Math Department Events Listing

This talk will be a crash course on some aspects of complex geometry and zeta function determinants of positve elliptic operators. This will review the background needed to fully appreciate the content of the talk I will give on Tuesday 4/5 in the Geometry Seminar (at 4:00pm in 402) on Quillen's construction of the determinant function for Cauchy-Riemann operators acting in a complex vector bundle over a Riemann surface. I will state (and partially justify) some facts from Kahler geometry, Hodge theory on Kahler manifolds, functional analysis of self-adjoint operators, and zeta-function determinants.

One-face maps can be uniquely represented by a special class of 3-regular graphs. Using this representation, one can examine the spectral density and scaled eigenvalue spacing distributions for random one-face maps. Numerical experiments have revealed interesting connections between the underlying geometry of the map and universal distributions from Random Matrix Theory. In this talk the graphical representation of one-face maps will be explained, and recent numerical results discussed.

The classical question of determining the whole numbers that can be expressed as a sum of two rational cubes leads to a study of the elliptic curves x³ + y³ = k. In this talk I will discuss aspects of the arithmetic of these curves, including some open questions about the distribution of ranks in this family. For these curves, descent via 3-isogeny yields an easily computed upper bound for the rank; a formula for this upper bound, as well as recent computational work, will also be presented.

The determinant of an operator acting in an infinite dimensional space is difficult to define. Even if the operator has only eigenvalues in its spectrum, the infinite product of these values need not converge. In this talk I will present Quillen's construction of the determinant function for Cauchy-Riemann operators acting in complex vector bundles over Riemann surfaces. We will carefully motivate and then state the problem which Quillen solved, outline his solution, and finish with some of the amazing details of the proof. I will give a preparatory talk on some of the background material in complex geometry and functional analysis in the Graduate geometry seminar, Monday 4/4 at 1:00pm in 402.

RSA, invented by Rivest, Shamir, Adleman in 1978, is the main public key cryptosystem in use today for securing various communications over the Internet. The mathematics behind RSA is basic modular arithmetic. Security of RSA depends on the difficulty of factoring integers, specifically factoring the public modulus. To withstand various attacks including factorization attacks RSA needs to be implemented properly with appropriate parameter choices. In this talk, we will describe the basic set up for RSA, and some of the implementation requirements to keep RSA secure against attacks.

The traditional 3 dimensional rigid body is a well understood integrable Hamiltonian system. The governing equations have both a physical and symplectic structure. The more general n-dimensional rigid body has no real physical significance, but one can exploit the symplectic structure to find a Hamiltonian framework, show it is integrable, and characterize generic equilibrium of the system. These results are mainly derived from the underlying Lie algebraic nature of the problem.

The Bose-Hubbard model is the next-simplest lattice model of interacting bosons, where the Hamiltonian involves hoppings and on-site repulsive interactions. In the limit of infinite repulsions we get the simplest model, which is the hard-core Bose system. The zero-temperature phase diagram involves Mott insulating phases and Bose-Einstein condensates. I will present a Feynman-Kac approach combined with standard cluster expansions that allows to study Mott phases. The transition line can then be established at lowest order in the ratio hopping/interaction.

Numerical modeling of wave propagation through inhomogeneous medium remains a challenging computational problem. Algorithms designed for problems of radar, fiber optics, and photonics should handle millions of unknowns in order to accurately represent complex material structures. Accurate modeling of waves on sharp material interfaces and high-order treatment of radiation conditions on infinity are of special interest in applications. I will present a set of novel computational techniques permitting high-order accurate solution of such problems, and will illustrate the performance of these methods by the results of actual computations. The talk will be accessible to graduate students.

Padé approximants have found many uses in the numerical solution of partial differential equations. Most often, Padé approximants of low order are used; use of Padé approximants of high order is less common. In this presentation we will discuss our own work on high- and low-order Padé approximation, which has helped tackle a range of distinct problems - including wave scattering by rough surfaces, resulting in very high accuracies with very limited computational costs for two- and three-dimensional scattering problems; high-frequency asymptotics and multiple scattering, leading to numerical methods that can resolve, with prescribed accuracies and in fixed computational times, scattering problems of arbitrarily high frequency; and, even, time evolution of nonlinear hyperbolic and parabolic PDEs (including wave equations) - giving rise to explicit time-marching methods that enjoy implicit-like stability properties for a wide range of problems.

Stacks were originally conceived by Grothendieck to study higher non-abelian cohomology. Later, Deligne–Mumford and Artin introduced algebraic stacks as geometric objects to study moduli problems. For a long time algebraic stacks were considered rather exotic and suffered an existence only at the fringe of geometry. This has changed since the 1990's. Algebraic (and other kinds) of stacks are now ubiquitous throughout mathematics. We will explain in simple terms what stacks are. Our guiding example will be the stack of triangles in the (Euclidean) plane. The talk will be elementary throughout.

A fundamental problem of cell biology is to understand how cells make measurements and then make behavioral decisions in response to these measurements. The full answer to this question is not known but there are some underlying principles that are coming to light. The short answer is that the rate of molecular diffusion contains quantifiable information that can be transduced by biochemical feedback to give control over physical structures. This principle will be illustrated by two specific examples of how rates of molecular diffusion contain information that is used to make a measurement and a behavioral decision.

Example 1: Bacterial populations of P. aeruginosa are known to make a decision to secrete polymer gel and become virulent on the basis of the size of the colony in which they live. This process is called quorum sensing and only recently has the mechanism for this been sorted out. It is now known that P. aeruginosa produces a chemical whose rate of diffusion out of the cell provides information about the size of the colony which when coupled with positive feedback gives rise to a hysteretic biochemical switch.

Example 2: Salmonella employ a mechanism that combines molecular diffusion with a negative feedback chemical network to “know” how long its flagella are. As a result, if a flagellum is cut off, it will regrow at the same rate with which it grew initially.

We present a new set of algorithms and methodologies for the numerical solution of problems of scattering by complex bodies in three-dimensional space. These methods, which are based on integral equations, high-order integration, fast Fourier transforms and highly accurate high-frequency methods, can be used in the solution of problems of electromagnetic and acoustic scattering by surfaces and penetrable scatterers --- even in cases in which the scatterers contain geometric singularities such as corners and edges. In all cases the solvers exhibit high-order convergence, they run on low memories and reduced operation counts, and they result in solutions with a high degree of accuracy. In particular, our algorithms can evaluate accurately in a personal computer scattering from hundred-wavelength-long objects by direct solution of integral equations --- a goal, otherwise achievable today only by supercomputing. A new class of high-order surface representation methods will be discussed, which allows for accurate high-order description of surfaces from a given CAD representation. A class of high-order high-frequency methods which we developed recently, finally, are efficient where our direct methods become costly, thus leading to a general and accurate computational methodology which is applicable and accurate for the whole range of frequencies in the electromagnetic spectrum.