New York Algebra Colloquium

Time: Fridays 10:30-11:30 a.m.
Place: CUNY Graduate Center
Room 3209
365 Fifth Ave. (at 34th St.),
New York, New York
Organizers
and Co-Founders: Andrew Douglas, and Delaram Kahrobaei.

Schedule

Fall 2009

9-25-2009 Stephen Majewicz (CUNY)
Title: Some Results on Extraction of Roots in Nilpotent R-Powered Groups.
10-9-2009 Aaron Lauda (Columbia University)
Title: Categorifying quantum groups
10-16-2009 Marcos Zyman (CUNY)
Title: IA-automorphisms of various classes of groups
11-6-2009 Jim Lepowsky (Rutgers University)
Title: From the theory of partitions to new structure in vertex operator algebra theory.
11-13-2009 Lev Shneerson (CUNY)
Title: On the growth of semigroups.
11-20-2009 Hubert de Guise (Lakehead University, Canada)
Title: A general approach to SU(n) quasi-distributions.
12-4-2009 Manhattan Group Theory Day!

Spring 2009
- 01-30-2009 Maria Wesslen (University of Toronto)
  Title: A Geometric Interpretation of Tensor Product Decompositions in su(3).
- 02-06-2009 Olga Kharlampovich (McGill University)
  Title: Fully residually hyperbolic groups.
- 02-20-2009 Yi-Zhi Huang (Rutgers University)
  Title: Algebras in tensor categories and open-closed conformal field theories.
- 02-27-2009 Russell Miller (CUNY)
  Title: Difficulty of Factoring Polynomials and Finding Roots
- 03-06-2009 Adrien Deloro (Rutgers University)
  Title: Ranked groups, the Cherlin-Zilber conjecture, and PSL2
- 03-13-2009 Margaret Dean (CUNY).
  Title: Embedding a wreath product in a skew field of fractions
- 03-20-2009 Yalin Celikler (CUNY)
  Title: Weierstrass Preparation Theorem in Analytic Geometry of Valued Fields
- 04-03-2009 Alexander Rozenblyum (CUNY)
  Title:Representations of the Group ISO(n) and Bessel Function with Matrix Index.
- 04-24-2009 Leigh Cobbs (Rutgers University)
  Title:Infinite descending chains of cocompact lattices in Kac-Moody groups
- 05-08-2009 Parisa Babaali (CUNY)
  Title: Generating Random Automata and analyzing their Combinatorial Properties
- 05-15-2009 Peter Hegarty (Chalmers University of Technology, Sweden)
  Title: Some fairly elementary problems concerning automorphisms of (mostly, but not only, finite) groups.
- 05-22-2009 Corina Calinescu (Ohio State Universtiy/Yale University)
  Title: On applications of vertex operator algebras in Lie theory and partition theory
- Thursday May 28. 4 to 5 pm Ramon Esteban-Romero (Universitat Politecnica de Valencia, Spain)
  Title: Groups in which Sylow permutability is transitive and groups which satisfy a converse of Lagrange's theorem
- Thursday May 28. 5 to 6 pm LUIS EZQUERRO (Universidad Pública de Navarra, Spain)
  Title: ON SUBGROUPS WHICH COVER OR AVOID CHIEF FACTORS OF A FINITE GROUP
Fall 2008
- 09-05-2008 Ben Fine (Fairfield University)
  Title: Surface Groups: Motivating Examples in Combinatorial Group Theory
- 09-12-2008 Stephen Majewicz (CUNY)
  Title: Nilpotent Q[x]-powered groups
- 09-19-2008 Sal Liriano (CUNY)
  Title: Algebro-geometric invariant of finitely generated groups
- 09-26-2008 Anthony Clement (CUNY)
  Title: The Baumslag-Solitar Groups: A solution for the Isomorphism Problem
- 10-03-2008 Alistair Savage (University of Ottawa)
  Title: Moduli spaces of sheaves and the boson-fermion correspondence
- 10-10-2008 Murray Bremner (University of Saskatchewan)
  Title: Lattice basis reduction and polynomial identities
- 10-17-2008 Lisa Carbone (Rutgers University)
  Title: Lattices, buildings and Kac-Moody groups
- 10-24-2008 Christophe Hohlweg (Université du Québec à Montréal)
  Title: Descent algebras and characters of finite Coxeter groups
- 11-07-2008 Roozbeh Hazrat (Queen’s University, Belfast)
  Title: Reduced K-theory for Azumaya Algebras, An overview
- 11-14-2008 Peter Brinkmann (CUNY)
- 11-21-2008 Earl Taft (Rutgers University)
  Title: The boson-fermion correspondence and one-sided quantum groups
- 12-05-2008 Andrew Douglas (CUNY)
  Title: Generalized Lie-Yamaguti structures on the sl(3,C)-modules V(n,n).
- 12-12-2008 Alex Dent (Royal Holloway and Bedford New College, UK)
  Title: Transitivity in Intransitive Groups
Abstracts

