The Hebrew University of Jerusalem

## Amitsur Algebra Seminar

#### Schedule for 2008-2009

Date Speaker Title (The abstracts appear below)
Nov 06 Bill Cassleman (UBC) Graphics tools for illustrating mathematics.
Nov 13 Lior Bary-Soroker (Hebrew University) On the characterization of Hilbertian fields.
Nov 20 Lucas Fresse (Weismann) Singular components of Springer fibers in the two-columns case
Jan 1 Moshe Jarden (Tel Aviv) The absolute Galois groups of semi-local fields (Talk will be in Hebrew!!)
Jan 15 Sidney A. Morris(University of Ballarat) An introduction to pro-Lie groups - a wide and well-behaved class of topological groups
Jan 29 Ehud Meir (Technion) Moore's conjecture and nilpotency of certain cohomology elements
Mar 19 Avinoam Mann (Hebrew University) Philip Hall's curious' formula for non-abelian groups
Mar 26 Lior Bary-Soroker (Hebrew University) Fully Hilbertian fields
Apr 23 Jack Sonn (Technion) On the minimal ramification problem for p-groups (abstract)
May 7 Louis Rowen (Bar-Ilan University) Supertropical algebras (joint work with Z. Itzhakian) (abstract)
May 14 Chen Meiri (Hebrew University) Ribes-Zalesskii product theorem (abstract)
June 11 Yonah Maissel (Ben-Gurion University) Betweenness relations which are determined by subgroups of their automorphisms group (abstract)
June 18 Ehud Friedgut (Hebrew University) Intersecting families of permutations, an algebraic approach (abstract)
Aug 27 Sergio Mendes (Lisbon) Base change and K-theory for GL(n) (abstract)

