UCSD Math Department Colloqium, 2013 - 2014.

### 2013 - 2014 UCSD Department of Mathematics Colloquium

Thursdays, 4:00 pm - 5:00 pm, AP&M 6402 (Unless Otherwise Stated)

Coordinator: Bo Li (Email: bli@math.ucsd.edu)

### Host

April 3, 2014
Special place:
AP&M 2402
Chris Bishop (SUNY Stony Brook) Title. Conformal Maps and Optimal Meshes

Abstract. The Riemann mapping theorem says that every simply connected proper plane domain can be conformally mapped to the unit disk. This result is over a 100 years old, but the study and computation of such maps is still an active area. In this talk, I will discuss the computational complexity of constructing a conformal map from the disk to an n-gon and show that it is linear in n, with a constant that depends only on the desired accuracy. As one might expect, the proof uses ideas from complex analysis, quasiconformal mappings and numerical analysis, but I will focus mostly on the surprising roles played by computational planar geometry and 3-dimensional hyperbolic geometry.
If time permits, I will discuss how this conformal mapping algorithm implies new results in discrete geometry, e.g., every simple polygon can be meshed in linear time using quadrilaterals with all new angles between 60 and 120 degrees. A closely related result states that any planar triangulation of n points can be refined by adding vertices and edges into a non-obtuse triangulation (no angles bigger than 90 degrees) in time O(n^(5/2)). No polynomial bound was previously known.

Peter Ebenfelt
Special time:
2:00 - 3:00,
Thursday, May 8
Vera Serganova
(UC Berkeley)
Title. Finite-dimensional representations of classical algebraic supergroups

Abstract. Studying Lie superalgebras and supergroups was initially motivated by applications in physics. In the recent years interesting connections with other branches of mathematics were discovered. The goal of the talk is to review some of these results.
I start with describing four series of algebraic supergroups, which are natural generalizations of general linear, orthogonal and symplectic groups. We shall see different superanalogues of Schur-- Weyl duality, which reveal connections with universal tensor categories constructedby Deligne. Then we discuss geometric methods in representation theory of algebraic supergroups: associated variety and Borel-Weil-Bott theory. Finally, I will talk about categorification and weight diagram technique and try to explain how they can be used for calculating the characters of irreducible representations of classical supergroups.

Efim Zelmanov
May 8 Burt Totaro
(UCLA)
Title. The fundamental group of an algebraic variety, and hyperbolic complex manifolds

Abstract. It is a mystery which groups can occur as fundamental groups of smooth complex projective varieties. It is conceivable that whenever the fundamental group is infinite, the variety has some "negative curvature" properties. We discuss a result in this direction, in terms of "symmetric differentials". There are interesting open questions even about the special case of compact quotients of the unit ball in C^n. (Joint work with Yohan Brunebarbe and Bruno Klingler.)

James McKernan
Special time:
4:00 - 5:00,
Friday, May 23
Ya-Xiang Yuan
Title.

Abstract.

Jiawang Nie
June 5 Robert Krasny
(Univ. Michigan)
Title. Lagrangian Particle Methods for Vortex Dynamics

Abstract. In this talk I'll discuss how Lagrangian particle methods are being used to study the dynamics of fluid vortices. These methods use the Biot-Savart integral to recover the velocity from the vorticity and they track the flow map using adaptive particle discretizations. I'll present computations of vortex sheet motion in 2D flow, with reference to Kelvin-Helmholtz instability, the Moore singularity, spiral roll-up, and chaotic dynamics. Other examples include vortex rings in 3D flow, and vortex dynamics on a rotating sphere.

Bo Li

### Host

January 9 Andrew Blumberg
(UT Austin)
Title. Algebraic K-theory and the geometry of module categories

Abstract. Algebraic K-theory is a deep and subtle invariant of rings and schemes, carrying information about arithmetic and geometry. When applied to the group ring of the loop space of a manifold, it captures information about the diffeomorphism group. Over the past 25 years, the study of algebraic K-theory has been revolutionized by the introduction of trace methods, which use trace (or Chern character) maps to the simpler but related theories of (topological) cyclic and Hochschild homology.
A unifying perspective on the properties of algebraic K-theory and these related theories is afforded by viewing the input as a category of compact modules (i.e., a piece of an enhanced triangulated category). This talk will survey recent work describing the structural properties of these theories using various models of the homotopical category of module categories.

Dragos Oprea
Special time
2:00 - 3:00,
Friday, January 10
Yaniv Plan
Title. Low-dimensionality in mathematical signal processing

Abstract. Natural images tend to be compressible, i.e., the amount of information needed to encode an image is small. This conciseness of information -- in other words, low dimensionality of the signal -- is found throughout a plethora of applications ranging from MRI to quantum state tomography. It is natural to ask: can the number of measurements needed to determine a signal be comparable with the information content? We explore this question under modern models of low-dimensionality and measurement acquisition.

