# Mathematics Colloquium Spring 2015

For further information please contact Professor Longzhi Lin or call the Mathematics Department at 459-2969

**March 31, 2015 *CANCELLED*
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**The Secondary Game: From Barbilian's Metrization Procedure to Gromov Hyperbolic Spaces****Bogdan Suceavă, California State University, Fullerton**

Introduced originally in 1934, Barbilian’s metrization procedure induced a distance on a planar domain by a metric formula given by the so-called logarithmic oscillation. The original idea was a generalization of the metric used in the Klein-Beltrami model for hyperbolic geometry. In 1959, Barbilian generalized this process to domains of a more general form, withstanding not necessarily on planar sets, but in a more abstract setting. We will show that there exists more general classes of distances than the ones produced by logarithmic oscillation. We present various extensions of Barbilian's metrization procedure and study several metrics defined by this process. We conclude our study by exploring the connection between these geometries and hyperbolicity in the sense of M. Gromov.

**April 7, 2015
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*Existence and uniqueness results for free boundary minimal surfaces*

**Ailana Fraser, University of British Columbia
** I will discuss joint work with R. Schoen on existence and uniqueness results for free boundary minimal surfaces in a Euclidean ball. These are proper branched minimal immersions of a surface into the ball which meet the boundary orthogonally. Such surfaces have been extensively studied and they arise as extremals of the area functional for relative cycles in the ball. They also arise as extremals of a certain eigenvalue problem.

**April 14, 2015
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*Analytic torsion and Borcherds modular forms*

**Xianzhe Dai, University of California, Santa Barbara
** The analytic torsion was introduced by Ray-Singer as an analytic analog for the Reidemeister torsion, the first topological invariant which is not a homotopy invariant (and thus can be used to distinguish homotopically equivalent structures). It is a combination of determinants of the Hodge Laplacian on differential forms of various degrees. Its complex analog, the holomorphic torsion, uses Dolbeault Laplacian on complex manifolds. Although there is no topological interpretation, the holomorphic torsion has found many remarkable applications, such as the Arithmetic Riemann-Roch theorem. In this talk we will focus instead on its fascinating connection with modular forms and report some recent work with Ken-ichi Yoshikawa.

** April 21, 2015
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**Moduli spaces of Geometric Structures****Given a topology S, how many ways (if any) are there of putting some kind of classical geometry on S? For example, the sphere has no compatible system of coordinates with Euclidean geometry. (There is no metrically accurate atlas of the world.) On the other hand, the 2-torus admits a rich supply of Euclidean structures, which form an interesting moduli space which itself enjoys hyperbolic non-Euclidean geometry.**

William Goldman, MSRI

William Goldman, MSRI

For other geometric structures, the moduli spaces are much more complicated and are best described by a dynamical system. This talk will survey some of the interesting dynamical systems which arise for simple examples of geometries on surfaces.

**April 28, 2015** ***SPECIAL COLLOQUIUM* McHenry Room 1240 - please note room change**

**Mathematics of the genome
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Steve Smale, University of Hong Kong and University of California, Berkeley

We will give some mathematical foundations for the dynamics of the genome with special attention to circadian rhythm.

**May 5****, 2015
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*The Deepest Interiors of Planets and UCSC's Interactions with NASA Ames: A Research and Administrative Tandem...*

*Quentin Williams, Earth & Planetary Sciences Department, University of California, Santa Cruz*

This will be a bifurcated talk: part research, on the internal structures of the solar system’s moons and terrestrial planets, and part institutional, with a bit of background on UCSC’s interactions with NASA, and prospective areas of possible engagement with math. On the research side, the topic of what’s inside planets has been of interest to natural scientists since (at least) Laplace’s seminal work on the topic. Indeed, the cores of moons and terrestrial planets typically reside at pressures between 2 x 10^{4} and 3 x 10^{6} bars, and temperatures of 1500-6000 K—extreme conditions that are experimentally attainable. The research side of this talk will cover experimental constraints on the compositions and dynamics of planetary interiors, and discuss inversions on the presence and size of the moon’s outer liquid and inner solid core, as derived from a reanalysis of seismic results from the stations deployed by the (long ago) Apollo missions, with some implications for the genesis (or non-genesis) of magnetic fields of moons and planets. On the institutional side, I will segue from planets to NASA, and mercilessly plug interactions with NASA Ames, with which UCSC has several long-standing relationships. While lots of applied work is done there, some areas of interaction with math may well exist, and I’ll try to describe those.

**May 12, 2015
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**Polynomial Pick forms for affine spheres and real projective polygons****Michael Wolf, Rice University** (Joint work with David Dumas.) Convex real projective structures on surfaces, corresponding to discrete surface group representations into SL(3, R), have associated to them affine spheres which project to the convex hull of their universal covers. Such an affine sphere is determined by its Pick (cubic) differential and an associated Blaschke metric. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found. Much of the talk will be a description of context and background; in particular, very few of the technical words in the previous sentences will be assumed.

**May 19, 2015
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*Families of polynomials appearing in the study of infinite dimensional Lie algebras.*Ben Cox, College of Charleston

Many types of polynomials arise naturally in the representation theory of Lie groups and Lie algebras. We will show how families of (nonclassical) orthogonal polynomials such as ultra spherical, Pollaczek, associated Legendre, associated Jacobi, and Cheybshev polynomials appear. Such polynomials arise when describing the universal central extension of particular families of Krichever-Novikov algebras and their automorphism groups. The associated Jacobi polynomials of Ismail and Wimp satisfy certain fourth order linear differential equations that also are related to the work of Kaneko and Zagier on supersingular j-invariants and Atkin’s polynomials. We will describe this family of differential equations. This is joint work with V. Futorny, J. Tirao, and R. Lu, X. Guo and K. Zhao.

**May 26, 2015**

**Isoperimetric inequalities for minimal submanifolds**

**Keomyko Seo, Sookmyung Women's University**

Isoperimetric inequality for a domain in a plane has been generalized in a various way. One natural extension is to consider minimal surfaces instead of a domain in the plane. We discuss isoperimetric inequalities for minimal surfaces and submanifolds. (If time permitted) We also discuss linear isoperimetric inequalities involving mean curvature term and first eigenvalue estimates for the spectrum of the Laplacian on minimal hypersurfaces in a Riemannian manifolds.

**June 2, 2015
** Although the Ricci flow with surgery has been used by Perelman to solve the Poincaré and Geometrization Conjectures, some of its basic properties are still unknown. For example it has been an open question whether the surgeries eventually stop to occur (i.e. whether there are finitely many surgeries) and whether the full geometric decomposition of the underlying manifold is exhibited by the flow as $t \to \infty$.

*There are finitely many surgeries in Perelman's Ricci flow*Richard Bamler, Univeristy of California, Berkeley

In this talk I will show that the number of surgeries is indeed finite and that the curvature is globally bounded by $C t^{-1}$ for large $t$. Using this curvature bound it is possible to give a more precise picture of the long-time behavior of the flow.