In this talk I will present recent results on ergodicity of Markov semigroups in large dimensional spaces including interacting Levy type systems as well as some R-D models.
Abstract: In this talk, I will give an overview of the new developments in the AdS_4/CFT_3 correspondence. I will present in detail an N=6 Chern-Simons-matter theory with gauge group U(N) x U(N) that is dual to N M2 branes in the orbifold C^4/Z_k. This theory can be derived from a construction involving D3 branes intersecting (p,q) fivebranes. I will also discuss various quantum mechanical aspects of this theory, including an enhancement of its supersymmetry algebra at Chern-Simons levels 1 and 2, and some novel phenomenon that arise in the U(N) x U(M) theory dual to configurations with N-M fractional branes. A generalization to N=3 CSM theories dual to AdS_4 x M_7, where M_7 is a 3-Sasakian 7-manifold, will be explained. The seminar will be based primarily on Aharony, Bergman, DJ, Maldacena; Aharony, Bergman, DJ; DJ, Tomasiello.
We prove existence of equilibrium in a continuous-time securities market in which the securities are potentially dynamically complete: the number of securities is at least one more than the number of independent sources of uncertainty. We prove that dynamic completeness of the candidate equilibrium price process follows from mild exogenous assumptions on the economic primitives of the model. Our result is universal, rather than generic: dynamic completeness of the candidate equilibrium price process and existence of equilibrium follow from the way information is revealed in a Brownian filtration, and from a mild exogenous nondegeneracy condition on the terminal security dividends. The nondegeneracy condition, which requires that finding one point at which a determinant of a Jacobian matrix of dividends is nonzero, is very easy to check. We find that the equilibrium prices, consumptions, and trading strategies are well-behaved functions of the stochastic process describing the evolution of information.
We prove that equilibria of discrete approximations converge to equilibria of the continuous-time economy
We show how the circle method with a suitably chosen Gaussian weight can be used to count unweighted zeros of polynomials. Tschinkel's problem asks for the density of solutions to Diophantine equations with S-unit and integral variables.
We consider the symmetric group S_n of degree n and an algebraically
closed field F of prime characteristic p.
As is well-known, many representation theoretical objects of S_n
possess concrete combinatorial descriptions such as the simple
FS_n-modules through their parametrization by the p-regular partitions of n,
or the blocks of FS_n through their characterization in terms of p-cores
and p-weights. In contrast, though closely related to blocks and their
defect groups, the vertices of the simple FS_n-modules are rather poorly
understood. Currently one is far from knowing what these vertices look
like in general and whether they could be characterized combinatorially
as well.
In this talk I will refer to some theoretical and computational
approaches towards the determination of vertices of simple FS_n-modules.
Moreover, I will present some results concerning the vertices of
certain classes of simple FS_n-modules such as the ones labelled by
hook partitions or two part partitions, and will state a series of
general open questions and conjectures.
The advent of multicore processors and other technologies like Graphical Processing Units (GPU) will considerably influence future research in High Performance Computing.
To take advantage of these architectures in dense linear algebra operations, new algorithms are
proposed that use finer granularity and minimize synchronization points.
After presenting some of these algorithms, we address the issue of pivoting and investigate randomization techniques to avoid pivoting in some cases.
In the particular case of GPUs, we show how linear algebra operations can be enhanced using
hybrid CPU-GPU calculations and mixed precision algorithms.
In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that the time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.
I show that the expansion of the real field by a total Pfaffian chain is model complete in a language with symbols for the functions in the chain, the exponential and all real constants. In particular, the expansion of the reals by all total Pfaffian functions is model complete.
The homological Hall lemma is a topological tool that has recently been used to derive Hall type theorems for systems of disjoint representatives in hypergraphs.
After outlining the general method, we.ll describe one such theorem in some detail. The main ingredients in the proof are:
1) A relation between the spectral gap of a graph and the topological connectivity of its flag complex.
2) A new graph domination parameter defined via certain vector representations of the graph.
Joint work with R. Aharoni and E. Berger
We classify 2-dimensional polygonal complexes that are simply connected, platonic (in the sense that they admit a flag-transitive group of symmetries) and simple (in the sense that each vertex link is a complete graph). These are a natural generalization of the 2-skeleta of simple polytopes.
