UC San Diego Geometry Seminar 2018-2019

Unless otherwise noted, all seminars are on Wed 2:00 - 2:50 pm in Room APM 5829

Fall 2018 Schedule

September 26th

Mike Eastwood, University of Adelaide
Title:How to recognise the geodesics of a metric
Abstract: A Riemannian metric gives rise to geodesics. As unparameterised curves, at each point there is one geodesic in every direction. Does this arrangement of curves determine the metric and will any such arrangement of curves determine a metric? These are classical questions considered, for example, by Roger Liouville in 1889 who made considerable progress in two dimensions. This talk will discuss these questions in two and three dimensions, mostly based on joint work with Robert Bryant and Maciej Dunajski.

October 9, 2018 (special date due to joint UCI-UCR-UCSD Southern California Differential Geometry Seminar, takes place at UCI)

Guangbo Xu, SUNY Stony Brook
Title: Bershadsky--Cecotti--Ooguri--Vafa torsion in Landau--Ginzburg models
Abstract: In the celebrated work of Bershadsky--Cecotti--Ooguri--Vafa the genus one string partition function in the B-model is identified with certain analytic torsion of the Hodge Laplacian on a K\"ahler manifold. In a joint work with Shu Shen (IMJ-PRG) and Jianqing Yu (USTC) we study the analogous torsion in Landau--Ginzburg models. I will explain the corresponding index theorem based on the asymptotic expansion of the heat kernel of the Schr\"odinger operator. I will also explain the rigorous definition of the BCOV torsion for homogeneous polynomials on ${\mathbb C}^N$. Lastly I will explain the conjecture stating that in the Calabi--Yau case the BCOV torsion solves the holomorphic anomaly equation for marginal deformations.
Yi Wang, Johns Hopkins
Title: Nonuniqueness for a fully nonlinear boundary Yamabe-type problem via bifurcation theory
Abstract: We consider $\sigma_k$-curvature equation with $H_k$-curvature condition on a compact manifold with boundary $(X^{n+1}, M^n, g)$. When restricting to the closure of the positive $k$-cone, this is a fully nonlinear elliptic equation with a fully nonlinear Robin-type boundary condition. We prove a general bifurcation theorem in order to study nonuniqueness of solutions when 2k is less than n. We explicitly give examples of product manifolds with multiple solutions. It is analogous to Schoen example for Yamabe problem on $S^1\times S^{n-1}$. This is joint work with Jeffrey Case and Ana Claudia Moreira.

October 17

Yongjia Zhang, UCSD
Title: On the equivalence between bounded entropy and noncollapsing for ancient solutions to the Ricci flow.
Abstract: At the beginning of section 11 in Perelman's celebrated paper "The entropy formula for the Ricci flow and its geometric applications", he made the assertion that for an ancient solution to the Ricci flow with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. We give a proof for this assertion.

October 24, 2018

Dmitri. Burago, Penn State
Title: Three scary math tales (see abstract).
Abstract: Small KAM perturbations of integrable systems which are entropy expansive. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that after a small perturbation of a non-degenerate completely integrable system it still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori remains a great mystery. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We are now able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations, answering an old-standing problem of Kolmogorov. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. In these modified examples positive metric entropy is generated in arbitrarily small tubular neighborhoods of one trajectory. Joint with S. Ivanov and Dong Chen.
Metric approximations of length spaces by graphs with uniformly bounded local structure. How well can we approximate an (unbounded) space by a metric graph whose parameters (degrees of vertices, lengths of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers are given (the most difficult case is $\R^2$) using dynamics and Fourier series. Joint with S. Ivanov.
On Busemann's problem on minimality of flats in normed spaces for the Buseman-Hausdorff surface area. Busemann asked if regions in affine subspaces of normed spaces are area minimizers with respect to the Busemann-Hausdorff measure. This has been known for long for hyperplanes (codim=1), this is a classic result in Convex Geometry. Sergei Ivanov and me were able to prove this for 2-dimensional subspaces.

October 24, 2018, Colloquium 4-5, APM 7421

Dmitri. Burago, Penn State
Title:Two fairy math tales (see abstract).
Abstract: Counting collisions. 20 years ago the topic of my talk at the ICM was a solution of a problem which goes back to Boltzmann and Ya. Sinai. The conjecture of Boltzmann-Sinai states that the number of collisions in a system of $n$ identical balls colliding elastically in empty space is uniformly bounded for all initial positions and velocities of the balls. The answer is affirmative and the proven upper bound is exponential in $n$. The question is how many collisions can actually occur. On the line, there can be $n(n-1)/2$ collisions, and this is he maximum. Since the line embeds in any Euclidean space, the same example works in all dimensions. The only non-trivial (and counter-intuitive) example I am aware of is an observation by Thurston and Sandri who gave an example of 4 collisions between 3 balls in $R^2$. Recently, Sergei Ivanov and me proved that there are examples with exponentially many collisions between $n$ identical balls in $R^3$, even though the exponents in the lower and upper bounds do not match.
A survival guide for a feeble fish and homogenization of the G-Equation. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water? This is related to G-Equation and has applications to its homogenization. G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.

