# Geometry & Analysis Seminar Fall 2015

McHenry Building
Thursdays 4:00pm
Room 4130

October 1, 2015

Dark Matter from the Geometric Point of View

Andrew Goetz, University of California, Santa Cruz

In two talks I will give an introduction to the astrophysical mystery of "dark matter" and discuss some mathematical properties of a theory known as "wave dark matter". The first talk will introduce topics such as the general theory of relativity, the observational evidence for dark matter, different theories of dark matter, and something called the "baryonic Tully-Fisher relation". The second talk will focus on the geometric PDEs (the Einstein-Klein-Gordon equations) underlying the theory of wave dark matter. I will talk about the family of spherically symmetric static solutions as models of dark galactic halos and possible connections to the baryonic Tully-Fisher relation.

October 8, 2015

Dark Matter from the Geometric Point of View

Andrew Goetz,
University of California, Santa Cruz

In two talks I will give an introduction to the astrophysical mystery of "dark matter" and discuss some mathematical properties of a theory known as "wave dark matter". The first talk will introduce topics such as the general theory of relativity, the observational evidence for dark matter, different theories of dark matter, and something called the "baryonic Tully-Fisher relation". The second talk will focus on the geometric PDEs (the Einstein-Klein-Gordon equations) underlying the theory of wave dark matter. I will talk about the family of spherically symmetric static solutions as models of dark galactic halos and possible connections to the baryonic Tully-Fisher relation.

October 15, 2015

Uniqueness of Critical Point of the Constant Mean Curvature Spacelike Surface in the Lorentz-Minkowski Space

Jingyong Zhu, University of Science and Technology of China

I will talk about some geometric properties of solutions to constant mean curvature (CMC) equation, and prove the uniqueness of the critical point of the height function of a spacelike CMC surface with respect to the plane in which the boundary (i.e. a convex curve) is included. Recently, Rafael Lopez and his co-authors gave an example to show there are points with positive Gaussian curvature for spacelike CMC surface spanning a convex planar curve, which must be a graph. As an application, our result shows those points can not be the critical point of the height function.

October 22, 2015

Recent developments in Arnold diffusion

Ke Zhang, University of Science and Technology of China

In 1963, Arnold asked whether a typical perturbation of completely integrable Hamiltonian systems admits orbits of instability. This phenomenon, later called "Arnold diffusion", complements the well known KAM theory, which asserts "most" orbits are stable. Solution to this problem in 2.5 degrees of freedom was announced by Mather in 2003, and with an alternative full proof given recently by V. Kaloshin and the speaker (independently, C.-Q. Cheng). In this talk, we will give a geometrical description of the ideas behind this proof, and discuss a plan to prove this conjecture in arbitrary degrees of freedom (joint with V. Kaloshin).

October 29, 2015

Closed minimal surfaces in some classes of hyperbolic three-manifolds

Zheng Huang, CSI and Graduate Center, CUNY

Minimal surfaces are fundamental objects in differential geometry. I will talk on some recent results on finding closed minimal surfaces and the multiplicity questions in several different classes of hyperbolic three-manifolds. A general theme is that such a canonical object always exists if one knows where to look.

November 5, 2015

On stability of the hyperbolic space form under normalized Ricci flow

Yucheng Lu, University of California, Santa Cruz

I will talk about the normalized Ricci flow on Riemannian manifolds with certain initial conditions and the stability results we can get using this method. For example, when the initial metric is a slight perturbation of the hyperbolic metric on H^n, if the perturbation is small and decays sufficiently fast at the infinity, then the normalized Ricci flow will converge exponentially fast to the hyperbolic metric.

November 12, 2015

Asymptotic formulas of determinants of Toeplitz matrices and Gaussian fluctuations of local statistics

Zheng Zhou, University of California, Santa Cruz

Toeplitz operators are of importance in connection with problems in physics and probability theory. I will give a brief introduction about the Toeplitz operators and the asymptotic formula of its determinants. Also, a point of view to the Gaussian fluctuations of local statistics on some certain matrix ensembles will be given based on the cumulant method and operator-theoretic method.

Wednesday, November 18, 2015 *Please note the date*

Kahler-Einstein metrics along the smooth continuity method.

Ved Datar, University of California, Berkeley

I will discuss some joint work with Gabor Szekelyhidi on an equivariant version of the Yau-Tian-Donaldson conjecture strengthening the result of Chen-Donaldson-Sun. Together with the recent work of Ilten-Suss, this yields new examples of Kahler-Einstein manifolds.

November 26, 2015 - Thanksgiving Holiday

No Seminar

December 3, 2015

Nodal degeneration of hyperbolic metrics and asymptotics of the Weil-Petersson metric on the moduli space

Xuwen Zhu, Stanford University

This is joint work with Richard Melrose. We analyze the behavior of the Laplacian on the fibres of a Lefschetz fibration and use it to describe the behavior of the constant curvature metric on a Riemann surface of genus >1 undergoing nodal degeneration.  In particular this applies to the universal curve over moduli space. The description of the regularity of the fibre hyperbolic metrics, up to the divisors forming the `boundary' of the Knudsen-Deligne-Mumford compactification of moduli space $\mathcal{M}_{g,n}$, implies boundary regularity for the Weil-Petersson metric.