Our classification is complete except for some existence questions for complexes made from squares and pentagons.
(Joint with Tadeusz Januszkiewicz, Raciel Valle and Roger Vogeler.)
In this talk, we present an extension of the theory of rough paths to partial differential equations. This allows a robust approach to stochastic partial differential equations, and in particular we can replace Brownian motion by more general Gaussian and Markovian noise. Support theorems and large deviation statements all become easy corollaries of the corresponding statements of the driving process. This is joint work with Peter Friz in Cambridge.
A polygonal complex $X$ is Platonic if its automorphism group $G$ acts transitively on the flags (vertex, edge, face) in $X$. Compact examples include the boundaries of Platonic solids. Noncompact examples $X$ with nonpositive curvature (in an appropriate sense) and three polygons meeting at each edge were classified by \'Swi\c{a}tkowski, who also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete. For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete. A Fuchsian building is a two-dimensional also determined when the group $G=Aut(X)$, equipped with the compact-open topology, is nondiscrete. For example, there is a unique $X$ with the link of each vertex the Petersen graph, and in this case $G$ is nondiscrete. A Fuchsian building is a two-dimensional hyperbolic building. We study lattices in automorphism groups of Platonic complexes and Fuchsian buildings. Using similar methods for both cases, we construct uniform and nonuniform lattices in $G=Aut(X)$. We also show that for some $X$ the set of covolumes of lattices in $G$ is nondiscrete, and that $G$ admits lattices which are not finitely generated. In fact our results apply to the larger class of Davis complexes, which includes examples in dimension > 2.
It is a classical result that the Alexander polynomial of a fibered knot has to be monic. But in general the converse does not hold, i.e. the Alexander polynomial does not detect fibered knots. We will show that the collection of all twisted Alexander polynomials (which are a natural generalization of the ordinary Alexander polynomial) detect fibered 3-manifolds.
As a corollary it follows that given a 3-manifold N the product S1 x N is symplectic if and only if N is fibered.
A new general approach to optimal stopping problems in L\'evy models, regime switching L\'evy models and L\'evy models with stochastic volatility and stochastic interest rate is developed. For perpetual options, explicit solutions are found, for options with finite time horizon, time discretization is used, and explicit solutions are derived for resulting sequences of perpetual options.
The main building block is the option to abandon a monotone payoff stream. The optimal exercise boundary is found using the operator form of the Wiener-Hopf method, which is standard in analysis, and interpretation of the factors as {\em expected present value operators} (EPV-operators) under supremum and infimum processes.
Other types of options are reduced to the option to abandon a monotone stream. For regime-switching models, an additional ingredient is an efficient iteration procedure.
L\'evy models with stochastic volatility and/or stochastic interest rate are reduced to regime switching models using discretization of the state space for additional factors. The efficiency of the method for 2 factor L\'evy models with jumps and for 3-factor Heston model with stochastic interest rate is demonstrated. The method is much faster than Monte-Carlo methods and can be a viable alternative to Monte Carlo method as a general method for 2-3 factor models.
Joint work of Svetlana Boyarchenko,University of Texas at Austin and Sergei Levendorski\v{i},
University of Leicester
We discuss the valuation problem for a broad spectrum of derivatives, especially in Levy driven models. The key idea in this approach is to separate from the computational point of view the role of the two ingredients which are the payoff function and the driving process for the underlying quantity. Conditions under which valuation formulae based on Fourier and Laplace transforms hold in a general framework are analyzed. An interesting interplay between the properties of the payoff function and the driving process arises. We also derive the analytically extended characteristic function of the supremum and the infimum processes derived from a Levy process. Putting the different pieces together, we can price lookback and one-touch options in Levy driven models, as well as options on the minimum and maximum of several assets.
The large-time behavior of solutions to Burgers equation with
small viscosity is described using invariant manifolds. In particular,
a geometric explanation is provided for a phenomenon known as
metastability,which in the present context means that
solutions spend a very long time near the family of solutions known as
diffusive N-waves before finally converging to a stable self-similar
diffusion wave. More precisely, it is shown that in terms of
similarity, or scaling, variables in an algebraically weighted $L^2$
space, the self-similar diffusion waves correspond to a one-dimensional
global center manifold of stationary solutions. Through each of these
fixed points there exists a one-dimensional, global, attractive,
invariant manifold corresponding to the diffusive N-waves. Thus,
metastability corresponds to a fast transient in which solutions
approach this ``metastable" manifold of diffusive N-waves, followed by
a slow decay along this manifold, and, finally, convergence to the
self-similar diffusion wave.
