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.
Speaker:
Uladzimir
Yahorau
(UO)
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.
Speaker:
Hadi Salmasian (U. Ottawa)
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.