Avraham (Rami) Aizenbud
אברהם (רמי) איזנבוד
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Avraham (Rami) Aizenbud |
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אברהם (רמי) איזנבוד |
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Seminar on Representation Theory and Algebraic Geometry
Fall 2004-- Summer 2005
Name: Ilya Tyomkin, Weizmann Title: Severi varieties and moduli spaces of curves in arbitrary characteristic.
Name: Shamgar Gurevich, Tel-Aviv Title: Proof of the Rudnick-Kurlberg conjecture.
Name: Istvan Heckenberger, WIS Title:Introduction to Nichols algebras. Abstract: Nichols algebras naturally appear in the classification of pointed Hopf algebras. Prominent examples are the symmetric and exterior algebras of a vector space, and the quantized enveloping algebras U_q(n^+). In this talk several results on the structure of Nichols algebras are introduced and the relation of Nichols algebras and semisimple Lie algebras is pointed out. Recently found tools for the classification of Nichols algebras are presented. Name: Daniela Nikolova, Institute of Mathematics and Informatics, Bulgaria Title: On the characterization of finite soluble groups satisfying some two-variable commutator laws. Abstract: It is a well-known fact that a finite group is nilpotent iff it satisfies an Engel law of some length. The subject of our recent research has been to establish a similar characterization for finite soluble groups. It is sufficient to check the problem on two-generated subgroups. For this purpose a sequence of two-variable commutator formulae are investigated (Conjecture of B.I.Plotkin). The solubility is checked by examining a counter-example of least possible order, i.e. we attempt to show that in minimal simple groups there are no such identities. Thus, we solve equations in matrix groups, especially for the generic case G=PSL(2,p) involving computer calculations. The solution of such equations is connected to some well-known problems in finite matrix groups, as for example the Ore conjecture. This is a joint work with E.B. Plotkin. Name: Leonid Fel, Technion Title: Frobenius Problem for 3-dim Semigroups. Abstract:ps-file Name: Dennis Gaitsgory, Chicago Title: On the construction of Wakimoto modules for affine algebras. Name: Maria Gorelik, Boris Noyvert Title: Examples of vertex algebras In this talk we will present simplest examples of vertex algebras which are subject of the Kac's lecture "What is a Vertex algebra and why?". Kac's lecture will be given in the framework of Distinguished Lecture Series on January, 18. No preliminary knowledge, except for linear algebra, is required.
Name: Victor Kac, MIT Title: An introduction to W-algebras Abstract: In the first lecture various approaches to vertex algebras are explained. In the second lecture I explain how the quantum Hamiltonian reduction in the framework of vertex algebras is used in representation theory.
Name: Victor Kac, MIT Title: What is a Vertex algebra and why? Abstract: Vertex algebra can be viewed as the simplest non-trivial example of a Quantum Field Theory, when the space-time is the formal neighbourhood of 0 on the line. Nevertheless this turned out to be a very rich mathematical structure. Several other points of view at this structure will be explained. At the end of the talk I will explain how the quantum Hamiltonian reduction in the framework of vertex algebras is used in representation theory.
Name: Dr. Michael Schein, Harvard University Title: Weights of Galois representations associated to Hilbert modular forms. Abstract: Let $F$ be a totally real field, $p$ a rational prime unramified in $F$, and $\wp$ a place of $F$ over $p$. Let $\rho:Gal(\bar F/F) \to GL_2(\bar{\fline}_p)$ be a two-dimensional mod $p$ Galois representation which is assumed to be modular of some weight and whose restriction to a decomposition subgroup at $\wp$ is irreducible. We specify a set of weights, determined by the restriction of $\rho$ to the inertia subgroup at $\wp$, which contains all the weights for which $\rho$ is modular (and, conjecturally, nothing else). This proves a special case of a conjecture of F. Diamond, which provides an analogue of Serre's epsilon conjecture for Hilbert modular forms mod $p$.
Name: Issai Kantor, Lund Title: Universal graded algebra Lie and related topics. The universal graded Lie algebra V^n=...+V_k+...+V_{-1}+V_{-1}+V_{0}+V_{1}+...+V_{k}+.. . is defined by n=dimV_{-1}. The subalgebra V_{-}=...V_{-k}+...+V_{-1} is a free Lie algebra generated by subspace V_{-1} In the positive direction the subspace V_{k} consists of all k+1-linear mappings of V_{-1} to V_{-1} with brackets defined in a special way. The universality means the that any graded Lie algebra ...+U_k+...+U_{-1}+U_{-1}+U_{0}+U_{1}+...+U_{k}+.. . with some natural restrictions and such that n=dimU_{-1} can be constructed by embedding in V^n followed by factorisation on some negative ideal. Many results on graded Lie algebras can be proved with the help of this constructon in a general way. Also many classes of simple Lie algebras can be considered with the help of this construction , particularly the algebras defined by Cartan matrix. One another application of the universal algebra is the mapping L which associate to any algebraical system, particularly with an algebra A of dimension n, a graded Lie algebra L(A) as a subalgebra of the universal algebra V^n generated by subspace V_{-1} and element A. (It has a sense because subspace V_{1} consists of all possible n-dimensional algebras).
