 

Tuesday, April 2, 2019 - 4:00pm to 5:00pm

Host: 

Location: 

RH 306

In the author's previous joint work with Hans-Joachim Hein, a mass formula for asymptotically locally Euclidean (ALE) Kaehler manifolds was proved, assuming only relatively weak fall-off conditions on the metric. However, the case of real dimension four presented technical difficulties that led us to require fall-off conditions in this special dimension that are stronger than the Chrusciel fall-off conditions that sufficed in higher dimensions. In this talk, I will explain how a new proof of the 4-dimensional case, using ideas from symplectic geometry, shows that Chrusciel fall-off suffices to imply all our main results in any dimension. In particular, I will explain why our Penrose-type inequality for the mass of an asymptotically Euclidean Kaehler manifold always still holds, given only this very weak metric fall-off hypothesis.

A promenade through the isoparametric story

Speaker: 

Quo-Shin Chi

Institution: 

Washington University in St. Louis

Time: 

Tuesday, April 23, 2019 - 4:00pm

Location: 

RH 306

Isoparametric hypersurfaces in the sphere are those whose
principal curvatures are everywhere constant with fixed multiplicities. In
some sense, such hypersurfaces represent the simplest type of manifolds we
can get a handle on. They have rather complicated topology and most of them
are inhomogeneous, and thus they serve as a good testing ground for
constructing examples and counterexamples. The classification of such
hypersurfaces was initiated by E. Cartan around 1938, and the completion of
the last case with four principal curvatures will appear soon in
publication. Since the classification is a long story covering a wide
spectrum of mathematics, I will highlight in this talk the decisive moments
and the key ideas engaged in the intriguing pursuit.

Pages

Subscribe to RSS - Differential Geometry