#### Abstracts

Date Abstract
Nov 6 Using pictures to explain mathematical ideas is a skill every mathematician should be trained to do well. It is a vastly underestimated tool and is not at all trivial, often requiring a combination of artistic and programming talent to do perfectly. But it is relatively easy to learn how to do an adequate job. I shall discuss both technical and higher level aspects of the task.
Nov 13 Hilbert's irreducibility theorem is one of the prominent theorems in number theorem. One of its formulations asserts that over Q (the field of rational numbers) for every irreducible polynomials f_1(X_1,...,X_n,Y_1,...,Y_m), ... , f_r(X_1,...,X_n,Y_1,...,Y_m) with coefficients in Q there is an n tuple of rational numbers a_1,...,a_n such that all f_i remain irreducible under the substitution X_i ---> a_i. A field that satisfies this condition is called Hilbertian. In this talk we present a recent relaxation of the condition needed for a field to be Hilbertian: It suffices to have irreducible substitutions for one (arbitrary) polynomial f(X,Y) that is absolutely irreducible.
Nov 20 The Springer fiber ${\cal B}_u$ over a nilpotent endomorphism $u$ is the variety formed by complete flags which are stable by $u$. We suppose $u$ of nilpotent order $2$. Fix a Jordan basis of $u$, and denote by $H$ the standard torus relative to this basis. We describe the elements of a given component of ${\cal B}_u$ which are fixed by $H$ for its linear action on flags. By using this result, we establish a necessary and sufficient condition of singularity for the components of ${\cal B}_u$.
Jan 15 This is an introductory talk based on the book "The Lie Theory of Connected Pro-Lie Groups" by Karl Heinrich Hofmann and the speaker, published by the European Mathematical Society Publishing House. The class of pro-Lie Groups is the smallest class of topological groups which contains all finite-dimensional Lie groups and is closed under products and closed subgroups. It contains all locally compact abelian groups, all compact groups, and all connected locally compact groups. In the book a full and rich Lie Theory is developed for pro-Lie groups, and it is used to describe the structure of connected pro-Lie groups.
Jan 29 Let G be a group and H a finite index subgroup. We consider the following question of Moore: Suppose that every nontrivial cyclic subgroup of G intersects H nontrivially. Is it true that a G-module which is projective over H is also projective over G? In many cases the answer is known to be positive, including for example finite groups and groups of finite cohomological dimension. A theorem of Aljadeff says that the conjecture also holds in case the group G has a finite index normal subgroup K<H, such that no cyclic subgroup of G/K intersects H/K trivially. This can be used also to prove that the conjecture is true for profinite groups. In this talk we will show that in all the cases mentioned above, the conjecture holds due to the nilpotency of a certain element (the Bockstein) in the cohomology ring of G (this was actually the way the conjecture was proved for finite groups). We will construct examples for pairs (G,H) of a group G and a finite index subgroup H such that the conjecture is true for G and H, eventhough the Bockstein element is not nilpotent. We will also generalize a result of Aljadeff, Cornick, Ginosar and Kropholler, and show that the conjecture is true for all groups G inside Kropholler's hierarchy LHF, and not just in case the module under consideration is finitely generated.
Mar 19 In 1938 Hall published the following identity, which he described as rather curious'. Fix a prime $p$, and let $G$ vary over all finite abelian $p$-groups. Then $$\sum 1/|G| = \sum 1/|Aut(G)|.$$ We will discuss identities of a similar type, which hold for families of non-abelian $p$-groups. These identities were also anticipated by Hall, but he did not write them explicitly, nor gave proofs. Our proof applies ideas and results of subgroup growth.
Mar 26 A Hilbertian field is defined by the property that every irreducible polynomial f(X,Y) with coefficients in K which is separable in Y has an irreducible specialization, i.e., there is a in K for which f(a,Y) is irreducible in K[X]. The set H(f) of all such a's is called a Hilbert set. A surprising fact is that although the cardinality of H(f) always equals the cardinality of K, the cardinality of the fields generated by a root of f(a,Y), where a runs over the elements of H(f), might be considerably smaller.
This talk is devoted to the new notion of fully Hilbertian fields. These are defined by the property that there are as many, and as distinct as possible, fields generated by roots of f(a,Y), where a runs over H(f). We will discuss their properties, application to modern Galois theory. If time permits, we will explain how this notion together with the notion of semi-free' profinite group shed new light to Jarden-Lubotzky twinning principle'.
Apr 23 (Joint work with H. Kisilevsky) Let p be a prime number. It is not known if every finite p-group of rank n>1 can be realized as a Galois group over Q with no more than n ramified primes. We prove that this can be done for the family of finite p-groups which contains all the cyclic groups of p-power order, and is closed under direct products, regular wreath products, and rank preserving homomorphic images. This family contains the Sylow p-subgroups of the symmetric groups and of the classical groups over finite fields of characteristic not p. On the other hand, it does not contain all finite p-groups, and is in fact contained in the family of "semiabelian" p-groups, which is known to be properly contained in the family of all finite p-groups. (Recently it has been proved by Danny Neftin that this family is in fact equal to the family of all semiabelian p-groups.)
May 7 The purpose of this talk is to introduce an algebraic structure rich enough to support algebraic formulations of many of the beautiful properties of tropical geometry. The difficulty with the max-plus algebra is that the lack of additive inverses vitiates many of the familiar techniques from commutative algebra. Accordingly, we consider a "cover" of the max-plus algebra which has a distinguished "ghost" ideal taking the place of the zero element in many of the theorems. This leads to natural notions such as roots of polynomials, and thus varieties, factorization, linear dependence of vectors, singularity of matrices, characteristic polynomial of matrices, eigenvalues, and resultants. In this talk, we focus on questions concerning polynomials and their roots.
May 14 Let F be a finitely generated non-abelian free group. Hall proved that a finitely generated subgroup of F is closed in the profinite topology. Ribes and Zalesskii extended this theorem, they showed that a finite product of finitely generated subgroups of F is closed in the profinite topology. Later Auinger and Steinberg gave a more constructive proof of the product theorem. In this talk we will present their proof.
June 11 One can think of a betweenness relation (B-relation) as a generalization of a tree in the sense of graphs. So in a B-relation, between any two points there is a unique path. However, whereas in a tree the path between two points is finite, in a B-relation such a path may be infinite and ! even dense. An exact definition will be given in the talk. We consider a class K of pairs of the form (M,G) where M is a B-relation and G is a subgroup of Aut(M). For (M,G) in K we say that G determines M within K, if for any (N,H) in K and any isomorphism F between G and H there is an isomorphism h between M and N such that F is conjugation&n! bsp;by g, that is, F(g) =& nbsp;h o g o h-1 for all g in G. A class K of (M,G)'s is faithful if every G determines M within K. Faithfulness results for classes of trees where obtained by H. Bass and A. Lubotzky, R.I. Grigorchok and V. Nekrashevych. Our main result is a faithfulness of a certain class of B-relations. We apply this result and obtain a faithfulness theorem for trees. This theorem strengthens some of the results of Bass and Lubotzky.
June 18 This talk will be similar to one that Haran Pilpel gave at the combinatorics seminar. I thought it might be of interest also to the algebra crowd. Much of extremal combinatorics deals with understanding families of sets given some information on their intersection pattern. The fundamental example of this is the Erdos-Ko-Rado theorem which
characterizes maximal intersecting families. In this talk we study analogous questions where the sets are replaced by
permutations, (with the appropriate definition of intersection), and prove some Erdos-Ko-Rado type results conjectured by Deza-Frankl and Cameron-Ku.The approach we use involves analyzing the spectral properties of weighted graphs that encode these structures, which leads us to problems related to  representations of the symmetric group.
Joint work with Haran Pilpel. The paper can be found on my webpage, and is joint with David Ellis, who achieved the same results independently.
Aug 27

We investigate base change E/F at the level of K-theory for the general linear group GL(n) when E/F is a Galois extension of local ﬁelds of characteristic zero.