Ery Arias-Castro
January 23 Jacob Bedrossian
(NYU)
Title. Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations

Abstract. We prove asymptotic stability of shear flows close to the planar, periodic Couette flow in the 2D incompressible Euler equations. That is, given an initial perturbation of the Couette flow small in a suitable regularity class, specifically Gevrey space of class smaller than 2, the velocity converges strongly in L2 to a shear flow which is also close to the Couette flow. The vorticity is asymptotically mixed to small scales by an almost linear evolution and in general enstrophy is lost in the weak limit. Joint work with Nader Masmoudi. The strong convergence of the velocity field is sometimes referred to as inviscid damping, due to the relationship with Landau damping in the Vlasov equations. Recent work with Nader Masmoudi and Clement Mouhot on Landau damping may also be discussed.

Bo Li
January 30 Rayan Saab
(UCSD)
Title. A compressive sampling of compressed sensing and related areas

Abstract. Compressed sensing is a signal acquisition paradigm that utilizes the sparsity of a signal (a vector in R^N with s << N non-zero entries ) to efficiently reconstruct it from very few (say n, s < n << N) generalized linear measurements. These measurements often take the form of inner products with random vectors drawn from appropriate distributions, and the reconstruction is typically done using convex optimization algorithms or computationally efficient greedy algorithms.
In this talk we cover some recent results in compressed sensing and related areas where the signal acquisition is also done via such generalized linear measurements. We discuss signal recovery from compressed sensing measurements using weighted $\ell_1$ minimization, when erroneous support information is available. We provide reconstruction guarantees that hold with high probability and improve on the standard results, provided the support information is accurate enough. TIme permitting, we also discuss some recent results on the digitization of generalized linear measurements, including compressed sensing measurements.

Peter Ebenfeld
February 6

(Columbia Univ.)

#### CANCELLED

Title. Ancient solutions to geometric flows

Abstract. We will discuss ancient solutions to non-linear parabolic equations, such as the semi-linear heat equation, the Ricci flow on surfaces, the Yamabe flow and the mean curvature flow. We will address the problem of classification of ancient solutions, the existence and classification of solitons as well as the construction or other ancient non-soliton solutions.

#### CANCELLED

Lei Ni
February 13 Claus Sorensen
(UCSD)
Title. A gentle introduction to local Langlands in families

Abstract. This talk will be geared towards a general audience. Assuming only familiarity with p-adic numbers, we will first discuss the local Langlands correspondence for GL(n), and explain why it's a wide-ranging generalization of local class field theory. Then we will follow Emerton and Helm in trying to extrapolate it to a correspondence between representations over more general local rings -- whereupon we reformulate the conjectural Ihara lemma in this language. The latter is a big open problem for n>2, which occurred in work of Clozel, Harris, and Taylor, in their attempt to mimic the proof of Fermat's Last Theorem in the context of GL(n).

Peter Ebenfelt
Joint Biostat/Math Colloq.
3:30 Friday, Feb. 21
Don Rubin
(Harbard)
Title.

Abstract.

Dimitris Politis
March 13 Gunther Cornelissen
(Utrecht and Caltech)
Title. Graph spectra and diophantine equations

Abstract. I will show how to find uniform finiteness results for certain diophantine equations in terms of the Laplace spectrum of an associated graph. The method is to bound the "gonality" of a curve (minimal degree of a map onto a line) by the "stable gonality" of an associated stable reduction graph, and then to bound this stable gonality of the graph (some kind of minimal degree of a map to a tree) in terms of spectral data. The latter bound is a graph theoretical analogue of a famous inequality of Li and Yau in differential geometry. An example of an application is to bound the degree of the modular parametrisation of elliptic curves over function fields. (Joint work with Fumiharu Kato and Janne Kool.)

Cristian Popescu

### Host

October 24 Todd Kemp
(UCSD)
Title. Brownian Motion on Lie Groups: Limits and Fluctuations in High Dimension

Abstract. Brownian motion is continuous random motion, discovered by early 19th Century botanist Robert Brown, studied by Albert Einstein in one of the three 1905 papers that led to his Nobel prize, and finally put on firm mathematical footing by Norbert Wiener in the 1920s. It is intimately tied to local and global geometry, and is an important tool in studying heat flow on more general manifolds.
In this talk, I will give an overview of some results on Brownian motion on classical Lie groups, focusing on unitary groups U_N and general linear groups GL_N. I will discuss my recent work on the large-N limit of Brownian motions on these groups, their fluctuations, and applications to random matrix theory and operator algebras.

