# Mathematics Colloquium Fall 2014

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

**October 14, 2014**

*Three circle theorems on Kahler manifolds and applications *

**Gang Liu, UC Berkeley**

The classical Hadamard Three Circle theorem is generalized to complete Kahler manifolds with nonnegative holomorphic sectional curvature.

Various applications will be discussed. For example, the connection with Yau's uniformization conjectures; the resolution of Ni's conjecture on complete Kahler manifolds with nonnegative bisectional curvature.

**October 21, 2014**

*40 years of Algebraic Methods in Numerical Analysis*

**Pedro Santos, University of Lisbon**

The year 2014 marks the fortieth anniversary of the PhD Thesis of A. V. Kozac, in which he proved the equivalence between the stability of an approximation sequence of operators and the invertibility of a certain element in a specially constructed Banach Algebra. His result unleashed a fruitful interplay between Numerical Analysis and Operator Algebras involving mathematicians from the former Soviet Union, United States, Germany, Israel, and Portugal, among others. I will present some of the basic results of the theory and describe interesting problems, phenomena, and results that were discovered during the past 40 years.

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**October 28, 2014**

*Mean curvature flow of hypersurfaces*

**Robert Haslhofer, Courant Institute, New York University**

A family of hypersurfaces $M_t\subset R^{n+1}$ evolves by mean curvature flow (MCF) if the velocity at each point is given by the mean curvature vector. MCF can be viewed as geometric heat equation, deforming surfaces towards optimal ones. For example, if the initial surface M_0 is convex, then the evolving surfaces M_t become rounder and rounder and converge (after rescaling) to the standard sphere S^n. The central task in the study of MCF for more general initial surfaces (not just convex ones) is to analyze the formation of singularities. For example, if M_0 looks like a dumbbell, then the neck will pinch off, preventing one from continuing the flow in a smooth way. To resolve this issue, one can either try to continue the flow as a generalized weak solution (and develop a structure and regularity theory for these weak solutions) or try to perform surgery (i.e. cut along necks and replace them by caps).

These ideas have been implemented successfully in the last 15 years in the deep work of White and Huisken-Sinestrari, and recently Kleiner and I found a streamlined and unified approach (arXiv:1304.0926,1404.2332).

In this lecture, I will survey these developments for a general audience.

**November 4, 2014 2:00pm - 3:00pm** ***Research Seminar Talk*****Eisenstein primes at composite level**** Kenneth A. Ribet, UC Berkeley**

I will summarize some joint work with Hawjong Yoo about the dimensions of Eisenstein kernels in Jacobians of modular curves. Recall that Barry Mazur treated this question in the case of prime level -- if p is a prime number and m is an "Eisenstein prime" in the ring of Hecke operators acting on J_0(p), Mazur proved that the kernel of m on J_0(p) is a 2-dimensional vector space over the residue field of m. Yoo and I replace p by a product of two distinct primes, say N=pq, and consider Mazur's question in this new case. For simplicity, we take m to be of residue characteristic prime to 6N. We find that the kernel can be 2-, 3- or 5-dimensional, depending on cases. To our annoyance, we are not (yet) able to give a complete characterization of the situations where the dimension is 3.

**November 4, 2014 *SPECIAL COLLOQUIUM***

*Discrete logs in mathematical cryptography*

**Kenneth A. Ribet, UC Berkeley**While I'm not really an expert in cryptography, I designed an upper-division cryptography course at Berkeley and have advised the work of graduate students who have specialized in this subject. My excuse for speaking about the subject at UCSC is the current work of my student Kim Laine, who is studying the discrete log problem on Jacobians of hyperelliptic curves of genus three in collaboration with Kristin Lauter of Microsoft Research.

I will begin by recalling the Diffie-Hellman problem and the discrete log problem in the multiplicative group of integers mod p (where p is a large prime). I will explain the index calculus attack that makes one worry about the security of encryption schemes that are based on this discrete log problem. Because of this attack, workers find it natural to move to elliptic curves mod p, where there is no analogous attack. Mathematicians might wonder whether it wouldn't be even better to work with Jacobians of curves of genus bigger than 1.

Cryptographers have understood that this is not necessarily a good idea: curves of high genus are to be avoided, whereas curves of genus 2 are fine and curves of genus 3 may or may not be OK.

