Caltech Analysis Seminar 2018-2019



Friday, October 12th, 2018
3-4pm, Linde 255
speaker: Christian Zillinger (USC)
title: Stabilization by mixing: On linear damping for the 2D Euler equations
abstract: In recent years, following the seminal works of Villani and Mouhot on Landau damping, phase-mixing as a damping mechanism and inviscid damping in fluids have attracted much interest. In this talk, I will provide an introduction to the underlying mechanism and discuss new linear stability and damping results near Taylor-Couette flow between concentric cylinders. This is based on joint work with Michele Coti Zelati at Imperial College, London.


Friday, October 26th, 2018
3-4pm, Linde 255
speaker: Dmitri Gekhtman (Caltech)
title: Holomorphic retracts of Teichmuller space
abstract: The Teichmuller space of a closed surface carries a natural complex structure, whose analytic properties reflect the topology and geometry of the surface. In this talk, we discuss the problem of classifying the holomorphic retracts of Teichmuller space. Our approach hinges on the analysis of two dynamical flows - one in the moduli space of half-translation surfaces, and the other in the space of bounded holomorphic functions on the polydisk.


Friday, November 9th, 2018
3-4pm, Linde 255
speaker: Franca Hoffmann (Caltech)
title: Nonlinear diffusion meets nonlocal interaction
abstract: We study interacting particles behaving according to a reaction-diffusion equation with nonlinear diffusion and nonlocal attractive interaction. This class of partial differential equations is a generalization of the Patlak-Keller-Segel model for bacterial chemotaxis, and has a nice gradient flow structure with respect to the Wasserstein-2 distance that allows us to make links to variations of well-known functional inequalities. Depending on the nonlinearity of the diffusion, the choice of interaction potential and the space dimensionality, we obtain different regimes. This talk will give an overview of recent advances in the fair-competition regime, when attractive and repulsive forces are in balance, as well as the diffusion-dominated and attraction-dominated regimes.


Friday, December 7th, 2018
3-4pm, Linde 255
speaker: Eden Prywes (UCLA)
title: A Bound on the Cohomology of Quasiregularly Elliptic Manifolds
abstract: A classical result gives that if there exists a holomorphic mapping $f\colon \mathbb C \to M$, then $M$ is homeomorphic to $S^2$ or $S^1\times S^1$, where $M$ is a compact Riemann surface. I will discuss a generalization of this problem to higher dimensions. I will show that if $M$ is an $n$-dimensional, closed, connected, orientable Riemannian manifold that admits a quasiregular mapping from $\mathbb R^n$, then the dimension of the degree $l$ de Rham cohomology of $M$ is bounded above by $\binom{n}{l}$. This is a sharp upper bound that proves a conjecture by Bonk and Heinonen. A corollary of this theorem answers an open problem posed by Gromov. He asked whether there exists a simply connected manifold that does not admit a quasiregular map from $\mathbb R^n$. The result gives an affirmative answer to this question.


Friday, January 18th, 2019
3-4pm, Linde 255
speaker: Sylvester Eriksson-Bique (UCLA)
title: Analysis of Carpets
abstract: I will discuss some recent work with J. Gong on large classes of ``fat'' carpets that possess Poincare inequalities. These generalize the examples of Tyson, Mackay and Wildrick. I will also discuss some related ideas and questions on ``thin'' carpets, such as the standard Sierpinski carpet. In addition to designing beautiful carpets for your living room, perhaps this talk also will give a sense of how natural objects carpets are and how they arise in different areas of geometry and topology.


Friday, February 1st, 2019
3-4pm, Linde 255
speaker: Haakan Hedenmalm (KTH)
title: Planar orthogonal polynomials and boundary universality in the random normal matrix model
abstract: We obtain a new asymptotic expansion of the orthogonal polynomials in the context of exponentially varying weights. This goes beyond the classical works of Carleman and Suetin, where the domain was fixed and the weight as well ("hard-edge"). Here, the domain is obtained implicitly from the problem using an energy approach, or alternatively from an obstacle problem. As a consequence, we obtain the boundary universality law for soft edges, given by the error function. This reports on joint work with A. Wennman.


Friday, February 15th, 2019
3-4pm, Linde 255
speaker: Zair Ibragimov (Cal State Fullerton)
title: Transfinite $\zeta$-metrics
abstract: I will discuss the concept of transfinite zeta-metrics. In some details I will discuss two specific metrics, namely the Apollonian and Cassinian metrics. I will discuss examples of domains where the transfinite Apollonian metric can be computed explicitly. This is a preliminary work.