Fall 2009

Stephen Majewicz
The study of extraction of roots in groups has a long history, going back to works of Mal'cev, Cernikov, Plotkin, Baumslag, and others. In this talk, I will discuss some recent results on the extraction of roots in the category of nilpotent R-powered groups. This is joint work with Marcos Zyman.

Aaron Lauda (Columbia University)
I'll explain joint work with Mikhail Khovanov on a categorification of one-half of the quantum universal enveloping algebra associated to a symmetrizable Kac-Moody algebra. This categorification is obtained from the graded representation category of a graded algebra that can be defined using a graphical calculus. I will also survey recent results and applications related to these algebras.

Marcos Zyman (CUNY)
The IA-automorphisms of a group G consist of those automorphisms that induce the identity on G abelianized. I plan to discuss some aspects of the IA-automorphisms of the following classes of groups: nilpotent, metabelian, groups with constant upper central series, and center-by-metabelian. In particuar, If G is a group with constant upper central series, I will offer necessary and sufficient conditions for its IA-automorphisms to equal a direct product of subgroups of AutG. (The second part of this talk is joint work with M.H. Dean and M. Bonanome).

Jim Lepowsky (Rutgers University)
I will sketch how partition identities such as the Rogers-Ramanujan identities have inspired new directions in vertex operator algebra theory. Recent developments include joint work with Corina Calinescu and Antun Milas on the description of certain structure related to such identities, and ideas about what this structure should lead to.

Lev Shneerson (CUNY)
We study the asymptotic behavior of the growth functions and connections between the types of growth and existence of identities in different classes of finitely generated semigroups.

Hubert de Guise (Lakehead University, Canada)
I will discuss the problem of constructing a phase space representation for elements of the Lie algebra su(n). Specific applications to su(2) will be discussed as an example, and the general framework will be illustrated for su(3).

G. Rosenberger (Germany)
Lyndon proved that in a free group F each solution set {x, y, z} of the quadratic equation x2y2z2 = 1 generates a cyclic group. We introduce a collection of Lyndon Properties and show that they hold under many amalgam constructions and onerelator groups with odd torsion. As an applicaton we get conditions on Baumslag doubles to contain surface groups.
Spring 2009

Maria Wesslen (University of Toronto)
Any tensor product of two irreducible representations in the Lie algebra su(3) can be decomposed into a direct sum. I will describe a geometric method for finding these tensor product decompositions which is similar to the method for drawing weight diagrams. This method is useful not only for working with specific examples, but also for studying tensor product decompositions in general. I will discuss several applications of the method and examine some surprising differences between tensor product decompositions and weight diagrams.

Olga Kharlampovich (McGill University)
We describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free group case, the group $H$ embeds into the Lyndon's completion $G^{\mathbb{Z}[t]}$ of the group $G$, or, equivalently, $H$ embeds into a group obtained from $G$ by finitely many extensions of centralizers. Conversely, every subgroup of $G^{\mathbb{Z}[t]}$ containing $G$ is universally equivalent to $G$ . Since finitely generated groups universally equivalent to $G$ are precisely the finitely generated groups discriminated by $G$ the result above gives a description of finitely generated groups discriminated by $G$. This is a joint work with A. Myasnikov.

