Mathematics Department - Topology/Geometry Seminar - Spring 2017

# Topology/Geometry Seminar - Spring 2017

### Organizer(s)

Steven Ferry, Feng Luo, Konstantin Mischaikow, Xiaochun Rong

### Website

http://www.math.rutgers.edu/~fluo/seminars17Spring.html

## Tuesday, April 25th

### "Embeddedness of the solutions to the H-Plateau Problem"

Time: 3:30 PM
Location: Hill 005
Abstract: In this talk, we will give a generalization of Meeks and Yau’s proof of Dehn’s Lemma for area minimizing disks, and prove an embeddedness result for constant mean curvature disks in 3-manifolds. arXiv:1504.00661

## Tuesday, April 18th

### "An introduction to semitoric integrable systems and the semitoric helix"

Time: 3:30 PM
Location: Hill 005
Abstract: Semitoric integrable systems, which generalize toric integrable systems in dimension four, were recently classified in terms of five invariants by Pelayo-V~{u} Ngd{o}c. In this talk we introduce semitoric integrable systems and give an overview of some recent work in the field. In particular we focus on the construction and interpretation of a new additional invariant, the semitoric helix, and explain how algebraic techniques and the semitoric helix can be used to construct the minimal models of semitoric systems. No knowledge of integrable systems is assumed.

Portions of the work presented are joint with Alvaro Pelayo, Daniel Kane, and Alessio Figalli.

## Tuesday, April 11th

Special Topology/Geometry Seminar

### "Pluri-lagrangian structure as integrability of variational systems"

Time: 4:50 PM
Location: Hill 525
Abstract: We propose a notion of a pluri-Lagrangian problem, which should be understood as an analog of multi-dimensional consistency for variational systems. This is a development along the line of research of discrete integrable Lagrangian systems initiated in 2009 by Lobb and Nijhoff, however having its more remote roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics and their quasiclassical limit, as well as in the theory of variational symmetries going back to Noether. We discuss main features of pluri-Lagrangian systems in dimensions 1 and 2, both continuous and discrete, along with the relations of this novel structure to more standard notions of integrability.

## Tuesday, April 11th

### " Boundary expansion for minimal graphs in the hyperbolic space"

Time: 3:30 PM
Location: Hill 005
Abstract: We develop the ODE iteration to study the boundary behavior of the solutions to Minimal graphs in the hyperbolic space. We will study how local and global geometry of the asymptotic boundary affect the regularity of the minimal graphs. Furthermore, we will discuss the convergence of the expansion. Other models of degenerate elliptic PDEs in geometry will be compared.

## Tuesday, April 4th

### "INTERIOR CURVATURE ESTIMATES AND THE ASYMPTOTIC PLATEAU PROBLEM IN HYPERBOLIC SPACE"

Time: 3:30 PM
Location: Hill 005
Abstract: In this talk, we will show for a very general class of curvature functions defined in the positive cone, there exists complete strictly locally convex hypersurfaces in $mathbb{H}^{n+1}$ with constant curvature. We will also show that, under certain assumptions, this constant general curvature hypersurface in hyperbolic space is unique. If we have time, we will show a strong duality theorem of hyperbolic space and De Sitter space.

## Wednesday, March 29th

### "Quotients of surface groups via TQFT"

Time: 3:30 PM
Location: Hill 101
Abstract: I will show how to construct linear quotients of surface groups which are infinite, and where each simple loop has finite order. As an application, I will construct finite covers of surfaces where the pullbacks of simple loops fail to generate the integral homology. We thus answer questions due to Looijenga.

This represents joint work with R. Santharoubane.

## Tuesday, March 7th

### "Poincare inequality and Gaussian heat kernel estimate for non-negatively curved graphs"

Time: 3:30 PM
Location: Hill 005
Abstract: We derive that if a graph has non-negative curvature then it has the volume doubling property, from this we can prove the Gaussian estimate for heat kernel, and then Poincare inequality and Harnack inequality. Under the assumption of positive curvature on graphs, we derive the Bonnet-Myers type theorem that the diameter of graphs is finite and bounded above in terms of the positive curvature.

