We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren.
We investigate in this talk the average root number (i.e. sign of the functional equa- tion) of one-parameter families of elliptic curves (i.e elliptic curves over Q(t), or elliptic surfaces over Q). For most one-parameter families of elliptic curves, the aver- age root number is predicted to be 0. Helfgott showed that under Chowla’s conjecture and the square-free conjecture, the average root number is 0 unless the curve has no place of multiplicative r eduction over Q(t). We then build families of elliptic curves with no place of multiplicative reduction, and compute the average root number of the families. Some families have periodic root number, giving a rational average, and some other families have an average root number which is expressed as an infinite Euler product. We also show several density results for the average root number of families of elliptic curves, and exhibit some surprising examples, for example, non- isotrivial families of elliptic curves with rank r over Q(t) and average root number −(−1)r, which were not found in previous literature.
Abstract: The combinatorial properties of large cardinals tend to clash with those satisfied by G\"odel's constructible universe, especially the square property (denoted $\square_\kappa$) isolated by Jensen in the seventies. Strong cardinal axioms refute the existence of square, but it is possible with some fine-tuning to produce models that exhibit some large cardinal properties together with weakenings of square. In this talk we will exhibit some results along these lines and will outline the techniques used to produce them.
A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.
A number of diverse biological systems involve diffusion in a randomly switching environment. For example, such processes arise in brain biochemistry, insect respiration, intracellular trafficking, and biochemical reaction kinetics. These processes present interesting mathematical subtleties as they combine two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment.
In this talk, we will demonstrate that these systems (a) arise naturally in several biological applications and (b) are mathematically rich. Special attention will be given to establishing mathematical connections between these classes of stochastic processes. In particular, we will use these connections to study certain random PDEs by analyzing the local time of a Brownian particle in a random environment.
Abstract:
In this talk, we discuss a compactness result on the space of compact Lagrangian self-shrinkers in R^4. When the area is bounded above uniformly, we prove that the entropy for the Lagrangian self-shrinking tori can only take finitely many values; this is done by deriving a Lojasiewicz-Simon type gradient inequality for the branched conformal self-shrinking tori. Using the finiteness of entropy values, we construct a piecewise Lagrangian mean curvature flow for Lagrangian immersed tori in R^4, along which the Lagrangian condition is preserved, area is decreasing, and the type I singularities that are compact with a fixed area upper bound can be perturbed away in finite steps. This is a Lagrangian version of the construction for embedded surfaces in R^3 by Colding and Minicozzi.This is a joint work with Jingyi Chen.
Reaction-diffusion models are widely used to study spatially-extended chemical reaction systems. The input parameters on which these models are predicated are experimentally derived. In order to understand how the dynamics of a reaction-diffusion model are affected by changes in input parameters, efficient methods for computing parametric sensitivities are required. In this talk, we focus on compartment-based stochastic models of spatially-extended chemical reaction systems, which partition the computational domain into voxels. Parametric sensitivities are often calculated using Monte Carlo techniques that are typically computationally expensive; however, variance reduction techniques can decrease the number of Monte Carlo simulations required. By exploiting the characteristic dynamics of spatially-extended reaction networks, we are able to adapt existing finite difference schemes to robustly estimate parametric sensitivities in a spatially-extended network. Our methods are tested for functionality and reliability in a range of different scenarios.
Lyapunov exponents measure the rate of exponential expansion or contraction in a dynamical system. For a given nonlinear dynamical systems, it turns out to be a very hard problem to prove the nonvanishing of Lyapunov exponents or even to estimate them quantitatively. For a given prototypical two-dimensional map, we introduce a small random perturbation to the map and give with relative ease the quantitative estimate of the Lyapunov exponents versus the size of the randomness. This is a joint work with A. Blumenthal and L.-S. Young.