Geometry and Topology

Speaker: Joel Kamnitzer (Toronto)

"Coherent sheaves on quiver varieties and categorification"

Time: 15:30 - 16:30

Room: MC 108

Nakajima defined a family of hyperKahler varieties called quiver varieties and showed that Kac-Moody algebras acted on their homology. I will explain a categorification of this construction, where we consider derived categories of coherent sheaves on quiver varieties. Conjecturally, we obtain a categorical Lie algebra action in the sense of Rouquier and Khovanov-Lauda.

Noncommutative Geometry

Speaker: Ali Motadelro (Western)

"Metric aspects of noncommutative geometry V"

Time: 14:00 - 15:30

Room: MC 106

Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

Analysis Seminar

Speaker: Tatiana Firsova (Toronto)

"Generic properties of holomorphic foliations of Stein manifolds: topology of leaves and Kupka-Smale property"

Time: 15:30 - 16:30

Room: MC 108

I'll talk about generic 1-dimensional foliations of Stein manifolds that are locally given by vector fields. (The foliations of $\mathbb{C}^n$ serve as the main example.) The leaves of such foliations are Riemann surfaces. I'll describe the topological type of leaves for a generic foliation. The main results can be summarized in the following theorems:

1) For a generic foliation all leaves except for a countable number are homeomorphic to disks, the rest are homeomorphic to cylinders. 2) Generic foliation is Kupka-Smale.

Multidimensional complex analysis, namely approximation theory on Stein manifolds, is the main tool used. All the results used will be referenced and explained.

Pizza Seminar

Speaker: Serge Randriambololona (Western)

"What if we had infinitely many fingers to count on ?"

Time: 17:30 - 18:30

Room: MC 107

Natural numbers encompasses at least two way of counting. The first one tells how many objects a collection has: there are 84 students in the class, 4 apples in my lunch box or 223,647,852 inhabitants in Indonesia. In the second way of counting, we care for the position of an event in a sequence of events: the final exam will be the 106th day of the academic year, "trois" is the name of the numeral that comes after "deux" in French and the 8,000,000,000th human birth has already happened. As far as we only consider finite collections, these two notions of counting lead to the same arithmetic. But when we try to generalize them to infinite collections, surprising phenomena appear.

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern-Weil's approach to Chern classes for vector bundles"

Time: 14:00 - 15:30

Room: MC 106

I will start from the definition of vector bundles over a manifold, basic operations on vector bundles, connections and curvature, Chern-Weil's approach to Chern classes of vector bundles, and basic properties of Chern classes.

Colloquium

Speaker: Reyer Sjamaar (Cornell)

"Induction of representations and Poincaré duality"

Time: 15:30 - 16:30

Room: MC 108

Let G be a group and H a subgroup. Frobenius showed in 1898 how to "enlarge" a representation of H to a representation of G. His method, now called induction, rapidly became a useful technical tool in algebra and harmonic analysis and was adapted by others in various ways. For instance, in 1965 Bott made a systematic study of induction methods based on invariant elliptic differential operators in the context of compact Lie groups, which led to generalizations of the Weyl character formula. I will review and update Bott's work and discuss some applications to K-theory.

Stacks Seminar

Speaker: Jeffrey Morton (Western)

"Groupoid Representation Theory"

Time: 11:30 - 12:30

Room: MC 107

The natural analog for a groupoid of the representation of a group on a vector space is a representation on a vector bundles or, with a little more structure, sheaves. This talk will introduce the representation theory of groupoids, and its relation to Morita equivalence.

Algebra Seminar

Speaker: Rajender Adibhatla (Carleton University)

"Local splitting behaviour of modular Galois representations"

Time: 14:30 - 14:30

Room: MC 108

This talk will discuss the local splitting behaviour of ordinary modular Galois representations and relate them to companion forms and complex multiplication. Two modular forms (specifically $p$-ordinary, normalized eigenforms) are said to be "companions" if the Galois representations attached to them satisfy a certain congruence property. Companion forms modulo $p$ play a role in the weight optimization part of (the recently established) Serre's Modularity Conjecture. Companion forms modulo $p^n$ can be used to reformulate a question of Greenberg about when a normalized eigenform has CM (complex multiplication).

Geometry and Topology

Speaker:

"No lecture"

Time: 15:30 - 16:30

Room: MC 108

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern-Weil's approach to Chern classes for vector bundles II"

Time: 14:00 - 15:30

Room: MC 106

I will start from the definition of vector bundles over a manifold, basic operations on vector bundles, connections and curvature, Chern-Weil's approach to Chern classes of vector bundles, and basic properties of Chern classes.

