Colloquium: Amanda Folsom (Yale University)

 

Title: q-series and quantum modular forms

 

Abstract: While the theory of mock modular forms has seen great advances in the last decade, questions remain. We revisit Ramanujan’s last letter to Hardy, and prove one of his remaining conjectures as a special case of a more general result. Surprisingly, Dyson’s combinatorial rank function, the Andrews-Garvan crank function, mock theta functions, and quantum modular forms, all play key roles. Along these lines, we also show that the Rogers-Fine false theta functions, functions that have not been well understood within the theory of modular forms, specialize to quantum modular forms. This is joint work with K. Ono (Emory U.) and R.C. Rhoades (Stanford U.).

Colloquium: Nathan Grigg (University of Washington)

 

Title: Moduli spaces, deformation theory, and noncommutative varieties

 

Abstract: In algebraic geometry, a moduli space is a space that parametrizes a class of objects, with each object represented by a point in the moduli space.

 

Deformation theory is the study of the neighborhood of a point of the moduli space. For example, the deformation theory of an algebraic variety classifies which varieties are "near" a given algebraic variety in the moduli space.

 

In this talk, I will discuss some examples of things you can learn from basic deformation theory. I will also discuss some recent results in this area which use the tools of modern algebraic geometry to deform an algebraic variety in a noncommutative direction.

Colloquium: Alan Parry (Duke University)

 

Title: Mathematics Meets Astrophysics: An Overview of General Relativity and Dark Matter in Galaxies

 

Abstract: One of the most beautiful intersections of mathematics and physics is Einstein’s theory of gravity, general relativity. It has successfully resolved problems with Newton’s notion of gravity and made incredible predictions about the universe that were later verified by observations. Recently, mathematical motivations for general relativity have presented a possible and tantalizing description of dark matter, a mysterious and exotic form of matter that makes up approximately 23% of the energy density in the universe. In this talk, I will introduce for general audiences the theory of general relativity and the mathematics behind it, discuss the major successes of general relativity, and describe this model for dark matter and the recent advances that have been made in its study.

Colloquium: Christopher Nelson (University of California, San Diego)

 

Title: A Perfect Positivstellensatz in a Free Algebra

 

Abstract: A positivstellensatz is an algebraic certificate for a given polynomial p to have a specific positivity property. Such theorems date back in some form for over one hundred years for conventional (commutative) polynomials. For example, given polynomials p, q, and r, one might ask

 

(Q)                                            Does q(x) ≥ 0 and r(x) = 0 imply p(x) ≥ 0?

 

In this talk I will focus on free noncommutative polynomials p, q, and r and substitute matrices for the variables xj. In the case that the positivity domain D ∶={X | p(X) ⪰ 0} is convex, there exists a "perfect positivstellensatz", which gives an elegant answer to (Q) with strict degree bounds. This non-commutative positivstellensatz has a much cleaner statement than analogous commutative results. In addition, from this result follow a few corollaries, including a non-commutative nullstellensatz as well as results relating to convex optimization.

Colloquium: Mark Allen (Purdue University)

 

Title:  Free Boundary on a Cone

 

Abstract:   The first problems a student encounters in a partial differential equations class involve looking for an unknown function which is a solution to an equation and has prescribed boundary data over the boundary of a fixed domain.However, in many applications in the physical sciences, biology, and even geometry, it becomes necessary to study a PDE with both an a priori unknown function and unknown domain. These types of problems have come to be called "free boundary problems". In this talk I will give some examples of free boundary problems that are notable both for their applications and for their importance in the historical development of this area of mathematics. I will discuss the importance of studying free boundary problems on manifolds that are not necessarily "smooth". I will also discuss recent work on studying the Alt-Caffarelli free boundary problem on a two dimensional cone generated by a smooth curve γ on the sphere. We show that when length(γ) < 2π the free boundary avoids the vertex of the cone. When length(γ) ≥ 2π we provide examples of minimizers such that the vertex belongs to the free boundary. This result is analogous to a well known result for distance minimizing geodesics on a cone. Indeed, the Alt-Caffarelli free boundary is known to behave similarly to minimal surfaces, and our result establishes another similarity between this free boundary problem and minimal surfaces. This is joint work with H. Chang Lara.

Colloquium: David McAllister

 

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.

Colloquium: Giovanni Forni (University of Maryland)

 

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.

Colloquium: Cristian Tomasetti

 

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.

Teaching Colloquium: Doug Corey

 

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.

Focus on Math: Dave Richeson (Dickinson College)

 

Title: Four Tales of Impossibility

 

Abstract: "Nothing is impossible!" It is comforting to believe this greeting card sentiment; it is the American dream. Human flight and the four-minute mile were proclaimed to be impossible, but both came to pass. Yet there are impossible things, and it is possible to prove that they are so. In this talk we will look at some of the most famous impossibility theorems—the so-called "problems of antiquity." The ancient Greek geometers tried and failed to square circles, trisect angles, double cubes, and construct regular polygons using only a compass and straightedge. So did future generations of great mathematicians. It took two thousand years to prove conclusively that all four of these are mathematically impossible. No, not even Chuck Norris can square the circle.