Algebra Seminar (Fall 2014 and Winter 2015) Department of Mathematics & Statistics University of Ottawa

 Date Speaker September 26 Kirill Zaynullin October 3 Baptiste Calmes October 6 Laurent Manivel October 31 Jeffrey Pike November 7 Colin Ingalls November 14 Bob Raphael November 21 Matthew Kennedy November 28 Uladzimir Yahorau December 4 Marc-Antoine Leclerc December 19 Brent Pym January 30 Kirill Zaynullin February 13 John Talboom February 27 Ying Zong March 6 Chris Kapulkin  CANCELLED March 13 Alistair Savage March 20 Lucas Calixto March 27 Hadi Salmasian April 28 Rick Jardine May 15 Saeid Molladavoudi

Speaker:  Kirill Zaynullin
Title: Hyperbolic root polynomials.
Time:  Friday September 26 from 1:30 to 2:20
Place:  KED  B004

Abstract:
An important combinatorial result in equivariant cohomology and K-theory
Schubert calculus is represented by the formulas of Billey and Willems for
the localization of Schubert classes at torus fixed points. These formulas
work uniformly in all Lie types, and are based on the concept of a root
polynomial. In this paper we define formal root polynomials associated
with an arbitrary formal group law (and thus a generalized cohomology
theory). We focus on the case of the hyperbolic formal group law
(corresponding to elliptic cohomology). We study some of the properties of
formal root polynomials. We give applications to the efficient computation
of the transition matrix between two natural bases of the formal Demazure
algebra in the hyperbolic case. As a corollary, we rederive in a simpler
and uniform manner the formulas of Billey and Willems. This is a report on
the joint project with C. Lenart (arXiv:1408.5952)

Speaker:  Baptiste Calmes
Title:  Chow-Witt correspondences and motivic cohomology
Time:  Friday October 3, from 1:15 to 2:20
Place:  KED B004

Abstract:
(Joint work with Jean Fasel.)  Voevodsky’s construction of motivic cohomology begins with the definition
of finite correspondences, an algebraic version of multivalued functions.
Although Chow groups do not appear explicitly at this point, I will
explain a way of rephrasing this definition in order to reveal them.
Then, I’ll show how to replace these Chow groups by Chow-Witt groups and
how to define new motivic cohomology groups, refining the ones of
Voevodsky. Finally, I’ll explain some cases in which these groups can be
computed, and in particular I will give an analogue of the
Nesterenko-Suslin-Totaro-Voevodsky isomorphism between the (n,n)-degree
part of motivic cohomology of a field and the n-th Milnor K-theory group
of that field.

Speaker: Laurent Manivel (CRM University of Montreal)
Title: The asymptotics of Kronecker coefficients
Time:  Monday October 6, from 4 to 5pm
Place:  KED B015

Abstract:
Kronecker coefficients are the multiplicities of tensor products
of irreducible representations of symmetric groups. Our understanding of
these coefficients is extremely poor, but our interest for them has been
exacerbated by the role they play in the Geometric Complexity Program
of Mulmuley and Sohoni. I will explain how to understand these
coefficients
geometrically, and how this geometric approach allows to understand
certain asymptotic phenomena like stabilization, recently studied by
Pak-Panova, Stembridge, Vallejo and others.

"Junior Algebra Seminar"

Speaker:  Jeffrey Pike (UO)
Title:  Quivers and Three-Dimensional Lie Algebras
Time:  October 31, from 1:15 to 2:20
Place:  KED B004

Abstract:
One of the first things encountered in an introduction to Lie algebras is a classification of all Lie algebras of dimension at most three. From this classification emerges an infinite family of non-isomorphic three-dimensional Lie algebras that depend upon a continuous parameter.  While these Lie algebras are easy to describe, very little is known about their representation theory. In this talk, I will use the theory of quivers, which are directed graphs, to obtain new results about certain subcategories of representations of these Lie algebras.  I will begin by going over the basics of the representation theory of quivers, and then I will relate these three-dimensional Lie algebras to suitable quivers, which will provide insight into their representation theory.  In particular, even though it is known that these Lie algebras are of wild representation type, I will show that if we impose certain restrictions on weight decompositions, we obtain full subcategories of the category of representations of these Lie algebras that are of finite or tame representation type.

Speaker:  Colin Ingalls (UNB)
Title:  Decorated Minimal Model Program
Time:  November 7, from 1:15 to 2:20
Place:  KED B004

Abstract:
We extend results of Chan and Ingalls concerning the minimal model program for orders over surfaces to all dimensions.  A decoration gives a number for all divisors of all models of a variety. We show that every decorated variety has a terminal resolution. We further show that if one carries out log contractions then decorated terminal varieties remain decorated terminal. As an application, one obtains a decoration from a Brauer class and that this can be used to give a minimal model program for orders over varieties in all dimensions.

This is the joint work of: Daniel Chan, Kenneth Chan, Louis de Thanhoffer de Volcsey, Colin Ingalls, Kelly Jabbusch, Sandor Kovacs, Rajesh Kulkarni, Boris Lerner, Basil Nanayakkara, Shinnosuke Okawa and Michel Van den Bergh.

