This talk will bring ideas from number theory, e.g., the Prime Number Theorem, Bertrand's Postulate, and Cebotar\"ev's Density Theorem, to study properties of infinite groups. In particular, we will introduce the notion of quantifying the extent to which a finitely generated group is residually finite. This asymptotic study connects word and subgroup growth via a function called the normal divisibility function that measures the size of the smallest finite quotient that a fixed group element maps to nontrivially. In this talk we will investigate such behavior for examples that include free groups and SL(n,O_K)$, where K is a number field. This talk uses ideas and methods from combinatorial group theory, number theory, the theory of profinite groups, and topology. Part of this talk covers joint work with B. McReynolds.

Semisimple Lie algebras are governed by the geometry and combinatorics of groups generated by reflections, acting on Euclidean space. Kac, Moody and others introduced more general Lie algebras, governed in the same way by groups acting on hyperbolic space. These had unexpected and spectacular application in Borcherds' proof of the "Monstrous Moonshine" conjectures. The problem of classifying all "interesting" such algebras may be within reach; I will explain recent and current work on the subject.

This is the second of a pair of expository talks. I'll sketch the construction of symplectic cohomology, commenting both on the features which are typical of Floer cohomology theories and the special adaptations needed in this setting.

The Monge transportation problem can be roughly stated as how to minimize the cost of transporting mass around on a space, where the cost to transport one unit of mass from location x to location y is a given function c(x,y). On this talk, we shall present a existence and uniqueness result by Robert McCann, where our space is a compact Riemannian manifold and the cost function is the distance squared.

Note the new room for the seminar this semester.

Abstract: We consider linear cocycles over hyperbolic diffeomorphisms and discuss criteria for such cocycles to be isometric or conformal in terms of their periodic data. We outline the proof of the result that if the return maps at the periodic points are, in a sense, conformal or isometric then so is the cocycle itself with respect to a continuous Riemannian metric.

The complex of curves is a nasty simplicial complex, and, accordingly, many original proofs of facts about this space are difficult and technical. In his thesis, Jason Behrstock proves a useful lemma on the behavior of distances in curve complexes of subsurfaces, but his proof relies on hierarchies, a complicated tool introduced by Masur and Minsky. I will present an elementary proof (observed by Chris Leininger) of Behrstock's Lemma using some constructions involving curve surgery and hyperbolic geometry. No prior knowledge of the curve complex will be assumed.

Abstract: Algebraic hyperbolic actions of Z^k and R^k with k at least 2 exhibit remarkable rigidity properties which are absent in rank k=1. The properties include measure rigidity, cocycle rigidity, and local smooth rigidity. We will give an overview of the area with emphasis on global smooth rigidity for non-algebraic hyperbolic actions.

Mathematicians derive immense pleasure from discovering elegant proofs of existing theorems. We are, after all, artists of reason. During this talk, we will investigate as many different proofs of the infinitude of primes as time allows. We will begin with Euclid's beautifully clever proof involving a few basic facts from arithmetic. From there we will venture into proofs involving calculus, infinite series, and topology. Each proof is truly elegant and each gives us new insight into the complexity of the prime numbers.

The determinant of the Laplacian is an important invariant of closed surfaces and has connections to the dynamics of geodesics, Ricci flow, and physics. Its definition is somewhat intricate as the Laplacian has infinitely many eigenvalues. I'll explain how to extend the determinant of the Laplacian to non-compact surfaces where one has to deal with additional difficulties like continuous spectrum and divergence of the trace of the heat kernel. On surfaces (even non-compact) this determinant has a simple variation when the metric varies conformally. I'll explain how to use Ricci flow to see that the largest value of the determinant occurs at constant curvature metrics.This is joint work with Clara Aldana and Frederic Rochon.

Three dimensional cryo-electron microscopy (3D cryo-EM, for short) is the problem of determining the three dimensional structure of a large molecule from the set of images, taken by an electron microscope, of randomly oriented and positioned identical molecular particles which are frozen in a thin layer of ice. A solution to this problem is of particular interest, since it promises to be an entirely general technique which does not require crystallization or other special preparation stages. Present approaches to the problem fail with particles that are too small, cryo-EM images that are too noisy or at resolutions where the signal-to-noise ratio becomes too small. In my talk, I will describe a novel algorithm, referred to as the intrinsic reconstitution algorithm, due to Amit Singer and Yoel Shkolnisky, which constitutes a basic step for the solution of the 3D cryo-EM problem. The appealing property of this new algorithm is that it exhibits remarkable numerical stability to noise. My main goal is to give a conceptual explanation, based on representation theory, for the admissibility (correctness) and the numerical stability of the intrinsic reconstitution algorithm. If time permits, I will mention some recent results concerning more elaborate aspects of the cryo-EM problem. This work is joint with Amit Singer (Princeton) and is a part of an ongoing project conducted jointly with Shamgar Gurevich (IAS), Yoel Shkolnisky (Tel-Aviv University) and Fred Sigworth (Yale).

We'll discuss some basic results of number theory from the viewpoint of the following question: How are the prime numbers distributed throughout the integers, and how are particular sets of primes distributed throughout the set of all primes? Most of the talk should be quite elementary. Time willing, we'll touch on Dirichlet's theorem on primes in an arithmetic progression, Chebotarėv's generalization to algebraic number fields, and how the latter can be used to determine the Galois groups of polynomials over Q.

It is well known that the stock price process is only a strict local martingale in certain stochastic volatility models. The lack of the martingale property gives multiple solutions to the pricing equation of European call options. In this talk we will show that the pricing equation has a unique solution if and only if the stock price process is a martingale. We will motivate our result by introducing an analogue result in local volatility models. In the second part of the talk, we will focus on the boundary condition of the pricing equations at vanishing volatility. This choice of boundary condition is a “thorny issue”. Different boundary conditions have been chosen for different models in numerical computation literature. We will give a probabilistic interpretation for these boundary conditions.