I shall give a gentle introduction to the cohomology of finite groups from the point of view of algebra, topology, group actions and number theory
One of the popular approaches to valuing options in incomplete financial markets is exponential utility indifference valuation. The value process for the corresponding stochastic control problem can often be described by a backward stochastic differential equation (BSDE). This is very useful for proving theoretical properties, but actually solving these equations is difficult. With the goal of obtaining more information, we therefore study BSDE transformations that allow us to derive upper and/or lower bounds, in terms of solutions of other BSDEs, that can be computed more explicitly. These ideas and techniques also are of independent interest for BSDE theory.
This is joint work with Christoph Frei and Semyon Malamud.
This purpose of this talk will be to introduce the idea that the spectrum of nonstandard models of a ``standard''
algebraic object can be used much like a microscope with which one may perceive and codify irrationality invisible within the standard model.
This will be done by examining the following three themes:
\item {\it Algebraic topology of foliated spaces} We define the fundamental germ, a generalization of fundamental group for foliations, and show that the fundamental germ of a foliation that covers a manifold $M$ is detected (as a substructure) by a nonstandard model of the fundamental group of $M$.
\item {\it Real algebraic number theory.} We introduce the group $(r)$ of diophantine approximations of a real number $r$, a subgroup of a nonstandard model of the integers, and show how $(r)$ gives rise to a notion of principal ideal generated by $r$.
The general linear group $GL(2, \mathbb{Z})$ plays here the role of a Galois group, permuting the real ideals of equivalent real numbers.
\item {\it Modular invariants of a Noncommutative Torus.} We use the fundamental germ of the associated Kronecker foliation as a lattice and define the notion of Eisenstein series, Weierstrass function, Weierstrass equation and j-invariant.
In the context of the linear theory of elasticity with eigenstrains, the radiated fields,
including inertia effects, and the energy-release rates (“driving forces”) of a spherically
expanding and a plane inclusion with constant dilatational eigenstrains are
calculated. The fields of a plane boundary with dilatational eigenstrain moving
from rest in general motion are calculated by a limiting process from the spherical
ones, as the radius tends to infinity, which yield the time-dependent tractions
that need to be applied on the lateral boundaries for the global problem to be
well-posed. The energy-release rate required to move the plane inclusion boundary
(and to create a new volume of eigenstrain) in general motion is obtained here for
a superposed loading of a homogeneous uniaxial tensile stress. This provides the
relation of the applied stress to the boundary velocity through the energy-rate balance
equation, yielding the “equation of motion” (or “kinetic relation”) of the plane
boundary under external tensile axial loading. This energy-rate balance expression
is the counterpart to the Peach-Koehler force on a dislocation plus the “self-force”
of the moving dislocation.
In the first part of the talk we briefly describe an algorithm which computes a relative algebraic K-group as an abstract abelian group. We also show how this representation can be used to do computations in these groups. This is joint work with Steve Wilson.
Our motivation for this project originates from the study of the Equivariant Tamagawa Number Conjecture which is formulated as an equality of an analytic and an algebraic element in a relative algebraic K-group. As a first application we give some numerical evidence for ETNC in the case of the base change of an elliptic curve defined over the rational numbers. In this special case ETNC is an equivariant version of the Birch and Swinnerton-Dyer conjecture
For the task of solving PDEs, finite difference (FD) methods are particularly easy to implement. Finite element (FE) methods are more flexible geometrically, but tend to be difficult to make very accurate. Pseudospectral (PS) methods can be seen as a limit of FD methods if one keeps on increasing their order of accuracy. They are extremely effective in many situations, but this strength comes at the price of very severe geometric restrictions. A more standard introduction to PS methods (rather than via FD methods of increasing orders of accuracy) is in terms of expansions in orthogonal functions (such as Fourier, Chebyshev, etc.).
Radial basis functions (RBFs) were first proposed around 1970 as a tool for interpolating scattered data. Since then, both our knowledge about them and their range of applications have grown tremendously. In the context of solving PDEs, we can see the RBF approach as a major generalization of PS methods, abandoning the orthogonality of the basis functions and in return obtaining much improved simplicity and flexibility. Spectral accuracy becomes now easily available also when using completely unstructured meshes, permitting local node refinements in critical areas. A very counterintuitive parameter range (making all the RBFs very flat) turns out to be of special interest. Computational cost and numerical stability were initially seen as serious difficulties, but major progress have recently been made also in these areas.
