# Mathematics Colloquium Winter 2015

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

**January 13, 2015**

*CANCELLED *

**Marco Sammartino, University of Palermo**

**January 20, 2015**

*Moduli space of K3 surfaces and Noether-Lefschetz conjecture*

**Zhiyuan Li, Stanford University**

The study of Picard groups of moduli problems is started by Mumford in 1960's. It is proved by Mumford and Harer that the Picard group of moduli space of curves of genus g>2 is a finitely generated group of rank one. In moduli theory, a K3 surface of genus g is a higher dimensional analogue of curves. However, the Picard group of their moduli spaces are much more complicated (shown by O'Grady in 1980's). According to Maulik and Pandharipande, there is a conjectural description of the Picard groups of moduli space of K3 surfaces of any given genus, called Noether-Lefschetz conjecture. In this talk, I will briefly review some classical results and talk about the most recent proof of this conjecture.

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**January 27, 2015**

*Poincaré's Legacy: Predictions on Time Scales Ranging from Milliseconds to Billions of Years *

**Greg Laughlin, University of California, Santa Cruz**

I will discuss and connect two long-standing, and at first glance unrelated, problems of prediction: (1) the long-term dynamical stability of the Solar System, and (2) price movements and volatility in financial markets. These phenomena have radically different governing mechanisms, and their characteristic time scales are vastly different, but they share a key common basis in the random walk, and the study of both can be traced directly back to the work of Henri Poincaré.

**February 3, 2015 **

Frank Bäuerle, University of California, Santa Cruz

Tony Tromba, University of California, Santa Cruz

*UC Calculus Online*Frank Bäuerle, University of California, Santa Cruz

Tony Tromba, University of California, Santa Cruz

We will describe UC's Calculus Online,hosted by UCSC's Mathematics Department, now available to all UC students through our new cross campus enrollment system as well as to all non matriculated students including foreign nationals. Calculus I for Science and Engineering Students has been successfully running for over a year and Calculus II since last Spring. Calculus III and IV are currently in development.

The courses have many components, from introductory welcome lectures, historical enrichment video lectures, online lecture videos ( all synchronized with an online interactive e-text originally developed for print by UCLA Professor Jon Rogawski), to an online discussion forum platform all accessible via UC's Canvas Learning Management System. We would very much welcome questions and suggestions.

**February 10, 2015 **

*Realizing all Free Homotopy classes within the Planar three body problem*Richard Montgomery, University of California, Santa Cruz

For 17 years I tried to prove the above result using variational methods.

Desperation led me to numerical experiments which led me to Carles Simo (U of Barcelona) who got me to change tack and look for a dynamical mechanism behind the alleged result. Once I took on the dynamical perspective, I saw that my collaborator Rick Moeckel (U of Minnesota) had unknowingly solved my problem over three decades ago!

Here is the problem, precisely. The configuration space of the planar three body problem, after dividing by rotations, is homotopic to a pair of pants. As such it has a huge set of free homotopy classes: the set of conjugacy classes for the free group on two letters. [These classes are encoded by sequences of eclipses between the bodies, or "syzygy sequences''.] The question, inspired by Riemannian geometry, is the following. Is ever free homotopy class ["reduced syzygy sequence''] realized by a collision-free solution to the Newtonian 3-body problem? The answer, in this joint work with Rick Moeckel [http://arxiv.org/pdf/1412.2263.pdf] is yes!

The answer raises more questions than it answers.

** February 17, 2015 **The (Morse) index of a minimal surface in R^3 is a poorly understood invariant. I'll discuss some recent work concerning a link between the topology and the index of the surface. This is joint work with Davi Maximo.

On the topology and index of minimal surfaces

Otis Chodosh, Stanford University

**February 24, 2015 *SPECIAL COLLOQUIUM*Theorems at the interface of number theory and representation theory**

**Ken Ono, Emory University**The speaker will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities are two peculiar identities which express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent work with Griffin and Warnaar, the speaker has obtained a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood q-series. This work characterizes those integral units that arise from this theory. In a related direction, the speaker revisits the classical Moonshine Theorem which asserts that the coefficients of the modular j-functions are dimensions of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Griffin and Duncan, the speaker has obtained exact formulas for these distributions. The speaker will conclude with a discussion of the Umbral Moonshine Conjecture.

**March 3, 2015**

**No Colloquium Schedule**

** March 10, 2015****An Exercise in Fourier Analysis on the Heisenberg Group****Persi Diaconis, Stanford University**

One version of the Heisenberg group is the 3 x 3 uni upper triangular matrices with x,y on the super diagonal and z in the upper right hand corner, say (x,y,z), so (x,y,z) (x',y',z') = (x+x',y+y',z+z' +xy'). This group appears in physics (through the quantum harmonic oscillator) in analysis, algebra and computer science (through it's connection with the fast Fourier transform). In joint work with Dan Bump, Angela Hicks, Laurent Miclo and Harold Widom we study the behaviour of simple random walk. We apply (non commutative) Fourier analysis and wind up having to bound the eigenvalues of some nice matrices. These same matrices come up in the applications above (e.g. in physics as Harpers operator, Hofstader's butterfly and the 10 Martinis problem). We need some new techniques which turn out to be useful in the applications and for other walks on other groups. I will try to explain all this to a general mathematical audience 'in English'.

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