Boston University

This will be an expository lecture intended for a general mathematical audience to illustrate, through examples, the theme of $p$-adic variation in the classical theory of modular forms.  Classically, modular forms are complex analytic objects, but because their Fourier coefficients are typically integral, it is possible to also do elementary arithmetic with them.   Early examples arose already in the work of Ramanujan.  Today one knows that modular forms encode deep arithmetic information about elliptic curves and Galois representations.  Our main goal will be to illustrate these ideas through simple concrete examples.   

Hall algebras of categories of quiver representations and coherent sheaves

on smooth projective curves over F_q recover interesting

representation-theoretic objects such as quantum groups and their

generalizations. I will define and describe the structure of the Hall

algebra of coherent sheaves on a projective variety over F_1, with P^2 as

the main example. Examples suggest that it should be viewed as a degenerate

q->1 limit of its counterpart over F_q.

We study a dynamic multi-asset economy with private information, a stock and a derivative. There are informed and uninformed investors as well as bounded rational investors trading on noise. The noisy rational expectations equilibrium is obtained in closed form. The equilibrium stock price follows a non-Markovian process, is positive and has stochastic volatility. The derivative cannot be replicated, except at rare endogenous times. At any point in time, the derivative price adds information relative to the stock price, but the pair of prices is less informative than volatility, the residual demand or the history of prices. The rank of the asset span drops at endogenous times causing turbulent trading activity. The effects of financial innovation are discussed. The equilibrium is fully revealing if the derivative is not traded: financial innovation destroys information.

Fix a prime $p$. In this talk, we will discuss the $p$-adic properties of the *coefficients* of the characteristic power series of $U_{p}$ acting on spaces of overconvergent $p$-adic modular forms. These coefficients are, by a theorem of Coleman, power series in the weight variable over $Z_{p}$.  Our first goal will be to show that in tame level one, the simplest case, every coefficient is non-zero mod $p$ and then to give some idea of the (finitely many) roots of each coefficient. The second goal will be to explain how it the previous result fails in higher levels, along with possible salvages. This will include revisiting the tame level one case. The progress we've made has applications, and lends understanding, to recent work being made elsewhere on the geometric structure of the eigencurve "near its boundary". This is joint work with Rob Pollack.

Particle-based stochastic reaction diffusion methods have become a 
popular approach for studying the behavior of cellular processes in 
which both spatial transport and noise in the chemical reaction process 
can be important. While the corresponding deterministic, mean-field 
models given by reaction-diffusion PDEs are well-established, there are 
a plethora of different stochastic models that have been used to study 
biological systems, along with a wide variety of proposed numerical 
solution methods.

In this talk I will motivate our interest in such methods by first 
summarizing several applications we have studied, focusing on how the 
complicated ultrastructure within cells, as reconstructed from X-ray CT 
images, might influence the dynamics of cellular processes. I will then 
introduce our attempt to rectify the major drawback to one of the most 
popular particle-based stochastic reaction-diffusion models, the lattice 
reaction-diffusion master equation (RDME). We propose a modified version 
of the RDME that converges in the continuum limit that the lattice 
spacing approaches zero to an appropriate spatially-continuous model. 
Time-permitting, I will discuss several questions related to calibrating 
parameters in the underlying spatially-continuous model.

Using Faltings' theory of the Hodge-Tate sequence of an abelian scheme we construct certain sheaves $\Omega^\kappa$, where $\kappa$ is a not-necessarily integral weight, over formal subschemes of modular varieties over which the canonical subgroup exists.   These sheaves generalize the integral powers, $\omega^k$, of the sheaf $\omega$ of relative differentials on a modular curve.   Global sections of $\Omega^\kappa$ provide geometric realizations of overconvergent automorphic forms of non-integral weight.  Applications of this approach to the theory of $p$-adic Hilbert modular forms will be given.   This is joint work with Fabrizio Andreotti and Adrian Iovita.

In this talk, we elaborate on the notions of no-free-lunch that have proved essential in the theory of financial mathematics---most notably, arbitrage of the first kind. Focus will be given in most recent developments. The precise connections with existence of deflators, numeraires and pricing measures are explained, as well as the consequences that these notions have in the existence of bubbles and the valuation of illiquid assets in the market.

Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.

As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.

Hedge fund managers receive a large fraction of their funds' gains, in addition to the small fraction of funds' assets typical of mutual funds. The additional fee is paid only when the fund exceeds its previous maximum - the high-water mark. The most common scheme is 20 percent of the fund profits + 2 percent of assets.

To understand the incentives implied by these fees, we solve the portfolio choice problem of a manager with Constant Relative Risk Aversion and a Long Horizon, who maximizes the utility from future fees.

With constant investment opportunities, and in the absence of fixed fees, the optimal portfolio is constant. It coincides with the portfolio of an investor with a different risk aversion, which depends on the manager's risk aversion and on the size of the fees. This portfolio is also related to that of an investor facing drawdown constraints. The combination of both fees leads to a more complex solution.

The model involves a stochastic differential equation involving the running maximum of the solution, which is related to perturbed Brownian Motions. The solution of the control problem employs a verification theorem which relies on asymptotic properties of positive local martingales.

Joint work with Jan Obloj.