The fifth Amitsur Memorial Symposium will take place on Wednesday and Thursday, June 23-24 1999, Tamuz 9-10 5759, at the Einstein Institute of Mathematics, the Hebrew University, Jerusalem. Here is the programme of the symposium, followed by abstracts for some of the talks. Wednesday, June 23rd 10:00 - 10:45: E.I.Zelmanov (Yale): On loop algebras and beyond. 10:45 - 11:15: coffee break. 11:15 - 12:00: S. Gelbart (Weizmann): On the holomorphy and boundedness of certain automorphic L-functions. 12:15 - 13:00: R.Griess (Michigan; Dozor Visiting Fellow, Ben-Gurion): Finite subgroups of Lie groups. 13:00 - 15:00: lunch break 15:00 - 15:45: A.R.Magid (Oklahoma and the Technion): The differential Galois group over the field of rational functions of the field of solutions of equations completely solvable by integrals. 16:00 - 16:45: U.Vishne (Bar-Ilan): Primitive algebras with arbitrary Gelfand-Kirilov dimension. 16:45 - 17:15: coffee break. 17:15 - 18:00: R.Meshulam (Technion): Extremal problems for matrix spaces. Thursday, June 24th 9:30 - 10:15: M.Raghunathan (Tata Institute): On the loop space of an affine algebraic homogeneous space. 10:15 - 10:45 - coffee break. 10:45 - 11:30: H.Bass (Columbia): An algorithm to determine discreteness of the automorphism group of a uniform tree. 11:45 - 12:30: R.Lawrence (Michigan and Jerusalem): Analytic structure in quantum invariants and representation theory. 12:30 - 13:45: lunch break. 13:45 - 14:30: B.Plotkin (Jerusalem): Algebraic geometry in first order logic. 14:45 - 15:30: M.Jarden (Tel-Aviv): Non-PAC fields. 15:30 - 16:00: coffee break. 16:00 - 17:00: E.I.Zelmanov: On some algebraic structures of conformal field theory. This last talk will also be the weekly department's colloquium talk. =========================================================================== Abstracts H.Bass: An algorithm to determine discreteness of the automorphism group of a uniform tree A uniform tree X is the universal cover of a finite connected graph A. How can we decide, directly in terms of A, whether the (locally compact) group G = Aut(X) is discrete? This happens, for example, when A itself is a tree (and G is finite) or when A is a circuit, and G is infinite dihedral. But there are many other (at first surprising) cases where A has big fundamental group. Tits and I have constructed an algorithm to decide this. It first constructs, algorithmically, the absolute quotient (A*,i*) of X mod G, as an "edge-indexed graph," where the indices on edges record ramification data about the action of G on X. Next, for any given edge -indexed graph (A,i), with universal covering tree X, we give a general criterion in order that the group H of deck transformations of X ---> A be discrete. (When (A,i) = (A*,i*) as above, we have H = G.) We call a finite graph A "pi-rigid" if its fundamental group is the full automorphism group of X (which is hence discrete). We determine the smallest such examples. ============================================================================= S.Gelbart: On the holomorphy and boundedness of certain automorphic L-functions About thirty years ago, R. P. Langlands introduced his famous program of automorphic forms and L-functions; eventually, part of this theory was further developed by F. Shahidi. We shall survey this entire program, and then zero in on certain boundedness criteria for L(s, \pi, Sym^3) and its generalizations (joint work with F. Shahidi). ============================================================================= R.L.Griess: Finite Subgroups of Lie Groups The classification of homomorphisms from finite groups to classical groups (general linear, orthogonal and symplectic) is treated by the well known theory of group characters. Since the early 80s, there has been a program to study the finite subgroups of the exceptional algebraic groups, i.e., those of types G_2, F_4, E_6, E_7 and E_8. Maps to these groups require a variety of special methods, including Lie theory, pure finite group theory, computer representation theory. At this point, we know which finite simple groups have a projective representation in one of the exceptional groups. We will survey some general theory about finite subgroups of algebraic groups and eventually concentrate on the status of embeddings of finite quasisimple groups in exceptional Lie groups. Easy arguments show that some finite groups do not embed, while serious computer work is needed to establish certain embeddings or classify them up to conjugacy. For maps to the general linear group, the character classifies maps up to conjugacy. Conjugacy of maps to exceptional groups is only partially resolved. ========================================================================== U.Vishne: Primitive algebras with arbitrary Gelfand-Kirillov Dimension We construct, for every real \beta \geq 2, a primitive affine algebra with Gelfand-Kirillov dimension \beta. Unlike earlier constructions, there are no assumptions on the base field. In particular, this is the first construction over the real or the complex field. Given a recursive sequence v_n of elements in a free monoid, we investigate the quotient of the free associative algebra by the ideal generated by all non-subwords in v_n. We bound the dimension of the resulting algebra in terms of the growth of