Algebra Seminar

Speaker: Sheldon Joyner (Western)

"On a certain $PSL(2, Z) 1$-cocycle"

Time: 14:30 - 15:30

Room: MC 107

Classically, if some manifold $M$ is equipped with an action of a subgroup $G$ of $PSL(2, Z)$ under which a certain space $F$ of functions on $M$ transforms via a $1$-cocycle, the latter is referred to as an automorphy factor, and the functions $F$ are said to be $G$-modular. In this talk, I will produce an injective $1$-cocycle of $PSL(2, Z)$ into a certain group of formal power series which extends the well-known identification of the fundamental group of $P^1\{0,1,\infty\}\;$ with associated Chen series. This cocycle may be regarded as a quasi-automorphy factor for sections of the universal prounipotent bundle with connection on $PSL(2, Z)$ - in particular for the polylogarithm generating series $Li(z)$. I will go on to show that the quasi-modularity of $Li(z)$ may be used to give a family of proofs of the analytic continuation and functional equation for the Riemann zeta function. Moreover, under this cocycle, the involutive generator of $PSL(2, Z)$ maps to the Drinfeld associator, while the infinite cyclic generator maps to an $R-matrix$, in Drinfeld's formal model of the quasi-triangular quasi-Hopf algebras, thereby producing a representation of $PSL(2, Z)$ into tensor products of certain $qtqH$ algebras. Underlying the whole story is a path space realization of $PSL(2, Z)$ using Deligne's idea of tangential basepoint.

Algebra Seminar

Speaker: Stefan Tohaneanu (Western)

"Minimum distance of linear codes and geometry of points"

Time: 14:30 - 15:30

Room: MC 108

Let $\mathcal C$ be an $[n,k,d]-$linear code with generating matrix $A$; this is assumed to be a rank $k$, $k\times n$ matrix with entries in a field $\mathbb K$. Computing $d$, the minimum distance, is in general an NP-hard problem and finding a good lower bound has been a major question in algebraic coding theory. If the matrix $A$ has no proportional nor zero columns, consider $\Gamma\subset\mathbb P^{k-1}$ the set of $n$ distinct points with homogeneous coordinates the entries of each column of $A$. In this talk we present a good lower bound for $d$ in terms of a certain shift in the last free module of the graded minimal free resolution of $\mathbb K$ $[x_1,\ldots,x_k]/I(\Gamma)$. We also present the De Boer-Pellikaan method to compute $d$. As a consequence of this method one can obtain a symbolic description of the variety of minimal codewords in terms of the top $Ext$ module of a certain ideal generated by products of linear forms.

Geometry and Topology

Speaker: Parker Lowrey (Western)

"Descent for derived categories"

Time: 15:30 - 16:30

Room: MC 107

Classically, in algebraic geometry, descent for quasi-coherent sheaves is valid for maps between schemes satisfying certain flatness and finiteness conditions. With the machinery developed in stable infinity categories, one can extend decent to the stable infinity category $QCoh(X)$ (whose homotopy category is the derived category of the scheme $X$) and to maps that are not necessarily flat. We will give some examples and discuss plausible conditions on maps to satisfy descent.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Real Submanifolds in a Complex Space I"

Time: 15:30 - 16:30

Room: MC 107

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

Colloquium

Speaker: Rick Jardine (Western)

"Path categories and calculations"

Time: 15:30 - 16:20

Room: MC 107

The path category P(X) of a space X is an invariant which is defined much like the fundamental groupoid, except that directions of paths are not formally reversed. This construction has applications in theoretical computer science, where it gives, in principle, a description of execution paths in geometric models for the behaviour of parallel processing systems. Path categories have resisted calculational analysis until just recently, in part because standard homotopy theoric methods are not applicable.

We now know that the path category P(K) of a finite simplicial complex K can be computed by an algorithmic method which is based on the existence of a finite 2-categorical resolution.

Methods of implementation and applications of this result will be discussed.

Algebra Seminar

Speaker: Andrey Minchenko (Western)

"Enhanced Dynkin diagrams"

Time: 14:30 - 15:30

Room: MC 107

The root system R of a complex semisimple Lie algebra is uniquely determined by its basis (also called a simple root system). It is natural to ask whether all homomorphisms of root systems come from homomorphisms of their bases. Since the Dynkin diagram of R is, in general, not large enough to contain the diagrams of all subsystems of R, the answer to this question is negative. We introduce a canonical enlargement of a basis (called an enhanced basis) for which the stated question has a positive answer. We use the name an enhanced Dynkin diagram for a diagram representing an enhanced basis. These diagrams in combination with other new tools (mosets, core groups) allow one to obtain a transparent picture of the natural partial order between Weyl orbits of subsystems in R.

Geometry and Topology

Speaker: Marcy Robertson (Western)

"Derived Morita Theory for Enriched Symmetric Multicategories"

Time: 15:30 - 16:30

Room: MC 107

Operads, multicategories, and their representations (also called operadic/multicategorical algebras) play a key role in organizing hierarchies of higher homotopies in any category with a good notion of homotopy theory. In this talk we show how one can generalize work of Toen and Rezk to provide a description of the derived category of any multicategorical algebra. Time permitting, we will discuss applications of this theory to problems in combinatorial representation theory.

We do not assume prior knowledge of the theory of operads and multicategories.

Analysis Seminar

Speaker: Ilya Kossovskiy (Western)

"Real Submanifolds in a Complex Space II"

Time: 15:30 - 16:30

Room: MC 107

The Theory of Real Submanifolds in a Complex Space (which is sometimes called, in some more general settings, "CR-geometry") goes back to H.Poincare and was deeply developed in further works of E.Cartan, N.Tanaka, S.Chern and J.Moser. In the present series of lectures we consider the classical aspects of this theory, as well as some recent results, focusing mainly on the holomorphic equivalence problem, groups of holomorphic symmetries and the holomorphic extension problem for real submanifolds in a complex space.

Colloquium

Speaker: Nantel Bergeron (York)

"Supercharacter theory of upper triangular matrices over finite fields and symmetric functions in non-commutative variables"

Time: 15:30 - 16:30

Room: MC 107

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent upper triangular matrices with coefficients in a finite field, and NCSym, the symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are Hopf isomorphic. This allows developments in each to be transfered. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

The resulting theory is very nicely behaved — there is a rich combinatorics describing induction and restriction along with an elegant formula for the values of superclasses on superclasses. The combinatorics is described in terms of set partitions (the symmetric groups theory involves integer partitions) and the combinatorics seems akin to tableau combinatorics. At the same time, supercharacter theory is rich enough to serve as a substitute for ordinary character theory in some problems.