Friday, March 1st, 2019
3-4pm, Linde 255
speaker: Oleksiy Klurman (KTH)
title: Multiplicative functions over \mathbb{F}_q[x] and the Erd\H{o}s Discrepancy problem.
abstract: We discuss analog of several classical results about mean values of multiplicative functions over \mathbb{F}_q[x] explaining some features that are not present in the number field setting. In the first part of the talk, which is based on the joint work with C. Pohoata and K. Soundararajan, we describe spectrum of multiplicative functions over \mathbb{F}_q[x]. In the second part of the talk (based on the joint work with A. Mangerel and J. Teravainen), we will focus on the "corrected" function field analog of the Erd\H{o}s Discrepancy Problem.


Friday, March 15th, 2019
3-4pm, Linde 255
speaker: Alexei Poltoratski (Texas A&M)
title: Inverse spectral problems for canonical Hamiltonian systems.
abstract: I will discuss recent progress in the area of inverse spectral problems for second order linear differential equations based on the use of truncated Toeplitz operators. The talk is based on joint work with N. Makarov


Friday, April 12th, 2019
3-4pm, Linde 255
speaker: Lenka Slavikova (Missouri)
title: A boundedness criterion for bilinear Fourier multipliers
abstract: We will present a sharp criterion for $L^2 \times L^2 \to L^1$ boundedness of bilinear operators associated with multipliers with $L^\infty$ derivatives, given in terms of $L^q$ integrability of the multiplier. More precisely, we will show that boundedness holds if and only if $q<4$. We will then discuss applications of this result concerning bilinear rough singular integrals. This is a joint work with L. Grafakos and D. He.


Friday, April 26th, 2019
3-4pm, Linde 255
speaker: Oleg Ivrii (Caltech)
title: Homogenization of random quasiconformal mappings and random Delauney triangulations
abstract: In this talk, I show that a random quasiconformal mapping is close to an affine mapping, while a circle packing of a random Delauney triangulation is close to a conformal map. This is joint work with Vlad Markovic.


Friday, May 10th, 2019
3-4pm, Linde 255
speaker: Sara Lapan (UC Riverside)
title: Local dynamics of maps tangent to the identity
abstract: In this talk, I will consider holomorphic self-maps of C^2 that fix the origin and are tangent to the identity (e.g., f(0) = 0 and df(0) = Id). An interesting topic to explore for such maps is how points near the origin move under iteration. Do they converge to the origin? If so, do they converge along a direction? I will discuss some results on the behavior of points near 0 under iteration by such a map. I will then focus on recent results of mine about what happens in a full neighborhood of the origin for a specific (degree two) map tangent to the identity. In addition, I will show how adding higher degree terms to this map will sometimes enable a domain of points to be attracted to the origin under iteration.


Friday, May 24th, 2019
3-4pm, Linde 255
speaker: Nam-Gyu Kang (KIAS)
title: Conformal field theory on the Riemann sphere and its boundary version for SLE
abstract: After presenting a gentle introduction to Schramm-Loewner evolution(SLE), I will explain how to construct a family of martingale-observables for chordal/radial SLE (with force points) from conformal field theory on the Riemann sphere and its boundary version. This is based on joint work with Nikolai Makarov.


Friday, June 7th, 2019
3-4pm, Linde 255
speaker: Mariusz Urbański (University of North Texas)
title: Dimension Spectrum of Conformal Iterated Function Systems
abstract: I will define conformal iterated function systems $S$ over a countable alphabet $E$ and their limit sets (attractors) $J_E$. I will discuss the formula for the Hausdorff dimension of this limit set, commonly referred to as a version of Bowen's formula, involving topological pressure. The main focus will be on the set $$ Sp(E)=\{HD(J_F): F\subset E\}, $$ called the dimension spectrum of the systenm $S$. I will prove that always $$ Sp(E)\supset (0,\theta_E), $$ where $theta_E$ is the finiteness parameter of $S$ (will be defined). I will also construct a system for which $Sp(E)$ is a proper subset of $(0,HD(J_E)]$. I will then discuss the property that $$ Sp(S)=(0,HD(J_E)], $$ called the full spectrum dimension property. In particular, I will discuss the conformal iterated function systems and their various subsystems, generated by real and complex continued fraction algorithms, and will show that many of them (subsystems) enjoy the full spectrum dimension property.