Yi-Zhi Huang (Rutgers University)
Various notions of algebra, including associative algebra, commutative associative algebra, Frobenius algebra and so on, can be defined in a braided tensor category. In the case that the braided tensor category is constructed from the category of modules for a vertex operator algebra, these algebras are equivalent to substructures in open-closed conformal field theories. In this talk, I will discuss results on such equivalences, including the results obtained by Huang-Kirillov-Lepowsky, Huang-Kong and Kong.

Russell Miller (CUNY)
For a polynomial p(X) over a field F, two basic questions arise: whether p(X) factors over F, and whether p(X) has a root in F. Clearly these questions are related, but it is not obvious which one is more difficult -- or even how one measures "difficulty" in this context. Computability theorists have several measures of relative difficulty of computation, and we will explain them and examine how they can be applied in this situation. The coarser measures indicate that the two questions are equally difficult, but there is a a finer measure, m-reducibility, quite common in computability theory, which indicates that for algebraic fields, one of these questions is always at least as hard as the other, and can be strictly harder. To find out which is the harder question, come to the talk!

Adrien Deloro (Rutgers University)
Ranked groups are abstract analogs of algebraic groups, where one would forget the Zariski topology and remember only the Zariski dimension and its properties. Cherlin and Zilber independently conjectured in the 70's that any simple, ranked group does come from (ie., is isomorphic as an abstract group to) an algebraic group. Much work has been done to attack this question, with a very interesting partial answer. Moreover, some methods in use are borrowed from >finite< group theory, so the ideas are likely to interact with CFSG. I shall motivate these questions and discuss the case of "small" ranked groups, the target structure being PSL2(K).

Margaret Dean (CUNY)
A wreath product of a free abelian group by a finitely generated torsion-free nilpotent group can be embedded in the skew field of fractions of the division ring of a nilpotent group. The trick is to find an appropriate nilpotent group to use in constructing the skew field. As an immediate but interesting corollary, we see that a finitely generated free metabelian group can likewise be embedded.

Yalin Celikler (CUNY)
One way of looking at Weierstrass Preparation Theorem is as a tool in understanding when power series behave like polynomials. Over valued fields certain power series are popular as sources of analytic functions and one considers the sets of points which satisfy certain equalities and inequalities of analytic functions. Such sets are very well-behaved geometrically. In my talk I will illustrate how one can obtain results in the geometry of these fields using Weierstrass Preparation to reduce the problem to the point where methods of algebraic geometry apply.

Alexander Rozenblyum (CUNY)
We consider unitary irreducible representations of the group ISO(n) - the group of motions in n-dimensional Euclidian space. It is well known that for n = 2 matrix elements (ME) of these representations can be reduced to classical Bessel functions. In the talk, we describe a construction of a certain square matrix consisting of ME of both equivalent and nonequivalent unitary irreducible representations of the group ISO(n). The constructed matrix has a number of interesting properties. In particular, using Casimir operator it is shown that this matrix can be described as a Bessel function with a matrix index. Spectral analysis of this index leads to orthogonal polynomials in many discrete and continuous variables that generalize classical Hahn, Racah Gegenbauer and Jacobi polynomials.

Leigh Cobbs (Rutgers)
A Kac-Moody group over a finite field is a topological group which acts cocompactly on a locally finite simplicial complex, namely its Tits building. The geometric and combinatorial properties of the action and corresponding quotient complex of groups give effective tools to construct cocompact lattice subgroups. For a rank 2 Kac-Moody group over the field of two elements, we use the method of extending coverings of edge-indexed graphs to covering morphisms of graphs of graphs to construct infinite descending chains of cocompact lattices. We give sufficient conditions for the existence of such chains over arbitrary finite fields. This is joint work with Lisa Carbone.

Parisa Babaali (CUNY)
In this talk we present an algorithm to generate a given finite deterministic automaton at random. We analyze the measure induced by the algorithm and study the combinatorial properties of automata. We show that certain properties such as minimality are not generic.

Peter Hegarty (Chalmers University of Technology, Sweden)
To describe the full automorphism group of a (finite) group is a task which tends to be either fairly straightforward or very difficult. Basically, this is because certain subgroups of the automorphism group present themselves very naturally. If these can be shown to exhaust all automorphisms then you're fine, otherwise you're often in trouble. In this talk I will discuss several ideas and problems connected to my own work on automorphism groups. These problems have an "inverse" character, i.e.: one starts with some assumption about the automorphism group and wants to deduce something about the group itself. Only quite elementary methods will be employed throughout.