Name: Issai Kantor, Lund Title: An introduction to Supermathematics Abstract: The Supermathematics is a branch of Mathematics which was started approximately 30 years ago in connection with existence of two types of elementary particles: bosons and fermions. While behavior of bosons could be described in ordinary variables (called in Supermathematics ``even" variables) for fermions one need so called ``odd" variables. Using both types of variables one can consider the Schrodinger and Pauli equations as one eqution in which both types of variables participate together. This situation has influence on many parts of Mathematics. The influes sometimes called a superisation of Mathematics. In the talk an elementary introduction will be given to Supermathematics mostly in terms of linear algebra.
Name: Amnon Yekutieli, Beer-Sheva Title: Deformation Quantization in Algebraic Geometry
Name: Maria Gorelik Title: On the character of the simple module of a generic highest weight at the critical level.
Name: Anthony Joseph Title:Eigenslices for Parabolic Actions
Name: Eitan Sayag Title:Explicit constructions of automorphic forms. Abstract: Motivated by arithmetic (class field theory) and harmonic analysis (spectral theory of adelic groups) R.P. Langlands envisioned conjectural relationships between automorphic representations on different groups, known as functorial lift/transfer. In particular these conjectures implies the Artin conjecture on holomorphicity of certain L-functions and the Ramanujan conjectures on Fourier coefficients of cusp forms. While these conjectures are far from being established, there are several techniques (trace formula, converse theorems, explicit constructions) which yields parial results. In the first lecture I hope to give a brief introduction to these realm of ideas. In the second lecture I will concentrate on the method of explicit constructions especially the theory of theta lifts and the method of descent hoping to describe some of my work on CAP representations on orthogonal groups. Name: Vadim Vologodsky, Chicago Title:On Voyevodsky motives. Name: Dmitry Kaledin, Moscow Title: Derived equivalences by quantization. Abstract: We show (joint work with R. Bezrukavnikov) that given a smooth symplectic resolution $X$ of a singular affine algebraic variety $Y$, one can described the derived category $D^b(X)$ of coherent sheaves on $X$ as the derived category of $R$-modules for some algebra. This algebra is finite over its center, and the center coincides with the algebra of functions on $Y$; generically on $Y$, $R$ is a matrix algebra. This equivalence of categories generates, among other things, a non-standard t-structure on $D^b(X)$. It generalizes to the (symplectic) higher dimensional case the results obtained in dim 3 by T. Bridgeland and M. Van den Berg. The proof proceeds by reduction to positive charactetistic; the algebra $R$ can be thought of as a certain generalization of the quantum group at a root of unity (but no groups enter into the construction at all, at least explicitly). Name: Adrian Iovita, Concordia University Title: On the Mazur-Tate-Teitelbaum conjecture. Name: Lucien Szpiro, City University of New York Title: Mahler measure for algebraic dynamical systems, diophantine approximation and equidistribution of preperiodic points. Name: Bernhard Kroetz (DFG, Germany) Title: Convexity theorems on the complex crown. Name: Kobi Kremnizer, Tel-Aviv Title: Quantum gerbes. Name: Shmuel Zelikson, Caen Title: Auslander-Reiten quivers and the Coxeter complex. Abstract: Auslander-Reiten (A.R.) quivers are a major combinatorial tool (oriented graphs) for codifying the structure of representations of finite-dimensional algebras, in particular of representations of quivers. In this lecture we consider quivers obtained by orienting Dynkin diagrams of type ADE, which are distinguished by having only a finite number of indecomposable representations (up to isomorphism). We present a new construction of A.R. quivers, realizing them as subcomplexes of the Coxeter complex ( the complex obtained from the arrangement of hyperplanes of the underlying root system). Our construction uses ideas coming from the combinatorics of the canonical basis of quantum groups. We hope to present a brief outline of how A.R. quivers appear in this setting. No seminar due to the conference on 60th Birthday of Professor Joseph Bernstein in Tel-Aviv (June, 5- June, 9) Name: Vera Serganova, Berkeley Title: Quantum groups and supersymmetry. Name: Arkady Vaintrob, EugeneTitle: Cohomological quantum field theories from spin curves and orbifolds.. Name: Shrawan Kumar, Chapel Hill Title:Towards a proof of the Cachazo-Douglas-Seiberg-Witten conjecture for simple Lie algebras. Name: Nikolaos Papalexiou, University of Aegean Title:Characteristic varieties of the ideals in the enveloping algebra of a semisimple Lie algebras. Name: Polyxeni Lamprou Title: Maximal Poisson commuting subalgebras (following Mischenko-Fomenko and Kostant). Name: Anthony Joseph Title: Invariants for truncated parabolic subalgebras and maximal Poisson commutative algebras. Name: Jacob Greenstein University of California, Riverside Title: An application of free Lie algebras to current algebras. Name: Jacob Greenstein University of California, Riverside Title: Free Lie algebras and extensions of finite dimensional representations of current algebras. Name: Jacob Greenstein University of California, Riverside Title: q,t-characters, braid group and vertex operators.
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