Title: CUDA: C for Data Parallel Problems
Abstract: I will provide a gentle introduction to GPU programming using Nvidia’s CUDA platform. CUDA offers a simple way to express data parallelism in C or C++. CUDA code executes on Nvidia GPUs in machines ranging from thin and light laptops to Titan, the world’s fastest supercomputer. I will survey a few application domains and give a taste of how to code with CUDA for optimal performance.
Title: Renormalization and deformation methods in parabolic dynamics
Abstract: Parabolic dynamics is characterized by slow, at most polynomial divergence of nearby orbits with time. Since most techniques of dynamical systems have been developed for systems with some hyperbolicity, that is, with exponential divergence of orbits, parabolic systems are often hard to study.
Important examples include homogeneous unipotent flows, for which Ratner proved fundamental results on the classification of invariant measures, and piecewise linear systems such as billiards in polygons. In this talk we will focus on the problem of establishing speed of convergence of ergodic averages for a class of nillflows closely related to Weyl sums of analytic number theory and on the problem of establishing ergodicity for billiards in polygons. The approach we will emphasize is derived from the idea of renormalization that applies to systems which are approximately self-similar at different scales. However, renormalization seems to fail in many cases (nilflows on higher step nilmanifolds, billiards in non-rational polygons) leading to extremely challenging problems. We will argue that renormalization can be generalized in an effective way drawing from ideas coming from the degeneration of Riemannian manifolds under deformations of the metric.
Title: Stochastic modeling of the accumulation of passenger and driver mutations in cancer
Abstract: Important progress has been made in our understanding of cancer thanks to the ever growing amount of data originated by sequencing technologies. One useful approach for better understanding the process of accumulation of somatic mutations in cancer is given by the integration of mathematical modeling with sequencing data of cancer tissues.
While it has been hypothesized that some of the somatic mutations found in tumors may occur prior to tumor initiation, there is little experimental or conceptual data on this topic. To gain insights into this fundamental issue, we formulated a new mathematical model for the evolution of somatic mutations in which all relevant phases of a tissue’s history are considered. The model provides a way to estimate the in-vivo tissue-specific somatic mutation rates from the sequencing data of tumors. The model also makes novel predictions, validated by our empirical findings, on the expected number of somatic mutations found in tumors of self-renewing tissues. Importantly, our analysis indicates that half or more of the somatic mutations in tumors of self-renewing tissues occur prior to the onset of neoplasia. Furthermore, a general principle for improving the detection of driver mutations by reducing the amount of "noise" caused by the passenger mutations will be introduced.
Our results have substantial implications for the interpretation of the large number of genome-wide cancer studies now being undertaken.
Title: Handling the "When-will-I-ever-use-this?" question.
Abstract: I wrote an essay for my calculus students this fall about this question and got a great response from students. The point of the essay was to help students realize that it is a very hard question for teachers to answer because of the way we actually use knowledge. We are not very good at predicting the kind of knowledge that will be useful to us in the future, for various reasons. A different philosophy about acquiring and using knowledge is more helpful than the view that is often at the root of the when-will-I-ever-use-this question. I explore some of these reasons to better understand the paradox that if mathematics is so useful, why is it so hard to help students see how it is useful.
Title: In search of the ultimate state of slippery Rayleigh-Benard convection
Abstract: For decades, experiments (and more recently numerical simulations) have attempted to determine how the effective transport of heat (measured by the non-dimensional Nusselt number Nu) scales with the driving force (as measured by the Rayleigh number Ra) –when said driving force is asymptotically strong–in Rayleigh-Benard convection where an incompressible, Boussinesq fluid is driven by an imposed temperature gradient. To date the results are inconclusive for experiments, and finite limitations on computational resources restrict the potential usefulness of direct numerical simulation. In contrast to these approaches, we derive rigorous upper bounds on the heat transport for slippery convection in which the velocity satisfies a stress-free (or free-slip) boundary condition on the vertical boundary. For 2d convection as well as 3d convection for infinite (or even large) Prandtl number (ratio of kinematic viscosity to thermal diffusivity) this bound takes the form Nu < C Ra5/12. At finite Prandtl numbers this scaling challenges some theoretical arguments regarding asymptotic high Rayleigh number heat transport by turbulent convection.
Title: How to Find Curves as Symmetric as the Klein Curve
Abstract: The complex curve x3y + y3z + z3x=0 is a nonsingular genus 3 curve whose automorphism group is famously isomorphic to the simple group G = PSL2(F7) of order 168. These 168 automorphisms are induced by (the projectivization of) one of the irreducible degree 3 representations of G. It is natural to wonder whether or not there might be other plane curves that have this unexpected symmetry property, and, if so, whether these curves have other interesting geometric characteristics. It will be possible to show that the answers to these questions display surprising patterns, and that these patterns are capable of considerable generalization.
Title: Unitary representations of Lie groups
Abstract: Unitary representations of Lie groups appear everywhere in mathematics: in harmonic analysis (as generalizations of the sines and cosines appearing in classical Fourier analysis); in number theory (as spaces of modular and automorphic forms); in quantum mechanics and more modern physical theories (as solutions of differential equations admitting symmetries); and in many other places. They have been the subject of intense study for decades, but only recently has their classification emerged. The answer relies, perhaps surprisingly, on an essentially combinatorial object — the finite Hecke algebra — and its relation to certain singular algebraic varieties. The purpose of this talk is to explain a little bit about unitary representations and offer a few hints about how Hecke algebras and algebraic geometry enter into their study.
Title: Gropes and their fundamental groups
Abstract: We consider open infinite gropes which are 2-dimensional CW complexes introduced by Stan’ko and Cannon. Gropes are classifying spaces of their fundamental groups. We show some properties of these groups and present some recent results.
Title: Mathematics of Voting Procedures
Abstract: We vote, but will the outcome accurately reflect what the voters want? We make decisions, but does the conclusion accurately reflect the data? While the importance of these concerns is obvious, it turns out that these issues are mathematically complex. With an emphasis on voting rules, the severity of the problem will be illustrated and some of the underlying mathematical structures will be described.
Title: Mathematics and the ‘Dark Matter’ Mystery
Abstract: Even after spending billions of dollars on experiments, nobody has been able to find that elusive, mysterious thing hiding out there in the heavens that is called "dark matter." Sounds ominous! But, what is it? Why do we believe it is important? As described in this expository lecture, mathematics now is shedding significant new light on this darkness to illuminate the mystery. Part of this puzzle includes mathematical aspects about the evolution of Newton’s universe, which will be briefly discussed.