Corina Calinescu (Ohio State University/Yale University)
Combinatorial identities and recursions have played, and continue to play, an important role in the theory of vertex operator algebras and representations of affine Lie algebras. In this talk we survey recent developments in the study of principal subspaces of standard modules for affine Lie algebras. We discuss generators-and-relations results for these spaces and recursions satisfied by the characters of the principal subspaces. Part of this talk is based on joint work with James Lepowsky and Antun Milas.

Ramon Esteban-Romero (Universitat Politecnica de Valencia, Spain)
In this talk we consider finite groups. We show the relation between the class of groups in which Sylow permutability is a transitive relation (the PST-groups) and the class of groups in which every subgroup possesses supergroups of all possible indices, the so-called $\mathcal{Y}$-groups. The parellelism between these classes in the soluble universe and the interest of the local study of PST-groups motivates a local study of $\mathcal{Y}$-groups. A group $G$ factorised as a product of two subgroups $A$ and $B$ is said to be a mutually permutable product whenever $A$ permutes with every subgroup of $B$ and $B$ permutes with every subgroup of $A$. We present some results concerning mutually permutable products of groups in the orbit of the above classes. This is a joint work with M. Asaad (Cairo, Egypt), A. Ballester-Bolinches (Val\`encia, Spain) and J. C. Beidleman (Lexington, KY, USA).

LUIS EZQUERRO
Abstract. A classical topic of research in Finite Group Theory is the following: What is the influence on the structure of the group of the fact that all members of some relevant family of subgroups enjoy a given embedding property?
The cover-avoidance property is an embedding property that has recovered much attention in the last few years. In this talk we present a number of results showing that its influence goes much further from the classical supersolubility.

Fall 2008

Ben Fine
What I will do is show how the theory of surface groups has motivated most of the development of combinatorial group theory.

Stephen Majewicz
In this talk, I will discuss several results on nilpotent Q[x]-powered groups. A nilpotent R-powered group is a nilpotent group which admits exponents from a binomial ring R. These groups have been studied in works by Malcev, Hall, Warfield, Baumslag, and various others. I will provide the audience with some background material and mention some of my contributions.

Sal Liriano
If G is a finitely generated group, and A a complex affine algebraic group such as PSL(2,C) or SL(2,C), then the space Hom(G,A) admits the structure of a complex algebraic variety, denoted briefly by R_A(G). In my talk I will present results concerning R_A(G) for cyclically pinched one relator groups and other groups, like parafree groups, that closely resemble free groups. The talk will also address torus knot groups, orientable surface groups in relation to a recent result of Liriano and Majewicz.

Alistair Savage
The boson-fermion correspondence is a fundamental result in mathematical physics that relates the state spaces of fermions (particles with half-integer spin) and bosons (particles with integer spin). It is also one of the basic examples of the use of vertex operators in representation theory. We will describe how one can give a geometric realization of the (r-colored) boson-fermion correspondence using the equivariant cohomology of the moduli space of framed torsion-free sheaves on the projective plane. In this framework, the vertex operators are realized as Chern classes of vector bundles. This is joint work with Anthony Licata.

Murray Bremner
Part 1 of this talk describes the Hermite Normal Form of a matrix over Z; to compute the HNF we use Gaussian elimination with division in Q replaced by GCDs in Z. I will explain how the HNF can be used to solve systems of linear Diophantine equations. Part 2 describes the LLL algorithm for lattice basis reduction and its applications to computer algebra, cryptography and number theory. I will explain how LLL can be used to find small solutions to systems of linear Diophantine equations. Part 3, based on joint work with Luiz A. Peresi of the University of Sao Paulo, describes the application of these techniques to polynomial identities for nonassociative structures.