Bruce Driver
October 31 Jesse Peterson
(Vanderbilt Univ.)
Title. Character rigidity for lattices in higher rank groups

Abstract. A character on a group is a class function of positive type. For finite groups, the classification of characters is directly connected to the representation theory of the group and plays a key role in the classification of finite simple groups. Based on the rigidity results of Mostow, Margulis, and Zimmer, it was conjectured by Connes that for lattices in higher rank Lie groups the space of characters should be completely determined by the finite dimensional representations of the lattice. In this talk, I will give an introduction to this conjecture (which has now been solved in a number of cases), and I will discuss its relationship to ergodic theory, abstract harmonic analysis, invariant random subgroups, and von Neumann algebras.

November 7 James McKernan
(UCSD)
Title. Boundedness of Fano varieties

Abstract. Fano varieties are in some sense the simplest type of algebraic varieties. They are the algebraic analogue of manifolds with positive curvature, such as spheres. In low dimensions one can classify Fano varieties (where for an algebraic geometer low means up to dimension three) and as the dimension increases, they form bounded families, so that one can in principle classify Fano varieties in all dimensions.
In this talk we explain some of the known and conjectured results, both for an explicit classification, and for some of the boundedness results.

Peter Ebenfeld
(UCSD)
Title. Hodge theory and abelian varieties

Abstract. I will give a brief introduction to Hodge theory and discuss Hodge-theoretic problems involving abelian varieties.

Peter Ebenfeld
November 21 Herbert Heyer
(Univ. Tuebingen, Germany)
Title. Limit theorems for probability measures on convolution structures of growing dimension

Abstract. Some central limit results on stochastic processes in a compact connected 2-point homogeneous space E(d) of growing dimension d are reformulated within the theory of polynomial convolution structures. This approach stresses the algebraic-topological relationship between those structures and the asymptotic properties of the stochastic processes under consideration, in particular of random walks and Gaussian processes on E(d) with d \to \infty.

Pat Fitzsimmons
3:00 - 4:00
(special time)

November 25
Elena Fuchs
(UC Berkeley)
Title. Thin groups: arithmetic and beyond

Abstract. In 1643, Ren\'{e} Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called \emph{thin} subgroup $\Gamma$ of $\textrm{GL}_4(\mathbb Z)$. This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs associated to $\Gamma$ -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.

Alina Bucur and Kiran Kedlaya
3:00 - 4:00
(special time)

December 5
James Demmel
(UC Berkeley)
Title. Communication Avoiding Algorithms for Linear Algebra and Beyond

Abstract. Algorithms have two costs: arithmetic and communication, i.e. moving data between levels of a memory hierarchy or processors over a network. Communication costs (measured in time or energy per operation) already greatly exceed arithmetic costs, and the gap is growing over time following technological trends. Thus our goal is to design algorithms that minimize communication. We present algorithms that attain provable lower bounds on communication, and show large speedups compared to their conventional counterparts. These algorithms are for direct and iterative linear algebra, for dense and sparse matrices, as well as direct n-body simulations. Several of these algorithms exhibit perfect strong scaling, in both time and energy: run time (resp. energy) for a fixed problem size drops proportionally to the number of processors p (resp. is independent of p). Finally, we describe extensions to algorithms involving arbitrary loop nests and array accesses, assuming only that array subscripts are affine functions of the loop indices.

Jiawang Nie
4:00 - 5:00
December 5
Lu Wang
(Johns Hopkins Univ.)
Title. Rigidity of Self-shrinkers of Mean Curvature Flow

Abstract. The study of mean curvature flow not only is fundamental in geometry, topology and analysis, but also has important applications in applied mathematics, for instance, image processing. One of the most important problems in mean curvature flow is to understand the possible singularities of the flow and self-shrinkers, i.e., self-shrinking solutions of the flow, provide the singularity models.
In this talk, I will describe the rigidity of asymptotic structures of self-shrinkers. First, I show the uniqueness of properly embedded self-shrinkers asymptotic to any given regular cone. Next, I give a partial affirmative answer to a conjecture of Ilmanen under an infinite order asymptotic assumption, which asserts that the only two-dimensional properly embedded self-shrinker asymptotic to a cylinder along some end is itself the cylinder. The feature of our results is that no completeness of self-shrinkers is required.
The key ingredients in the proof are a novel reduction of unique continuation for elliptic operators to backwards uniqueness for parabolic operators and the Carleman type techniques. If time permits, I will discuss some applications of our approach to shrinking solitons of Ricci flow.

Ben Chow and Lei Ni

Last updated by Bo Li on March 14, 2014.