November 1, 2018 (Colloquium)

T. Colding, MIT
Title: Singularities and dynamics of flows
Abstract: Parabolic flows are smoothing for short time however, over long time, singularities are typically unavoidable and can be very nasty. The key to understand such flows is to understand their singularities and the set where those singularities occur. We begin with discussing mean curvature flow and will explain which singularities are generic and what one can say about the short and long time dynamics near singularities. After that we turn to the question of optimal regularity of geometric flows in general. We will see that these seemingly different questions turn out to be related. The ideas draws inspiration from an number of different fields, including Geometry, Analysis, Dynamical Systems and Real Algebraic Geometry.


November 7, 2018

Lei Ni, UCSD
Title:Two curvature notions on K\"ahler manifolds and some questions
Abstract: Here I shall introduce the two curvatures and discuss their relations with existing ones and their implications, including comparison theorems, vanishing theorems, projective embeddings and hyperbolicity. If time permits, I shall also discuss some open problems related.

November 14, 2018

Sean Curry, UCSD
Title: Strictly pseudoconvex domains in C^2 with obstruction flat boundary
Abstract: A bounded strictly pseudoconvex domain in C^n, n>1, supports a unique complete Kahler-Einstein metric determined by the Cheng-Yau solution of Fefferman's Monge-Ampere equation. The smoothness of the solution of Fefferman's equation up to the boundary is obstructed by a local CR invariant of the boundary called the obstruction density. In the case n=2 the obstruction density is especially important, in particular in describing the logarithmic singularity of the Bergman kernel. For domains in C^2 diffeomorphic to the ball, we motivate and consider the problem of determining whether the global vanishing of this obstruction implies biholomorphic equivalence to the unit ball. (This is a strong form of the Ramadanov Conjecture.)

November 20, 2018; 3-4pm at APM 6402 (Special date, time and place)

Teng Fei, Columbia
Title: Hull-Strominger system and Anomaly flow over Riemann surfaces
Abstract:The Hull-Strominger system is a system of nonlinear PDEs describing the geometry of compactification of heterotic strings with torsion to 4d Minkowski spacetime, which can be regarded as a generalization of Ricci-flat Kähler metrics coupled with Hermitian Yang-Mills equation on non-Kähler Calabi-Yau 3-folds. The Anomaly flow is a parabolic approach to understand the Hull-Strominger system initiated by Phong-Picard-Zhang. We show that in the setting of generalized Calabi-Gray manifolds, the Hull-Strominger system and the Anomaly flow reduce to interesting elliptic and parabolic equations on Riemann surfaces. By solving these equations, we obtain solutions to the Hull-Strominger system on a class of compact non-Kähler Calabi-Yau 3-folds with infinitely many topological types and sets of Hodge numbers. This talk is based on joint work with Zhijie Huang and Sebastien Picard.

November 28, 2018; 4-5pm at APM 6402 (Special date, time and place)

Luca Spoloar, MIT
Title: Singularities for the Plateau Problem
Abstract:In this talk I will introduce two different notions of solutions to the Plateau Problem, called Area and Size minimizers, due respectively to Federer-Fleming and Almgren. The fundamental difference between them is wether multiplicity/orientation plays a role or not, and they were originated respectively to describe integral homology class and soap films. I will then explain how different types of singularities arise in both formulation and some recent progress made on the structure of the singular set and of minimizers near singularities. If time permits I will also explain some possible future developments.

November 29, 2018; 2-3pm at APM 7218 (Special date, time and place)

Luca Spoloar, MIT
Title: (Log) -Epiperimetric Inequality and the Regularity of Free-Boundaries
Abstract:In this talk I will present a new method for studying the regularity of minimizers of some variational problems, including in particular some classical free-boundary problems. Using as a model case the so-called Obstacle problem, I will explain what regularity of the free-boundary means and how we obtain it by using a new tool, called (Log) -epiperimetric inequality. This technique is very general, and much like Caffarelli's 'improvement of flatness' for regular points, it allows for a uniform treatment of singularities in many different free-boundary problems. Moreover it is able to deal with logarithmic regularity, which in the case of the Obstacle problem is optimal due to an example of Figalli-Serra. If time permits I will explain how such an inequality is linked to the behavior of a gradient flow at infinity.

November 29, 2018 (Special Colloquium)

Otis Chodosh, Princeton University
Title:The multiplicity one conjecture on 3-manifolds
Abstract: Minimal surfaces are critical points of the area functional on the space of surfaces. Thus, it is natural to try to construct them via Morse theory. However, there is a serious issue when carrying this out, namely the occurrence of "multiplicity." I will explain this issue and recent joint work with C. Mantoulidis ruling this out for generic metrics.

November 30, 2018 (Special place: APM 6402; 2-3pm)

Otis Chodosh, Princeton University
Title: A splitting theorem for scalar curvature
Abstract: I'll discuss joint work with M. Eichmair and V. Moraru in which we prove a natural minimal surface analogue of the splitting theorem for 3-manifolds with non-negative scalar curvature.

December 5, 2018


Winter 2018 Schedule

Questions: lni@math.ucsd.edu