This is a joint work with Horn, Liu and Yau.

## Tuesday, February 28th

### "Analysis of Velázquez's solution to the mean curvature flow with a type II singularity"

Time: 3:30 PM
Location: Hill 005
Abstract: J.J.L. Velázquez in 1994 used the degree theory to show that there is a perturbation of Simons' cone, starting from which the mean curvature flow develops a type II singularity at the origin. He also showed that under a proper time-dependent rescaling of the solution around the origin, the rescaled flow converges in the C^0 sense to a minimal hypersurface which is tangent to Simons' cone at infinity. In this talk, we will present that the rescaled flow actually converges locally smoothly to the minimal hypersurface, which appears to be the singularity model of the type II singularity. In addition, we will show that the mean curvature of the solution blows up near the origin at a rate which is smaller than that of the second fundamental form.

This is a joint work with N. Sesum.

## Tuesday, February 21st

### "Minimal surfaces from deformations of circle patterns"

Time: 3:30 PM
Location: Hill 005
Abstract: William Thurston introduced circle packings to approximate holomorphic functions. Burt Rodin and Dennis Sullivan proved the convergence of the analogue of Riemann maps for circle packings. Oded Schramm further extended the idea by considering circle patterns, where circles are allowed to intercept with each other. We present a discrete analogue of the Weierstrass representation for minimal surfaces in terms of discrete holomorphic quadratic differentials. Given a triangle mesh in the plane, a circle pattern is induced by the circumscribed circles. We investigate infinitesimal deformations of the triangle mesh that preserve the intersection angles of the circumscribed circles. We then deduce discrete holomorphic quadratic differentials from the change in cross ratios, which yield polyhedral surfaces with vanishing mean curvature. A similar story is obtained if we replace the intersection angles by length cross ratios.

This is joint work with Ulrich Pinkall.

## Tuesday, February 14th

### "Free Seifert fibered pieces of pseudo-Anosov flows"

Time: 3:30 PM
Location: Hill 005
Abstract: We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In this talk we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. We introduce almost k-convergence groups, and an associated convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we define what is a hyperbolic blow up.

This is joint work with Thierry Barbot.

## Tuesday, February 7th

### "A Few Math Fairy Tales"

Time: 3:30 PM
Location: Hill 005
Abstract: The format of this talk is rather non-standard. It is actually a combination of several mini-talks. They would include only motivations, formulations, basic ideas of proof if feasible, and open problems. No technicalities. Each topic would be enough for 2+ lectures. However I know the hard way that in diverse audience, after 1/3 of allocated time 2/3 of people fall asleep or start playing with their tablets. I hope to switch to new topics at approximately right times. I include more topics that I plan to cover for I would be happy to discuss others after the talk or by email/skype. I may make short announcements on these extra topics. The topics will probably be chosen from the list below. I sure will not talk on topics I have spoken already at your university. “A survival guide for feeble fish”. 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 flow. This is related to homogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov. How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably “nice” mm–spaces. A notion of ho-Laplacian and its stability. Joint with S. Ivanov and Kurylev. One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodic dynamics. What happens outside KAM tori has been remaining a great mystery. The main quantative invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however, on a zero measure set. We were able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations. Furthermore, a slightly modified construction resolves another long–standing problem of the existence of entropy non-expansive systems. These modified examples do generate positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. The technology is based on a metric theory of “dual lens maps” developed by Ivanov and myself, which grew from the “what is inside” topic. Quite a few stories are left in my left pocket. Possibly: On making decisions under uncertain information, both because we do not know the result of our decisions and we have only probabilistic data.

## Tuesday, January 31st

### "Computing twisted harmonic maps"

Time: 3:30 PM
Location: 005 Hill Basement
Abstract: I will present an ongoing project to develop a computer software whose main purpose is to compute and investigate equivariant harmonic maps between hyperbolic surfaces or more general symmetric spaces. This is joint work with Jonah Gaster. Basic information and screenshots of this software can be found here:

http://math.newark.rutgers.edu/~bl498/software.html#hitchin

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