Lisa Carbone
A Kac-Moody Lie algebra is the most natural generalization to infinite dimensions of a finite dimensional simple Lie algebra. Kac-Moody groups and algebras of affine type are known to have wide applications in physical theories and have concrete mathematical and physical realizations. Hyperbolic Kac-Moody theory naturally generalizes the theory of finite dimensional and affine Kac-Moody groups and algebras, and their symmetries in high energy physics have recently been discovered, though many fundamental questions regarding these structures remain open. In this talk we discuss a program to study forms of hyperbolic Kac-Moody groups over finite fields, their lattices and actions on their hyperbolic buildings.

Christophe Hohlweg
In 1976, Solomon discovered an algebra, the descent algebra, which turns out to be an efficient tool for studying the representations of a finite Coxeter group W. Several surprising properties and generalizations of the descent algebra have been discovered through the last twenty years, linking the descent algebra to different theories such as the theory of the free Lie algebra, the probabilistic theory of card shuffling, and the structure and representation theory of the underlying group W. In this talk, I will give an overview of certain properties and generalizations of descent algebras, and their applications to the study of the characters of W. This lecture will also be suitable for students at the graduate level.

Roozbeh Hazrat
The theory of Azumaya algebras developed parallel to the theory of central simple algebras. However the latter are algebras over fields whereas the former are algebras over rings. One wonders how the K- theory of these objects compare to each other. We look at higher K- theory and reduced K-theory of these objects. We ask nice questions!

Earl Taft
Recently, A. Lauve, S. Rodriguez-Romo and E. J. Taft have constructed a class of one-sided quantum groups. They are modeled after the quantum special linear groups, but have a left antipode which is not a right antipode. The construction turns out to be related to a recent quantization of the boson-fermion correspondence of physics. The classical boson-fermion correspondence for a matrix A with commuting entries is(algebraic version) that the power-series whose n-th coefficient is the trace of the n-th symmetric power of A and the polynomial whose n-th coefficient is (-1)^n times the trace of the n-th exterior power of A are inverses of each other. This correspondence originated in combinatorics as the MacMahon master theorm.

Andrew Douglas
The Lie algebra sl(3) of dimensions 8 has an irreducible representation V(2,2) of dimension 27. By projecting the exterior square of V(2,2) onto itself and onto the adjoint representation, we may define a binary-ternary structure on V(2,2). We will describe how computer algebra was implemented to determine the polynomial identities satisfied by this structure in degrees less than or equal to 7, and see that it precisely satisfies the defining identities of Lie-Yamaguti algebras. We will also discuss the extension of the computations to V(3,3) and V(4,4) whose structures are shown to be generalizations of Lie-Yamaguti algebras. This is joint work with Murray Bremner.

Alex Dent
A permutation group is a group in which the elements acts to permute a set of objects -- i.e. each group element describes a permutation on the set of objects. An orbit is a subset such that, for any two elements of the subset, there exists a group element which maps one set element to the other. This can be thought of as the members of the set which are "connected" by the group. A permutation group is transitive if the set contains only a single orbit (i.e. for any two elements in the set, there exists a group element that maps one set element to the other). By definition, it is impossible for a permutation group to be transitive if it induces more than one orbit on a set. For transitive permutation group, we may also consider extended notions of transitivity such as t-transitivity (where, for any two t-vectors of set elements, there exists a group element that maps one vector to the other component-wise) and t-homogeneity (where, for any two t-sets of set elements, there exists a group element that maps one set to the other). By definition, it is impossible for a permutation group to be t-transitive or t-homogeneous if it is not transitive. We investigate a more general notion of transitivity and homogeneity, termed orbit-t-transitivity and orbit-t-homogeneity, which can be applied to intransitive groups. A group is orbit-t-transitive if, for any two t-vectors in which equal components of each vector lie in the same orbit, there exists a group element that maps one vector to the other vector. A group is orbit-t-homogeneous if, for any two t-sets which contain the same number of elements of each orbit, there exists a group element that maps one set to the other. We show that every permutation group has a maximal orbit-t-transitive/orbit-t-homogeneous subgroup and this subgroup is normal. Furthermore, these subgroups former a chain with certain properties. This is joint work with Prof. Peter Cameron (QMUL).

Last Update: September 9, 2009
Webmaster: Andrew Douglas

New York Algebra Colloquium

Schedule

Abstracts