|Center for Advanced
Studies in Mathematics
Ben Gurion University of the Negev
Department of Mathematics
המרכז ללימודים מתקדמים במתמטיקה
אוניברסיטת בן גוריון בנגב
|Held at the Department of Mathematics,
Ben Gurion University
Building 58, room -101
Prof. Shimshon A. Amitsur (1921 - 1994)
|יתקיים במחלקה למתמטיקה
אוניברסיטת בן גוריון
בניין 58, חדר מינוס 101
אלי אלחדף, הטכניון
Mia Cohen (mia at math.bgu.ac.il)
Ami Braun (abraun at math2.haifa.ac.il)
Uri Onn (urionn at math.bgu.ac.il)
- Wednesday, May 25th
- 9:30 Gathering
10:00 Ilya Tyomkin : Zariski's theorem via tropical geometry (Abstract)
11:00 Yuval Ginosar : On Groups of I-Type and Involutive Yang-Baxter Groups (Abstract)
12:00 Coffee break
12:20 Darrell Haile : A graph theoretic approach to graded identities on matrices (Abstract)
- 13:15 Lunch break
14:45 Eli Aljadeff : Polynomial Identities, Graded Algebras and (ONE APPLICATION TO) Division Algebras (Abstract)
15:45 Coffee break
16:00 Udi Meir : On simple quotients of finite dimension Hopf algebras (Abstract)
17:00 Paul Baum : Expanders and K-Theory for discrete groups (Abstract)
The Symposium dinner will be held in the evening.
- Thursday, May 26th
- 10:00 Serge Skryabin : Structure of artinian H-module algebras (Abstract)
- 11:00 Sara Westreich : Conjugacy classes and class sums for Hopf algebras (Abstract)
12:00 Coffee break
12:20 Steffen Koenig : Symmetric powers, Brauer algebras and Schur algebras (Abstract)
- 13:15 Lunch break
14:45 Elad Paran : An elementary solution for the inverse Galois problem over C(x) (Abstract)
15:40 Coffee break
16:00 Chen Meiri : Sieve methods in group theory (Abstract)
17:00 Ami Braun : Unique factorisation versus Cohen-Macaulay - a tale of two properties (Abstract)
- Eli Aljadeff (Technion) : Polynomial Identities, Graded Algebras and (ONE APPLICATION TO) Division Algebras
- Abstract: Amitsur showed (in the early 70's) the existence of non crossed products division algebras by means of "generic" constructions.
- This establishes a clear connection between polynomial identities and division algebras. Twenty years later the theory of G-graded polynomial
- identities was developed. In the main part of the lecture I will recall some fundamental facts on polynomial identities and present some new
- results in the G-graded context. Then, towards the end of the lecture, I'll present an application of the theory of G-graded polynomial identities
- to division algebras in the same spirit as Amitsur.
- Paul Baum (The Pennsylvania State University University) : Expanders and K-Theory for discrete groups
- Abstract: Let G be a locally compact topological group which is Hausdorff and second countable. TheBaum-Connes (BC) conjecture proposes
- an answer for the problem of calculating the K-theory of the reduced C* algebra of G. At the present time there is no known counter-example to BC.
- A natural generalization of BC is Baum-Connes with coefficients (BCC). This talk will consider discrete groups and will make the point that
- a discrete group which "contains" an expander in itsCayley graph is a counter-example to BCC. Here "contains" means "contains in a weak coarsegeometry sense". Does such a discrete group exist? M. Gromov began the construction of sucha group, and the construction has now been
- completed due to the work of several mathematicians.The talk will describe the basics of BC and BCC for discrete groups and will then indicate
- why the Gromov group provides a counter-example to BCC. The key point is that the Gromov group is not exact.
- Ami Braun (Haifa University) : Unique factorisation versus Cohen-Macaulay - a tale of two properties
- Abstract: In 1961 Pierre Samuel asked whether the U.F. property of Netherian commutative local domain forces it to be CM.
This turned out to be false in general but true in low-dimensional (geometric, zero char.) cases. This is due to works of Raynaud,
Boutot, Hartshorn-Ogus, Freitag-Kiehl and S.Mori, in the mid 1970's.
The story in prime characteristic is less complete and the results here are due to Bertin ,Serre and Fossum-Griffith.
We shall discuss all of this as well as some new examples ,arising in enveloping algebras of prime char., which complete the story.
This is joint work with my student Oz Ben-Shimol.
- Yuval Ginosar (Haifa University) : On Groups of I-Type and Involutive Yang-Baxter Groups
- Abstract: It is known that groups which admit a bijective 1-cocycle are solvable. We give a cohomological approach to the following
converse problem: does every finite solvable group admit a bijective 1-cocycle? A joint work with Nir Ben David.
- Darrell Haile (Indiana University) : A graph theoretic approach to graded identities on matrices
- Abstract: (file)
- Steffen Koenig (University of Stuttgart) : Symmetric powers, Brauer algebras and Schur algebras
- Abstract: Classical Schur-Weyl duality relates Schur algebras of infinite general linear groups with finite symmetric groups via commuting
actions on tensor space. Similarly, Schur algebras of symplectic or orthgonal groups are related with Brauer algebras. Now replace tensor
space by a direct sum of tensor products of symmetric powers. For general linear groups, the endomorphism ring is the classical Schur algebra.
