The p-adic numbers are usually defined analytically, as a completion of the rationals. They can also be constructed using the language of categories and inverse limits. We will discuss both approaches and use this as motivation to learn about these “categories” that keep coming up in class.

The is a Ph. D qualifying exam

I will discuss Witten's proof of the positive energy theorem of general relativity, involving Dirac operators and spinors. Most of the talk is based on Parker and Taubes' rigorous account. Emphasis will be on the geometric structures involved. (If you liked Dr. Freed's course on the index theorem last spring, you will love this.)

Obstacle problems consist of studying the least supersolution of some equation which stays above a given function. We study the corresponding obstacle problem for the fractional laplacian and achieve regularity results for the solution that are almost optimal. This problem has applications to option pricing in mathematical finance. This is a thesis defense.

The inner ear (cochlea) can detect pressure fluctuations that are 1/2,000,000,000 that of atmospheric pressure. How does one model such a sensitive organ? We will first look at the physiology of the ear and discuss the current models, most of which are linear. We then introduce a nonlinear and nonlocal treatment of the cochlea. There will be simulations, a discussion of numerical methods, and a plethora of results to discuss. This is a thesis defense.

This is work done in a VIGRE research project.

The quot scheme parameterizes quotient sheaves of the rank n free sheaf over P^1 of given rank k and degree d. This smooth variety is a compactification of the space of rational curves of degree d in the Grassmannain of rank k quotients of C^n. The ranks of its homology groups were determined by Stroome, who used the action of a natural torus on the quot scheme. In joint work with Linda Chen and Tom Braden, we give a presentation for the equivariant Chow ring of the quot scheme. While we use the formalism of Goresky, Kottwicz, and Macpherson, this formalism does not strictly apply, as the 1-dimensional torus orbits on the quot scheme are not isolated. In this talk, I will describe some of the geometry of the quot scheme and this torus action, and then explain how to extend the formalism of Goresky-Kottwicz-MacPherson to obtain a presentation of the equivariant Chow ring of the quot scheme.

Let K be an imaginary quadratic field. Modular forms for K are related to the cohomology of arithmetic 3-manifolds. This absence of algebraic geometry makes calculating the dimensions of such cohomology groups difficult. Using Galois representations, we give an approach to calculate such spaces in certain explicit towers of congruence subgroups. (Joint work with Nathan Dunfield)

A crossed product of an algebra with a group of automorphisms encodes the group action in a larger algebra, and deformations of these algebras have been of much recent interest. For example, Drinfel'd's graded Hecke algebras are deformations of crossed products of polynomial rings with finite groups. These have been studied for particular types of groups by many people, for example Lusztig (real reflection groups), Ram-Shepler (complex reflection groups), and Etingof-Ginzburg (symplectic reflection groups). We will discuss some of these examples and generalizations.

The market of Derivatives was created thirty years ago when Black and Scholes proposed the risk replication approach. Probability, Analysis and Mathematical Physics contributed to its further development. A new academic field, Mathematical Finance, was born. We will go through its major milestones and we will discuss some of the current challenges in global markets beyond Derivatives.

We will discuss the "Nearest Point Map". First, we will build an intuiton of all the necessary conditions in order to guarantee the existence and uniqueness of such map. Then we will talk about some properties of the function, according to some characteristics of the space and the set itself. We finish with a more general notion: retraction.

I will explain how I solved a 20 year old problem in computational complexity using elementary differential and algebraic geometry. The problem in question was to determine if the best known algorithm for multiplying two by two matrices is the best possible, even after allowing for a small error. I will then explain a more general program for studying the asymptotic exponent for matrix multiplication - the number x such that asymptotically one can multiply n by n matrices using n^x multiplications. (Using naive row-column multiplication technique we know x is at most 3 and much work over the years has brought it close to 2.) This program involves studying higher secant varieties of compact Hermitian symmetric spaces, a subject of interest in its own right. Most of the talk will be elementary, acessible to anyone who knows how to multiply matrices.

Many people are familiar with the Fourier Transform over the real numbers. I will talk about how the Fourier Transform can be applied to finite abelian groups and discuss some of the results that follows from this. I will also talk about the Reed-Muller codes and their connection to the Fourier Transform. Snacks by Silvia.

I will discuss the problem of propagation of bounds for solutions of the nonlinear Boltzmann equation which have one-dimensional (plane-wave) symmetry. Such bounds are sufficient to guarantee uniqueness and propagation of smoothness and to obtain classical solutions if the initial data are smooth. The results are based on a certain regularization property of the averaged "gain" term and on integral inequalities techniques that allow us to restrict the blowup behavior of the pointwise bounds. I will present two approaches: one for large data and one for small data, and discuss their advantages and limitations.

