Schedule-at-a-Glance

38 PRIMA 2013 Abstractsthe noise is realized as a subordination of the Brownianmotion. In particular, the estimates are sharp forα-stable type noises. To derive these estimates, a newderivative formula is established for the semigroup by usingthe Malliavin calculus and a finite-jump approximationargument.Viscosity solutions of path dependent PDEsJianfeng ZhangUniversity of Southern California, USAjianfenz@usc.eduPath Dependent PDEs (PPDEs, for short) is a convenienttool to characterize the value functions of varioustypes of stochastic control problems in non- Markovianframework. Its typical examples include Backward SDEs(semi- linear PPDEs), second order BSDEs (path dependentHJB equations), path dependent Bellman-Isaacsequations, and Backward Stochastic PDEs. PPDEs canrarely have classical solutions. In this talk we shall proposea notion of viscosity solutions for PPDEs and establishits wellposedness. Our definition relies heavily on theFunctional Ito formula initiated by Dupire. Unlike theviscosity theory of standard PDEs, the main difficultyin path dependent case is that the state space is not locallycompact. To overcome such difficulty, we replacethe pointwise maximization in standard theory with anoptimal stopping problem under Peng’s nonlinear expectation.The talk will be based on joint works with NizarTouzi, and Ibrahim Ekren, Christian Keller, Triet Pham.Special Session 19Representation Theory and CategorificationGeometric reciprocity for algebraic tori overnon-Archimedean local fieldsClifton CunninghamUniversity of Calgary, Canadacunning@math.ucalgary.caRecent work with David Roe has shown that the geometrizationof admissible characters of T (K), where Tis an algebraic torus over a non-Archimedean field K, isachieved by introducing the tensor category of charactersheaves on the Greenberg transform of the Neron modelof T . On the other hand, earlier worth with Achar, Kamgarpourand Salmasian, closely related to ideas of Vogan,shows that the geometrization of Langlands parametersfor T is achieved by passing to the category of equivariantperverse sheaves on an ind-variety formed from L T .In this talk I will use the class field theory of Serre andHazewinkel to relate these two categories.Schubert calculus and cohomology of LiegroupsHaibao DuanInstitute of Mathematics, Chinese Academy of Sciences,Chinadhb@math.ac.cnThe problem of determining the cohomology of Lie groupswas raised by E. Cartan in 1929, and has been a focus ofalgebraic topology for the fundamental roles of Lie groupsplaying in geometry and topology. On the other handSchubert calculus begun with the intersection theory ofthe 19 century, and clarifying this calculus had been amajor theme of the 20 century algebraic geometry.We bring a connection between these two topics bothwith distinguished historical backgrounds, and demonstratehow Schubert calculus is extended as to give anexplicit and unified construction of the integral cohomologyrings of all compact and 1-connected Lie groups.Vanishing properties of Jack polynomialsStephen GriffethUniversidad de Talca, Chilesgriffeth@inst-mat.utalca.cl(Joint work with Christine Berkesch and Steven Sam)We describe some interactions between representationsof rational Cherednik algebra, especially unitary representations,and questions arising in combinatorial commutativealgebra and mathematical physics. These questionsall have to do with highly symmetric linear subspacearrangements and the ideals of polynomials vanishingon them. Among other things, we will indicate howto use representation theory to prove a number of conjecturesof Bernevig and Haldane on the order of vanishingof certain Jack polynomials along these arrangements,and present our own conjecture describing a Bernstein-Gelfand-Gelfand type resolutions of unitary representationsof the Cherednik algebra. Any such resolution isautomatically a minimal free resolution for the ideal ofthe corresponding linear arrangement, so we predict acombiantorial formula for the graded equivariant Bettinumbers of these ideals.The modular generalized Springer correspondenceAnthony HendersonUniversity of Sydney, Australiaanthony.henderson@sydney.edu.auGiven a connected reductive algebraic group G with Weylgroup W , the Springer correspondence realizes the categoryof representations of W as a quotient of the categoryof G-equivariant perverse sheaves on the nilpotentcone. In the original definition, the representations andsheaves were over a field of characteristic zero, but wehave recently shown that the same formalism works withmodular coefficients, where the categories are no longersemisimple. In the characteristic-zero case, Lusztig defineda generalized Springer correspondence to interpretthe whole category of G-equivariant perverse sheaves onthe nilpotent cone in terms of representations of relativeWeyl groups. We define and determine a modular generalizedSpringer correspondence in the case G = GL(n).This gives a geometric explanation for the fact that, inthe modular case, the category of modules over the Schuralgebra can be obtained by successive recollements of categoriesof representations of suitable products of symmetricgroups.Knot invariants and their categorifications viaHowe dualityAaron LaudaUniversity of Southern California, USAlauda@usc.eduIt is a well understood story that one can extract linkinvariants associated to simple Lie algebras. These invariantsare called Reshetikhin-Turaev invariants and thefamous Jones polynomial is the simplest example. Kauffmanshowed that the Jones polynomial could be describedvery simply by replacing crossings in a knot diagram byvarious smoothings. In this talk we will explain Cautis-Kamnitzer-Licata’s simple new approach to understandingthese invariants using basic representation theory andthe quantum Weyl group action. Their approach is basedon a version of Howe duality for exterior algebras calledskew-Howe duality. Even the graphical (or skein theory)description of these invariants can be recovered in an elementaryway from this data. The advantage of this approachis that it suggests a ‘categorification’ where knothomology theories arise in an elementary way from higherrepresentation theory and the structure of categorifiedquantum groups.Center at the critical level and commutativesubalgebras