(joint work with E. Hrushovski and A. Pillay)
If G is a definably compact, connected group definable in an o-minimal structure then, as is known, G/Z(G) is semisimple (no infinite normal abelian subgroup).
We show, that in every o-minimal expansion of an ordered group:
If G is a definably connected central extension of a semisimple group then it is bi-intepretable, over parameters, with the two-sorted structure (G/Z(G), Z(G)). Many corollaries follow for definably connected, definably compact G.
Here are two:
1. (G,.) is elementarily equivalent to a compact, connected real Lie group of the same dimension.
2. G can be written as an almost direct product of Z(G) and [G,G], and this last group is definable as well (note that in general [G,G] is a countable union of definable sets, thus not necessarily definable).
In 1989, Happel raised the following question: if the Hochschild cohomology
groups of a finite dimensional algebra vanish in high degrees, then does the
algebra have finite global dimension? This was answered negatively in a
paper by Buchweitz, Green, Madsen and Solberg. However, the Hochschild
homology version of Happel's question, a conjecture given by Han, is open.
We give a positive answer to this conjecture for local graded algebras,
Koszul algebras and cellular algebras. The proof uses Igusa's formula for
relating the Euler characteristic of relative cyclic homology to the graded
Cartan determinant. This is joint work with Dag Madsen.
This is not intended to be a systematic History, but a selection of highlights, with some digressions, including:
The early days of the Computing Lab;
How the coming of the Computer changed some of the ways we do Computation;
A problem from the Study Groups;
Influence of the computing environment (hardware and software);
Convergence analysis for the heat equation, then and now.
A function $u: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is one-homogeneous if $u(ax)=au(x)$ for any positive real number $a$ and all $x$ in $\R^{n}$. Phillips(2002) showed that in two dimensions such a function cannot solve an elliptic system in divergence form, in contrast to the situation in higher dimensions where various authors have constructed one-homogeneous minimizers of regular variational problems. This talk will discuss an extension of Phillips's 2002 result to $x-$dependent systems. Some specific one-homogeneous solutions will be constructed in order to show that certain of the hypotheses of the extension of the Phillips result can't be dropped. The method used in the construction is related to nonlinear elasticity in that it depends crucially on polyconvex functions $f$ with the property that $f(A) \to \infty$ as $\det A \to 0$.
We identify a natural way of ordering functions, which we call the interval dominance order, and show that this concept is useful in the theory of monotone comparative statics and also in statistical decision theory. This ordering on functions is weaker than the standard one based on the single crossing property (Milgrom and Shannon, 1994) and so our monotone comparative statics results apply in some settings where the single crossing property does not hold. For example, they are useful when examining the comparative statics of optimal stopping time problems. We also show that certain basic results in statistical decision theory which are important in economics - specifically, the complete class theorem of Karlin and Rubin (1956) and the results connected with Lehmann's (1988) concept of informativeness – generalize to payoff functions that obey the interval dominance order.
How can one guarantee the presence of an oriented four-cycle in an oriented graph G? We shall see, that one way in which this can be done, is to demand that G contains no large `biased. subgraphs; where a `biased. subgraph simply means a subgraph whose orientation exhibits a strong bias in one direction.
Furthermore, we discuss the concept of biased subgraphs from another standpoint, asking: how can an oriented graph best avoid containing large biased subgraphs? Do random oriented graphs give the best examples? The talk is partially based on joint work with Omid Amini and Florian Huc.
There is a non-degenerate 2-form on S^6, which is compatible with the almost complex structure that S^6 inherits from its inclusion in the imaginary octonions. Even though this 2-form is not closed, we may still define Lagrangian submanifolds. Surprisingly, they are automatically minimal and are related to calibrated geometry. The focus of this talk will be on the Lagrangian submanifolds of S^6 which are fibered by geodesic circles over a surface. I will describe an explicit classification of these submanifolds using a family of Weierstrass formulae.