This is a qualifying exam

The talk will describe recent results by Y. Pinchover (Technion, Israel) and the speaker. Liouville properties of elliptic equations on non-compact Riemannian manifolds have been studied extensively for the past 30 years, since the work done by S. T. Yau. Here by the Liouville property we mean finite dimensionality of the space of solutions of a given polynomial growth (in particular, of the space of bounded solutions). This is a non-generic property in the sense that small variations in the operator's coefficients destroy it, so the study must be operator specific. Major results have been achieved in work by T. Colding and W. Minicozzi, as well as by P. Li with co-authors. A parallel activity has existed in complex analysis, where one is interested in finite dimensionality of the spaces of holomorphic functions of a polynomial growth with respect to a suitable Riemannian metric. An interesting case was discovered more than a decade ago by M. Avellaneda and F. Lin and by J. Moser and M. Struwe, who showed that for divergent type $\bf periodic$ elliptic second order equations in $\Bbb R^n$ the Liouville property holds, and the corresponding dimensions are the same as for the Laplace equation. We will present the recent results that provide necessary and sufficient conditions for the Liouville theorem to hold for periodic elliptic equations (including overdeterminate systems), as well as explicit formulas for the dimensions of the spaces of solutions. The answers are given in terms of the notions borrowed from the solid state physics: dispersion relations and Fermi surfaces. New results are also obtained concerning Liouville theorems for holomorphic functions on abelian coverings of compact complex manifolds and for solutions of periodic equations on abelian coverings of finite combinatorial and quantum graphs.

Dilute force-free systems of dissipatively colliding particles - Granular Gases - reveal interesting phenomena such as non-Maxwellian velocity distribution, anomalous diffusion, self-organized vortexes and, at late stages of their evolution, density inhomogeneities and clusters. Similar to molecular gases, Granular Gases may be described by the concepts of classical Statistical Mechanics, such as temperature, velocity distribution function etc. Once initialized with a certain velocity distribution, during their evolution Granular Gases cool down due to inelastic collisions of their particles, characterized by the coefficient of restitution $\varepsilon$. We consider the first step of the evolution of a Granular Gas, where no spatial structures have emerged yet, starting from an initially homogeneous distribution. For simplicity of the mathematical analysis, in most studies the coefficient of restitution is assumed to be constant. It is known, however, that this assumption contradicts experiments and even a dimension analysis. Instead, for realistic materials it is a function on the impact velocity, $\varepsilon=\varepsilon(g)$ It is shown that the model assumptions $\varepsilon=$const and $\varepsilon=\varepsilon(g)$ (viscoelastic particles) lead to qualitatively different results. Whereas the velocity distribution function for the case $\varepsilon=$const. has a simple scaling form, the distribution function of gases of viscoelastic particles depends explicitely on time. This property allows to assign a Granular Gas an age. Moreover, in the long-time behavior, for viscoelastic particles the formation of clusters is suppressed or may occur only as a transient phenomenon. Ref: N.V. Brilliantov and T. P\"oschel: "Kinetic Theory of Granular Gases", Oxford University Press (2004)

An almost complex structure on a real manifold is a smooth choice of isomorphisms, J, on each tangent space, such that J squares to -1. Given a complex manifold, there is a natural almost complex structure induced on the underlying real manifold. The Newlander-Nirenberg theorem decides when the converse is true: that is, given an almost complex structure, when is it induced by an actual complex structure on the manifold? I will give a brief introduction to complex manifolds and sketch a proof of this important theorem.

We will define the usual and p-adic absolute values on the rationals and begin to classify all absolute values on an arbitrary number field. Mahler Measure and height will NOT make an appearance.

The notion of "sameness" pervades all areas of mathematics: homeomorphic surfaces in topology and isomorphic groups in algebra just being two examples. In this talk we will define equidecomposability, which is another notion of sameness, and show that certain sets that are the same in the usual sense are very different in this sense. We will conclude with the Banach-Tarski Theorem that shows that we can "cut" any ONE solid ball into five pieces, apply certain rigid motions to them (basically rotate them), and put them together to get TWO solid balls of exactly the same size as the original!

Lie groups are prevalent in many areas of geometry and physics. The goal of this talk is to introduce some basic theory of Lie groups as well as to give many examples of Lie groups. Some specific topics to be covered are the exponential function, parallelization (say that five times fast) and the Adjoint and Co-Adjoint actions.

John will introduce us to the language of modern algebraic geometry, sheaves and stalks. If nobody stops him, he might even get to schemes.