In this talk, we try to construct a dynamical model for the basket credit products in the credit market under the structural-model framework. We use the particle representation for the firms' asset value and investigate the evolution of the empirical measure of the particle system. By proving the convergence of the empirical measure we can achieve a stochastic PDE which is satisfied by the density of the limit empirical measure and also give an explicit formula for the default proportion at any time t. Furthermore, the dynamics of the underlying firms' asset values can be assumed to be either driven by Brownian motions or more general Levy processes, or even have some interactive effects among the particles. This is a joint work with Dr. Ben Hambly.
The Cosserat brothers’ ingenuous and powerful idea (presented in several papers in the Comptes Rendus at the turn of the 20th century) of solving elasticity problems by considering the homogeneous Navier equations as an eigenvalue problem is presented. The theory was taken up by Mikhlin in the 1970’s who rigorously studied it in the context of spectral analysis of pde’s, and who also presented a representation theorem for the solution of the boundary-value problems of displacement and traction in elasticity as a convergent series of the ( orthogonal and complete in the Sobolev space H1) Cosserat eigenfunctions. The feature of this representation is that the dependence of the solution on geometry, material constants and loading is provided explicitly. Recent work by the author and co-workers obtains the set of eigenfunctions for the spherical shell and compares them with the Cosserat expressions, and further explores applications and a new variational principle. In cases that the loading is orthogonal to some of the eigenfunctions, the form of the solution can be greatly simplified. Applications, in addition to elasticity theory, include thermoelasticity, poroelesticity, thermo-viscoelasticity, and incompressible Stokes flow; several examples are presented, with comparisons to known solutions, or new solutions.
Many classical results and conjectures in representation theory of finite groups (such as
theorems of Thompson, Ito, Michler, the McKay conjecture, ...) address the influence of global properties of representations of a finite group G on its p-local structure. It turns out that several of them also admit real, resp. rational, versions, where one replaces the set of all complex representations of G by the much smaller subset of real, resp. rational, representations. In this talk we will discuss some of these results, recently obtained by the speaker and his collaborators. We will also discuss recent progress on the Brauer height zero conjecture for 2-blocks of maximal defect.
I will survey the recent work of Haskins and myself constructing new special Lagrangian cones in ${\mathbb C}^n$
for all $n\ge3$ by gluing methods. The link (intersection with the unit sphere ${\cal S}^{2n-1}$) of a special Lagrangian cone is a special Legendrian $(n-1)$-submanifold. I will start by reviewing the geometry of the building blocks used. They are rotationally invariant under the action of $SO(p)\times SO(q)$ ($p+q=n$) special Legendrian $(n-1)$-submanifolds of ${\cal S}^{2n-1}$. These we fuse (when $p=1$, $p=q$) to obtain more complicated topologies. The submanifolds obtained are perturbed to satisfy the special Legendrian condition (and their cones therefore the special Lagrangian condition) by solving the relevant PDE. This involves understanding the linearized operator and its small eigenvalues, and also ensuring appropriate decay for the solutions.
The Gilbert model of a random geometric graph is the following: place points at random in a (two-dimensional) square box and join two if they are within distance $r$ of each other. For any standard graph property (e.g. connectedness) we can ask whether the graph is likely to have this property. If the property is monotone we can view the model as a process where we place our points and then increase $r$ until the property appears. In this talk we consider the property that the graph has a Hamilton cycle. It is obvious that a necessary condition for the existence of a Hamilton cycle is that the graph be 2-connected. We prove that, for asymptotically almost all collections of points, this is a sufficient condition: that is, the smallest $r$ for which the graph has a Hamilton cycle is exactly the smallest $r$ for which the graph is 2-connected. This work is joint work with Jozsef Balogh and B\'ela Bollob\'as
This is the second (of two) talks concerning the Birch--Swinnerton-Dyer Conjecture.
In this talk we will review some recent results on the long-time/large-scale, weak-friction asymptotics for the one dimensional Langevin equation with a periodic potential. First we show that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We also show that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. Furthermore we prove that the same result is valid for a whole one parameter family of space/time rescalings. We also present a new numerical method for calculating the diffusion coefficient and we use it to study the multidimensional problem and the problem of Brownian motion in a tilted periodic potential.