v_n. In particular, if |v_n| is less than doubly-exponential then the dimension is 2. the work was published in J. of Algebra 211(1), 1999. ============================================================================= R.Lawrence: Analytic structure in quantum invariants and representation theory The SO(3) Witten-Reshetikhin-Turaev invariant of 3-manifolds defines, from a manifold M, a complex number Z_M(q) (in fact algebraic integer) for each root of unity q. Their definition is as an expectation value of the coloured Jones polynomial of the surgery link, in other words, it is a weighted expectation value over all ways of labelling the components of the link by (a finite family of) representations of the (reduced) quantum group U'_q(sl_2). For certain manifolds, it has been shown in work of Jeffrey, L.-Rozansky, L.-Zagier and others, that these numbers are closely connected to analytic functions in the unit disc, some with almost modular properties. In this talk we will survey some of these connections and demonstrate how these functions are related to certain families of representations of quantum groups. ============================================================================== A.R.Magid: The differential Galois group over the field of rational functions of the field of solutions of equations completely solvable by integrals Let F = C(t) denote the differential field of rational functions in one variable over the complex numbers with derivation df/dx. A monic linear homogeneous differential equation L=0 over F is completely solvable by integrals if the Picard--Vessiot (differential Galois) extension E for L over F is contained in a differential (but not necessarily Galois) extension F(a_1,...,a_n) where a_1^\prime \in F and a_{i+1}^\prime \in F(a_1,...,a_i). The compositum of all such is the (infinite) differential Galois extension of the title, and we will show that the corresponding (pro) algebraic differential Galois group is free (prounipotent). ========================================================================== B.Plotkin ALGEBRAIC GEOMETRY IN FIRST ORDER LOGIC We consider a mathematical area which accumulates algebraic logic and algebraic geometry. This combination is going to be used in knowledge science. The category of elementary knowledge and knowledge base is defined. Logic and geometry are considered in respect to an arbitrary but fixed variety of (universal) algebras \Theta. This \Theta plays a role of data type in applications. Affine space in the given \Theta is represented in the form Hom(W,G), where W=W(X) is the free in \Theta algebra with the finite X and G is an algebra from \Theta. Points in this space are the points of type \Theta, represented as homomorphisms. Equations have the form w=w', w, w' \in W, and we consider their solutions in the algebra G. A point \mu: W\to G is a solution of the given equation, if w^\mu=w'^\mu in G. This means, that the point \mu belongs to the value of the formula w=w' in algebra G, i.e. the formula itself belongs to Ker \mu. We consider also a system of symbols of relations \Phi and the corresponding models (G,\Phi,f), where G\in \Theta and f is a realization of the set \Phi in G. Now we are able to replace the equations in the setting above by the arbitrary first order formulas in \Theta-logic, which is based on the sets \Phi. We are looking for solutions of these ``equations'' in the models (G,\Phi,f). A point \mu: W\to G is a solution of an ``equation'' u if \mu belongs to the value Val_f(u) of the formula u in the given model, or, what is the same, u belongs to the logical kernel LogKer(\mu), which depends on the realization f. Thus, algebraic sets or algebraic varieties in the space Hom(W,G) are defined by a set of formulas, which are not necessarily equalities. This leads to an algebraic geometry in logic, which differs substantially from the standard equational geometry. In equational geometry the category \Theta^0 of all free in \Theta algebras W=W(X) with finite X is used. The similar role for algebraic geometry in logic plays a special structure of algebraic logic. =========================================================================== M.Raghunathan: On the loop space of an affine algebraic homogeneous space Let G be a connected complex algebraic group and H a connected algebraic subgroup of G such that X = G/H is an affine variety. Let L(X) denote the loop space of X, namely the set of all continuous maps of the circle S (identified with complex numbers of modulus 1) that map 1 in S to the identity coset in, X equipped with the compact open topology. Let M(X) be the set of all morphisms of C^ (= non-zero complex numbers) considered as an algebraic varity in X that map 1 in C^ to the identity coset. This set M(X) has a natural identification with the inductuve limit of an increasing family of affine algebraic varieties. Each of the varieties in the family has a natural hausdorff topology and M(X) is given the inductive limit topology. The main result to be discussed in the talk is the following theorem: the inclusion of M(X) in L(X) (got by restricting morphisms of C^ to S) is a homotopy equivalence. --------------------------------------------------------- >>Technion Mathematics Net 2 [TECHMATH2] (Editor: Michael Cwikel)<< Announcement from: Avinoam Mann