For orthogonal and symplectic groups, however, the endomorphism ring is a new algebra, which at the same time plays the role of a Schur
algebra for Brauer's algebra. (This is joint work with Anne Henke.)
- Ehud Meir (Cambridge) : On simple quotients of finite dimension Hopf algebras
- Abstract: Let G be a finite group, and let V be an irreducible complex represntation of G. Schur have asked what are the subfields K of C
- over which it is possible to realize V (In other words - what scalars do we need in order to realize V).
The Brauer-Witt Theorem proves that all simple quotients of group algebras (the so called Schur algebras) are Brauer equivalent
to cyclotomic algebras. From this we deduce that any irreducible representation of any finite group is realizable over a cyclotomic extension
- of the rational numbers (and so, all the scalars we need in order to realize representations are roots of unity).
In this talk we will consider the question of what are the Brauer equivalence classes of simple quotients of finite dimensional Hopf algebras.
- By applying techniques from Galois cohomology to Hopf algebras, and by considering some very basic properties of representation of Hopf algebras, we will be able to show that in fact all central simple algebras over a field K are Brauer equivalent to quotient of Hopf algebras over the same field.
Also, we will be able to construct some new examples of finite dimensional Hopf algebras over non algebraically closed fields.
- Chen Meiri (The Hebrew University) : Sieve methods in group theory
- Abstract: (file)
- Elad Paran (Tel-Aviv University) : An elementary solution for the inverse Galois problem over C(x)
- Abstract: It is a long standing conjecture that for every field K and finite group G there exists a Galois extension of K(x) with Galois group G.
- Over fields that are complete w.r.t. an archimedean absolute value (that is, the real or complex numbers) the conjecture is known by a deep
- result -- Riemann's Existence Theorem. In 1987 Harbater proved the conjecture for fields that are complete with respect to non-archimedean absolute
- values (e.g. the p-adic numbers), using another deep result -- Grothendieck's Existence Theorem. There was an analogy between the constructions
- made in the archimedean and non-archimedean cases, but it was only an analogy. In this talk we will present a unified proof for all complete valued fields.
- Moreover, the proof is elementary, relying on no deep theorems.
- Serge Skryabin (Chebotarev Research Institute, Kazan) : Structure of artinian H-module algebras
- Abstract: It has recently become evident that existence of a Hopf algebra action on another algebra implies unexpectedly strong ring-theoretic
properties. Especially, every right artinian H-module algebra without nonzero H-stable nilpotent ideals turns out to be a quasi-Frobenius ring
and decomposes into a direct product of finitely many H-simple H-module algebras. In the case of a cosemisimple Hopf algebra one obtains
the ordinary semisimplicity of the module algebra. All H-equivariant modules turn out to be projective. These results extending fundamental facts
in the classical theory of artinian rings to H-module algebras have been proved even for an infinite dimensional Hopf algebra, sometimes under
a restriction on the growth.
The artinian case is very useful also in a more general situation when it is possible to pass to artinian rings of fractions. It appears that good
properties of the Hopf algebra itself and its coideal subalgebras to a large extent depend on the existence of artinian rings of fractions.
In my talk I am going to discuss several ideas used in dealing with these questions.
Ilya Tyomkin (Ben-Gurion University of the Negev) : Zariski's theorem via tropical geometry
- Abstract: In 1982, Zariski gave a dimension-theoretic characterization of Severi varieties on the complex projective plane by showing that the
dimension of the locus of plane curves of degree d and geometric genus g is bounded by 3d+g-1; and proved that a general curve of degree d and
geometric genus g is nodal. Zariski's theorem played an important role in Harris’s proof of the irreducibility of Severi varieties, and in a series of
enumerative results. It was generalized to other rational surfaces (e.g. weighted projective planes) in characteristic zero, but no known proof covered
the case of positive characteristic.
In my talk, I will give examples showing that the second assertion of Zariski's result is false in positive characteristic (at least for certain weighted
projective planes). I will also explain how to use tropical geometry to obtain the first part of Zariski's theorem in arbitrary characteristic. For this, I will,
first, show that if the ground field is non-Archimedean then any algebraic curve comes with a canonical combinatorially defined object associated
to it (called tropical curve), and, second, will explain how to reduce the bound on the dimension to a combinatorial statement (Mikhalkin's lemma)
that can be verified relatively easily.
- Sara Westreich (Bar-Ilan) : Conjugacy classes and class sums for Hopf algebras
- Abstract: We extend the notion of conjugacy classes and class sums from nite groups to semisimple Hopf algebras and show
that the conjugacy classes are obtained from the factorization of H as irreducible left D(H)-modules. For quasitriangular semisimple
Hopf algebras H we prove that the product of two class sums is an integral combination of the class sums up to 1/d^2 where d = dimH.
We show also that in this case the character table is obtained from the S-matrix associated to D(H).
- About Prof. Amitsur:
- Shimshon A. Amitsur / The Mathematics Genealogy Project
Shimshon Avraham Amitsur (1921 - 1994) / The MacTutor History of Mathematics archive