Speaker:  Bob Raphael (Concordia)
Title:  Limit closures of some classes of commutative rings
Time:  November 14, from 1:15 to 2:20
Place:  KED B004

Abstract:
The category C of all commutative rings without nilpotent elements is complete in the categorical sense.  However, this is not the case for the subcategories of fields, integral domains or integrally closed domains.  The completion of the subcategory of fields in C has long been known but not that of the other two subcategories.  These completions and the corresponding reflector functors will be described.   This is joint work with M. Barr and J. Kennison.

Speaker: Matthew Kennedy (CU)
Title: Operator algebras and analytic group theory
Time:  November 21, from 1:30 to 2:30
Place:  Herzberg 4351 (Carleton U)

Abstract:
It has been known since the work of von Neumann that many questions about the analytic properties of groups are most naturally studied within an operator-algebraic framework. In this talk, I will give an overview of some problems relating the structure of a group to the structure of a corresponding algebra of operators, and a method of attack which utilizes a new approach to the theory of group boundaries. In particular, I will discuss the recent solution of the following very natural problem: For which groups is this algebra simple?  Joint work with E. Breuillard, M. Kalantar and N. Ozawa.

Title:  Conjugacy theorem for extended affine Lie algebras
Time:  November 28, from 1:15 to 2:20
Place:  KED B004

Abstract:
An extended affine Lie algebra (EALA) is a generalization of an affine Kac-Moody Lie algebra to higher nullity (in a sense that can be made precise). It is a pair consisting of a Lie algebra and its maximal adjoint-diagonalizable subalgebra (MAD), satisfying certain axioms. It is natural to ask if a given Lie algebra admits a unique structure of an extended affine Lie algebra, i.e. if two MADs which are parts of two different structures are conjugate. In a joint work with V. Chernousov, E. Neher and A. Pianzola we proved that if the centreless core of an EALA (E,H) is a module of finite type over its centroid then such MADs are conjugate, thereby obtaining a positive answer to this question.

In this talk I will give the definition and construction of an EALA. I will then discuss the proof of the conjugacy theorem for EALAs.

Speaker:  Marc-Antoine Leclerc (UO)
Title:  A hyperbolic Demazure algebra for a Kac-Moody root system
Time:  Thursday December 4, from 10:30 to 11:20
Place:  KED B004

Abstract:
In a recent paper in 2013, A. Hoffnung, J. Malagon-Lopez, A. Savage and K. Zainoulline constructed a generalization of an Hecke algebra starting from a formal group law and a finite root system. In this talk we discuss how to generalize their construction to a Kac-Moody root system in the case of a hyperbolic formal group law. This is joint work with E. Neher and K. Zainoulline.

Speaker:  Brent Pym (Oxford)
Title:  Hypersurface singularities on log symplectic manifolds
Time:  December 19, from 1:15 to 2:20
Place:  KED B004

Abstract:
Log symplectic manifolds are holomorphic Poisson manifolds that are symplectic on an open dense set, but degenerate along a reduced hypersurface.  Examples include Hilbert schemes of del Pezzo surfaces, compactified moduli spaces of SU(2) monopoles, moduli spaces of decorated vector bundles on elliptic curves, and the linear duals of certain Frobenius Lie algebras.  The hypersurfaces that arise in this way are typically highly singular, and I will describe several results that indicate a remarkable degree of rigidity in their local and global structure.  Time permitting, I will outline some applications to noncommutative ring theory by way of deformation quantization.

Speaker:  Kirill Zaynullin
Title:  From cobordism-motives of twisted flag varieties to integer/modular representations of Hecke-type algebras.
Time:  Friday January 30 from 1:15 to 2:20
Place:  KED  B015

Abstract:
This is a report on the joint work in progress with A. Neshitov, N. Semenov and V. Petrov.  Motivated by the motivic Galois group approach, we relate the category of cobordism-motives of twisted flag varieties for a linear algebraic group $G$ with the category of integer (or modular) representations of the associated Hecke-type algebra $H$ for $G$ introduced and studied in a series of papers by Calm\'es, Savage, Zhong and others. In this way, we translate various motivic discrete invariants (e.g. $J$-invariant of linear algebraic groups), results about indecomposable motives and upper motives (Karpenko-Merkurkev-Vishik), etc, into the language of respective integer/modular representations of $H$.

Speaker:  John Talboom (Carleton)
Title:  Irreducible Modules for the Lie Algebra of Divergence Zero Vector Fields on a Torus
Time:  Friday February 13 at 11:30
Place:  room HP 4325 (Carleton U)

Abstract:
In his 1996 paper, S. Eswara Rao investigates "Irreducible Representations of the the Lie-algebra of the Diffeomorphisms of a d-Dimensional Torus."  The current paper considers the restriction of these representations to the subalgebra of divergence zero vector fields. It is shown here that Rao's results transfer to similar irreducibility conditions for the Lie algebra of divergence zero vector fields.

