Avraham (Rami) Aizenbud
אברהם (רמי) איזנבוד
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Avraham (Rami) Aizenbud |
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אברהם (רמי) איזנבוד |
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Young Algebra Day 2017Thursday, July 27, 2017Weizmann Institute of Science,Ziskind Building (Mathematics), Room 1 (map)
Organizers: Avraham Aizenbud, Inna Entova-Aizenbud, Maria Gorelik Contact: Inna Entova-Aizenbud, entova [at] bgu ac il Program
Abstract: We shall discuss theory of symmetric functions. We will introduce basic examples, namely, elementary, power sum and Schur functions from a combinatorial point of view. We will show that the ring of symmetric functions is in fact isomorphic to another ring arising in representation theory, namely the Grothendieck ring of the category of finite-dimensional modules over semisimple Lie algebras. Prerequisites: linear algebra, group and ring theory Abstract: The field of p-adic numbers was first defined about 120 years ago, and since then has been an indispensable tool in number theory. In this lecture we will construct the field of p-adic numbers Q_p and discuss the properties of (algebraic) groups defined over the field of p-adic numbers. We will then see an instance of Bruhat-Tits theory: that for the group GL_2 of invertible 2x2 matrices whose entries are p-adic numbers over one naturally has a p+1-regular tree on which this group acts. In the remaining time we will mention some applications of this theory. Prerequisites: group theory, linear algebra, notion of ring. Abstract: It is a famous theorem in harmonic analysis that any (nice) periodic function on the real line R can be written as a (possibly infinite) sum of sines and cosines, known as a Fourier series. It turns out that the Fourier series is a direct manifestation of the representation theory of the circle group S^1, when we think of periodic functions on R as functions on the circle S^1. A similar classic result is that any (nice) function on the sphere S^2, can be written as a sum of certain functions called spherical harmonics. Here, one can obtain this decomposition using the representation theory of a pair of groups: the rotation group SO(3) and the circle group S^1, where S^2=SO(3)/S^1. The latter example (and in fact both!) is an instance of relative representation theory- the representation theory of a group G, with respect to some subgroup H. In this talk I will give an introduction to representation theory and relative representation theory, and discuss some examples. Prerequisites: linear algebra, group theory, topology. Abstract: Let K be a the field Q_p of p-adic numbers (which will be discussed in Yotam's lecture) or the field F_p((t)) of Laurent series over a finite field F_p, where p is a prime number. These are examples of non-archimedean local fields. A prominent object in the study of smooth representations of algebraic groups over K, such as GL_n(K), is the analysis of irreducible characters of their maximal compact subgroups, e.g. G=GL_n(o), where o is the valuation ring of K (e.g. the ring of p-adic integers, in the case when K = Q_p). Regular characters of GL_n(o) form a considerable subset of the set of irreducible continuous characters, and are currently the largest subset amenable to explicit construction. The definition of such characters goes back to the works of Shintani and Hill. In my talk I will present the explicit definition of regular characters of GL_n(o) and present the construction of such characters. Furthermore, I will report on new results regarding the case where G is arises as a maximal compact subgroup of a symplectic or an orthogonal group. Prerequisites: introduction to representation theory, number theory. (a more advanced version of the abstract can be found here) Abstract: For the last several decades, the main tools to attack central problems in knot theory (first among them being classification of knots) were polynomial invariants such as Jones and Alexander polynomials. These invariants gave only a partial classification: for instance, they could not distinguish some non-trivial knots from the trivial one. More recently a new approach to the problem was developed, due to Khovanov's groundbreaking work. This approach is called "categorification of knot invariants" and the idea behind it is to construct a functor which attaches a complex of vector spaces to every knot in an invariant way, and encodes the previous invariants. In this talk I will present some basic ideas from knot theory and discuss extensively the Khovanov caetforification of the Jones polynomial. If time permit, we will discuss the theoretical and practical difficulties in computing this homology. Abstract: E'tale obstruction theory is a recently developed method to prove that a variety defined over a field F has no F-rational points. It is based on a general method to attach a "topological space" to each scheme X, using refinement of the notion of C'ech complex of a cover. Then the existence of a rational points translates to the existence of a fixed point of the Galois group of F on the corresponding "space". I will describe these obstructions for points on F-forms of the standard sphere x_0^2 + ... + x_n^2 = 1, namely solutions to equations of the form a_0 x_0^2 + ... + a_n x_n^2 = 1. The calculation method compare these obstructions to the obstruction for a fixed point of the group of orthogonal transformations of R^n on the standard sphere in R^n of radius 1. This is a joint work with Edo Arad. No previous knowledge of E'tale homotopy theory or obstruction theory for fixed points will be assumed! |