Analysis Seminar

Speaker: Debraj Chakrabarti (Notre Dame)

"The Cauchy-Riemann equations on Product Domains"

Time: 14:30 - 15:30

Room: MC 108

It is well-known that the $\overline{\partial}$--Neumann operator on a product domain does not preserve smoothness up to the boundary; however, the canonical solution operator does preserve smoothness up to the boundary. We try to understand this phenomenon, and derive some estimates for the canonical solution in Sobolev spaces. (Joint work with Mei-Chi Shaw.)

Noncommutative Geometry

Speaker: Ali Moatadelro (Western)

"Metric aspects of noncommutative geometry VI"

Time: 14:00 - 15:30

Room: MC 106

My aim was to explain four formulas of Riemannian geometry due to Alain Connes which have analogues in NCG. So far we discussed a dual version formula of geodesic distance on a manifold which is stated in terms of algebra of smooth functions, Dirac operator in the Hilbert space of spinors. We also talked about Weyl's formula which gives us the integration against the volume form in this set up. Next we introduced Connes' differential forms and their inner product to bring in Yang-Mills action. In the upcoming talk, I shall try to reach the third formula which states the relation between Connes differential graded algebra and de Rham algebra of differential forms on the manifold.

Algebra Seminar

Speaker: READING WEEK

"(NO TALK SCHEDULED)"

Time: 14:30 - 14:30

Room: MC 108

Geometry and Topology

Speaker:

"No lecture"

Time: 15:30 - 16:30

Room: MC 108

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern-Weil's approach to Chern classes for vector bundles III"

Time: 14:00 - 15:30

Room: MC 106

I will start from the definition of vector bundles over a manifold, basic operations on vector bundles, connections and curvature, Chern-Weil's approach to Chern classes of vector bundles, and basic properties of Chern classes.

Colloquium

Speaker: Dror Bar-Natan (Toronto)

"Homomorphic expansions and w-knots"

Time: 15:30 - 16:30

Room: MC 108

Even though little known, the notion of a "homomorphic expansion" is extremely general; it makes sense in the context of practically any algebraic structure, be it a group, or a group homomorphism, or a quandle, or a planar algebra, or a circuit algebra with unzip operations, or whatever.

Even though little known, w-knots make a cool generalization of ordinary knots. They contain ordinary knots and are contained in 2-knots in 4-space and are easier than the latter. They are a quotient of "virtual knots" and are easier then those.

My talk will be about these two notions, homomorphic expansions and w-knots, and about what happens when the two are put together. Lie algebras arise, and Lie groups, and the Kashiwara-Vergne statement, which is one of the deeper statements about the relationship between Lie groups and Lie algebras.

There are also u-knots, and v-knots, and f-knots, and other things which are not knots at all, and there are equally nifty things to say about homomorphic expansions for all those. But not today.

(For more information and handouts, click on the title above.)

Stacks Seminar

Speaker: Peter Oman (Western)

"Toposes and Groupoids"

Time: 11:30 - 12:30

Room: MC 107

We will show how localic groupoids model a generalized notion of 'topological space' or topos. This talk will introduce toposes, monadic descent, and give an overview of extended Grothendieck-Galois theory developed by A. Joyal and M. Tierney.

Stable Homotopy

Speaker: Sam Isaacson (Western)

"The algebraic Whitehead conjecture"

Time: 13:30 - 14:30

Room: MC 106

Algebra Seminar

Speaker: David Wehlau (Queen's)

"Invariants for the modular cyclic group of prime order via classical invariant theory"

Time: 14:30 - 14:30

Room: MC 108

Let $F$ be any field of characteristic $p$ and let $C_p$ denote the cyclic group of order $p$. It is well-known that there are exactly $p$ inequivalent indecomposable representations $V_1,V_2,\dots,V_p$ of $C_p$ defined over $F$. It is also well-known that there is a unique (up to equivalence) $d+1$ dimensional irreducible complex representation of $\textrm{SL}_2(\mathbb{C})$ given by the action on the space $R_d$ of $d$ forms. In this talk I will describe my recent result which reduces the computation of the ring of $C_p$-invariants of a $C_p$-representation $V=\oplus_{i=1}^k V_{n_i+1}$ to the computation of the classical ring of invariants (or covariants) $C[R_1 \oplus (\oplus_{i=1}^k R_{n_i})]^{\textrm{SL}_2(\mathbb{C})}$.

This allows us to compute for the first time the ring of invariants for many representations of $C_p$.