# Mathematics Colloquium Spring 2014

Tuesdays - 4:00 p.m.
McHenry Building -  Room 4130
Refreshments served at 3:30 - McHenry Building 4161

### Yi Wang Stanford University

A well-known question in differential geometry is to prove the isoperimetric inequality under intrinsic curvature conditions. In dimension 2, the isoperimetric inequality is controlled by the integral of the positive part of the Gaussian curvature. In my recent work, I prove that on simply connected conformally flat manifolds of higher dimensions, the role of the Gaussian curvature can be replaced by the Branson's Q-curvature. The isoperimetric inequality is valid if the integral of the Q-curvature is below a sharp threshold. Moreover, the isoperimetric constant depends only on the integrals of the Q-curvature. The proof relies on the theory of $A_p$ weights in harmonic analysis.

### Torsten EhrhardtUniversity of California, Santa Cruz

Dimer models are some of the classical basic models in statistical physics. As a rule they describe multi-particle configurations placed on a lattice. Besides explaining the very classical background, I will consider a concrete dimer problem which contains a parameter interpolating between triangular and square lattices. The computation of the monomer-monomer correlation function leads to evaluating a certain block Toeplitz determinant.  In contrast to the scalar case, where the classical theory of Toeplitz determinants is well developed, in the block case no general formula exists. I will indicate how an answer can be obtained in this case, which in physical terms tells whether the model is confining or deconfining.

### Dispersive Curve Flows

Chuu-Lian Terng

University of California, Irvine

A number of important model linear dispersive equations give rise to interesting curve flows in differential geometry. In this talk we will discuss some of these including the Schrodinger curve flow on the two sphere,  the Hodge star mean curvature curve flow in Euclidean andLorentzian 3-space, and the geometric Airy curve flow on Euclidean space and the affine space.  The equations of  curvatures of these curve flows turn out to be soliton equations. Hence we can use techniques from soliton theory to study these curve flows.  In particular, we can construct infinitely many families of explicit solutions and solve the periodic Cauchy problem.

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Special Tea time is from 3:15 pm - 3:45 pm.

Professor Richard Palais will visit our Department and meet with faculty, graduate students, and Palais Prize recipient from 3:15 to 3:45 pm.

Richard Palais is one of America's most renowned mathematicians. He is a founder of the field of Global Non Linear Analysis and his work has had a great influence on the research of several UCSC faculty. He was a member of our Department for several years. He is currently Professor Emeritus at Brandeis University and Professor of Mathematics at University of California, Irvine.

The Palais family created an endowment for Mathematics and Molecular and Cellular Biology students. The award enables meritorious students to pursue research projects while establishing important mentoring relationships with faculty.

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April 29, 2014

### University of California, Berkeley

We will discuss some joint work with Daniel Smith. On compact sets where the Chern scalar curvature is bounded from above, we will show that the flow converges smoothly in infinite time. This implies that singularities for the flow form precisely where Chern scalar curvature blows up.

### University of Glasgow/MSRI

One of the best known examples of a polynomial Hopf algebra is associated to the (graded) ring of symmetric functions. It has the remarkable property that it and its dual are commutative and in fact isomorphic as Hopf algebras. It appears in many contexts including representation theory of symmetric groups and the study of Chern classes and multiplicative sequences. It is in fact the cohomology ring of the classifying space $BU$.

A less well known example is obtained from a non-commutative version of this with commutative dual, the ring of quasi-symmetric functions. Again this appears in many contexts and is actually the cohomology of an $H$-space $\Omega\Sigma \mathbb{C}P^\infty$. It was conjectured in the 1970s that this ring was polynomial and after several incomplete proofs, this Ditters conjecture was proved around 2000 by Hazewinkel.

I will explain some of the algebraic background on Hopf algebras, then discuss a strategy for proving this sort of result for cohomology rings of certain loop spaces, generalising an earlier joint proof with Birgit Richter. The methods used involve the Eileneberg-Moore spectral sequence and various standard topological tools.

### University of California, San Diego

Since ancient times, beginning with the famous Dido problem, isoperimetric problems, in which one seeks to find the smallest perimeter enclosing a given volume, have stimulated much mathematical research. In the early twentieth century, such problems became wellunderstood in constant curvature spaces. In the 1960's with the advent of geometric measure theory, tools became available to study isoperimetry in more general spaces, yet characterising optimal domains, beyond the fact that they must have constant mean curvature, only began to be understood in the late twentieth century, initially by Benjamini and Cao, through the application of the curve shortening flow. One reason to expect that curvature flows should prove useful in studying isoperimetric problems is that the prototypical flow, the mean curvature flow is the gradient flow of the area functional, hence one expects it to converge to an optimal domain. There is a deep interplay between geometric flows and isoperimetric problems, and I will endeavour to describe some on the fascinating aspects of this relationship, in particularly by exploring the connection between the isoperimetric profile and curvature flows such as the Ricci mean curvature flows.

### May 27, 2014RIGIDITY AND FLEXIBILITY PHENOMENA IN GENERAL RELATIVITYAlessandro CarlottoStanford University

In this colloquium talk, I will give a broad spectrum overview of some of my most recent results concerning the large scale structure of asymptotically flat initial data sets for the Einstein equation in General Relativity. This will be aimed at the general mathematical audience.

A well-known corollary of the positive mass theorem by Schoen-Yau is that if an asymptotically flat manifold (of non-negative scalar curvature) is exactly flat outside of a compact set, then it has to be globally flat: in other terms any such metric can never be localized inside a compact set. So what is the "optimal" localization of those metrics? For instance, can one produce scalar non-negative metrics that have positive ADM mass and still are trivial in a half-space?
In recent joint work with Schoen, we answer these questions by giving a systematic method for constructing solutions to the Einstein constraint equations that are localized inside a cone of  arbitrarily small aperture. This sharply contrasts with various scalar curvature rigidity phenomena both in the closed and in the free-boundary setting.
Obviously, the manifolds we get have plenty of complete (non-compact), stable minimal hypersurfaces, which is a remarkable fact since I proved that in  dimension less than eight an asymptotically Schwarzschildean manifold containing an unbounded, stable minimal hypersurface has to be isometric to the Euclidean space.
Moreover, the gluing scheme that we adopt allows to produce a new class of exotic N-body solutions for the Einstein equation, which patently exhibit the phenomenon of gravitational shielding: for any large T we can engineer solutions where any two massive bodies do not interact at all for any time 0<t<T in striking contrast with the Newtonian gravity scenario.