**November 11, 2014 *Veterans Day - No Colloquium***

**November 18, 2014**

*High-order Numerical Methods for Predictive Science on Large-scale High-performance Computing Architectures*

**Dongwook Lee, Professor of Applied Mathematics and Statistics, UC Santa Cruz**

Modeling diverse physical processes using mathematical algorithms has become a successful tool in modern science and engineering. The underlying mathematical models are carefully designed to perform large-scale computer simulations that involve disparate scales of space and time. Such complexities often arise when incorporating various multi-physics components that can be represented by classes of partial differential equations.

In the first part, I will discuss key issues in seeking computational solutions on large-scale high-performance Computing (HPC) architectures, and the need for using high-order numerical algorithms on HPC.

I describe mathematical algorithms with special attention to two numerical approaches: first, the traditional formulations based on high-order polynomials; second, a new innovative exponentially converging formulation based on Gaussian Process Modeling. Moreover, I will show the importance of fast convergent, high-order accurate numerical methods and how they are crucial for future high-performance exascale computing architectures.

In the second part, I will present laboratory astrophysics scientific simulations using the numerical algorithms introduced in the first part. They include large-scale computer simulations of astrophysics and high-energy-density plasma physics, with special emphasis on laser-driven shock experiments to shed lights on the processes behind magnetic field generation and amplification.

The key ideas and challenges of computational mathematics in this talk have been developed within the framework of the University of Chicago's FLASH code.

FLASH is a highly capable, massively parallel, publicly available open source scientific code with a wide user base in the fields of astrophysics, cosmology, and high-energy-density physics.

**November 25, 2014**

*Functoriality, Smith theory, and the Brauer homomorphism*

**David Treumann, Boston College and MSRI**I will discuss some results relating mod p cohomological automorphic forms on groups G and H, when H is the fixed points of an automorphism of G of order p. The results are obtained via Smith theory, an old technique in algebraic topology. No knowledge of automorphic forms or Smith theory will be assumed. Based on joint work with Venkatesh.

**December 2, 2014****Affine Lie algebras, current algebras, and representations**

** Michael Lau, Université Laval**Affine Lie algebras burst onto the mathematical scene in the late 1960s as the most important “new” examples of the recently discovered Kac-Moody algebras. From the beginning, it was understood that they have a powerful alternative interpretation as extensions of loop algebras, families of maps from the circle to finite dimensional Lie algebras. The traditional representation theory for these algebras is infinite dimensional, based on the Kac-Moody presentation, and led to the theory of vertex operator algebras. Their finite dimensional representation theory is based on the loop approach and is somewhat less well known. After discussing some of the interesting features of affine Lie algebras, we will describe their finite dimensional representations and discuss generalisations to current algebras on n-tori and other varieties. This talk will be accessible to a general mathematical audience.

**December 9, 2014 *SPECIAL COLLOQUIUM*****The Future of Solar Energy after the Big Crash**

** Sue Carter, Associate Dean of Graduate Studies, Professor of Physics, UC Santa Cruz**

The precipitous crash of photovoltaic (PV)module prices over the last 5 years has made the dream of generating renewable electricity a cost comparable to coal a potential reality (as well as bankrupting much of the US solar industry!). However, while PV module prices are close to meeting the goals set by the DOE’s Sunshot program 5 years early, the costs related to the balance of systems, namely the costs associated with the land, mounting hardware, grid electronics, installation labor and permitting have remained stubbornly high. This dramatic shift of costs has transitioned the solar energy field from one focused solely on reducing module costs to one focused on increasing power efficiency beyond the Shockley-Queisser Limit (set over 40 years ago) and developing PV modules that can be directly into existing building infrastructure.** **

In this talk, I will provide an overview of the approaches we, and others, are pursing to both increase power efficiency and reduce balance of systems costs in order to make solar energy cost competitive with energy generated from fossil fuels. By utilizing quantum confinement effects, more then one electron can be collected for each incident photon (i.e. over 100% quantum efficiency), allowing us to exceed the long standing Shockley-Queisser limit. By taking advantage of energy efficiencies across the solar spectrum, wavelength-selective PV modules can be directly installed into building windows or over agriculture crops or greenhouses to simultaneously grow food, generate electricity, and harness thermal energy. Computational modeling is critical to understand the basic physics behind how the technologies work and how to rapidly optimize the technology to transition it to potential deployment. I will show some of the modeling work we, and others, have done using Poisson’s equations, transfer matrices, and geometric ray tracing to understand charge and light transport in these next generation PV. I will conclude by discussing the importance of developing models that accurately include thermal modeling with this electrical and optical modeling to develop technologies that can efficiently utilize the entire solar energy spectrum.** **