Speaker:  Ying Zong (Montréal)
Title:  Almost non-degenerate abelian fibrations
Time:  Friday February 27 at 11:30
Place:  room 205 in building UCU (University Center, uottawa)

Abstract:  A criterion is provided to recognize/characterize non-degenerate abelian vibrations.

Speaker:  Chris Kapulkin   ---CANCELLED
Title:
Time:  Friday March 6 at 11:30
Place:  room 205 in building UCU (University Center, uottawa)

Abstract:

Speaker:  Alistair Savage
Title:  Twisted Frobenius extensions
Time:  Friday March 13 at 11:30
Place:  room 205 in building UCU (University Center, uottawa)

Abstract:   Frobenius algebras are finite-dimensional unital associative algebras with a certain type of bilinear form giving the algebras nice duality properties.  They are of vital in importance in topological quantum field theory.  Frobenius extensions are generalizations of Frobenius algebras, where one does not require the base ring to be a field.  We will introduce an even more general concept, that of a twisted Frobenius extension, that involves automorphisms of the base ring and the extension.  In the case that these automorphisms are trivial, we recover the usual notion of a Frobenius extension.  The motivation for our definition comes from categorification, where one is often interested in the adjointness properties of induction and restriction functors.  We show that A is a twisted Frobenius extension of B if and only if induction of B-modules to A-modules is twisted biadjoint to restriction of B-modules to A-modules.  A large (non-exhaustive) class of examples is given by the fact that any time A is a Frobenius algebra and B is a subalgebra that is also a Frobenius algebra, then A is a twisted Frobenius extension of B.  This is joint work with Jeffrey Pike.

Speaker:  Lucas Calixto
Title:  Equivariant map queer Lie superalgebras
Time:  Friday March 20 at 11:30
Place:  room 205 in building UCU (University Center, uottawa)

Abstract:  Equivariant map Lie superalgebras are Lie superalgebras consisting of equivariant maps from an algebraic variety or scheme to a "target" Lie superalgebra, that are equivariant with respect to the action of some finite group.  They form a large class of Lie superalgebras that generalize the well-known loop and current Lie superalgebras.  The cases when the "target" is a finite dimensional (non-super) Lie algebra or a basic classical Lie superalgebra have been studied in some depth.  However, beyond these cases, not much is known.  We will address the case when the target is the queer Lie superalgebra.  In particular, we will present a classification of the irreducible finite-dimensional modules in this case.  This is joint work with Adriano Moura and Alistair Savage.

Title:  Spherical polynomials and the spectrum of invariant differential operators for the supersymmetric pair GL(m,2n)/OSp(m,2n)
Time:  Friday March 27 at 11:30
Place:  room 205 in building UCU (University Center, uottawa)

Abstract:  The algebra of invariant differential operators on a multiplicity-free representation of a reductive group has a concrete basis, usually referred to as the Capelli basis. The spectrum of the Capelli basis on spherical representations results in a family of symmetric polynomials (after \rho-shift) which has been studied extensively by Knop and Sahi since the early 90's. In this talk, we generalize some of the Knop-Sahi results to the symmetric superpair GL(m,2n)/OSp(m,2n). As a side result, we show that the qualitative Capelli problem (in the sense of Howe-Umeda) for this superpair has an affirmative answer. Finally, we prove that in the Frobenius coordinates of Sergeev-Veselov, our polynomials turn into the shifted super Jack polynomials. This talk is based on joint work with Siddhartha Sahi.

Speaker:  Rick Jardine (U. Western Ontario)
Title:  Path categories and algorithms
Time:  Tuesday April 28 at 4:15 pm
Place:  KED, room B005

Abstract:  The theory of path categories and path 2-categories for finite oriented cubical and simplicial complexes will be reviewed.  There is an algorithm for computing the path category P(K) of a finite complex K which is based on its path 2-category. This 2-category algorithm will be  displayed, and complexity reduction methods for the algorithm will be  discussed.

The 2-category algorithm works well only for toy examples. The size of the path category P(K) of a complex K can be an exponential function of the size of K. The algorithm has so far resisted parallelization.

One wants combinatorial local to global methods for addressing examples that are effectively infinite. The time variable gives a coarse measure of distance between states, but it is probably only locally defined in the right big picture. The existence of paths between states is an issue in large examples.

Speaker:  Saeid Molladavoudi (U. of Ottawa)
Title:  Topological invariants for abelian quotients of multi-particle quantum states
Time:  Friday May 15 at 2:30 pm
Place:  KED, room B005

Abstract:  We present a mathematical framework to study geometry and topology of quotients for multi-particle quantum systems. In particular, we are interested in geometrical and topological properties of symplectic quotients of pure multipartite states, as complex projective spaces, which are acted upon by maximal tori of the compact semi-simple Lie groups. We discuss that the existing geometrical methods equip us with a powerful set of tools to compute topological invariants, such as Poincare polynomials and Euler characteristics of these abelian symplectic quotients. The analogy can be made with the space of pure states of a composite quantum system containing ''r'' quantum bits under action of the maximal tori of Local Unitary operations.