Geometry and Topology

Speaker: Victor Snaith (Sheffield)

"Monomial resolutions of locally $p$-adic groups"

Time: 14:30 - 15:30

Room: MC 108

In the 1980's (at UWO) I gave a local construction of the Deligne-Langlands epsilon factors attached to representations of Galois groups of local field extensions. The method was to resolve an arbitrary representation by monomial representations, for which the construction was straightforward. At the time my idea was to attack the Langlands programme by making a similar resolution of an arbitrary admissible representation of $GL_{n}K$ where $K$ is a local field.

Returning to this with a bit more knowledge, I now more or less have the correct definition and the outline of the construction to the extent that I can handle $GL_{2}K$!

The entire Langlands programme has many features which were suggested by properties of representations of finite groups such as $GL_{n}{\mathbb F}_{q}$.

So I shall spend a lot of the time illustrating the constructions and properties in the case of finite groups - looking at some or all of: (i) Weil representations, (ii) cuspidality and monomial resolutions, (iii) local L-functions and (iv) Shintani descent.

Geometry and Topology

Speaker: Julie Bergner (UC/Riverside)

"Homotopy-theoretic approaches to higher categories"

Time: 15:30 - 16:30

Room: MC 107

Several models for $(\infty, 1)$-categories have been defined and shown to be equivalent, and they are all being used in different areas of algebra and topology. More recently, there has been interest in more general $(\infty, n)$-categories, especially with Lurie's recent work on the Cobordism Hypothesis. Comparison of different definitions is still work in progress by several authors. In this talk, we will go over some of the models for $(\infty, 1)$-categories and discuss some of the methods for inductively generalizing them to models for $(\infty, n)$-categories.

Colloquium

Speaker: joe blow (Western)

"TBA"

Time: 08:30 - 08:30

Room: MC 108

Symplectic Learning Seminar

Speaker: M. Mousavi (Western)

"Schur-Horn-Kostant theorem : The kernel"

Time: 13:30 - 15:00

Room: MC 104

In this talk, we will construct a kernel that will give a new characterization of doubly stochastic operators. This is needed in the proof of the Horn theorem.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"An Introduction to the Heisenberg Group"

Time: 12:30 - 13:30

Room: MC 108

I will introduce the Heisenberg group, and will elaborate on the Stone-von Neumann Theorem which characterizes the unitary dual of this group. I will then prove the Plancherel Theorem for the Heisenberg group.

Colloquium

Speaker: Eckhard Meinrenken (Toronto)

"Group-valued moment maps and Verlinde formulas"

Time: 15:30 - 16:30

Room: MC 107

The theory of group-valued moment maps provides a natural framework for moduli spaces of flat G-bundles over surfaces. In this talk, I will describe a quantization procedure for such moment maps. An application to the moduli space example gives the Verlinde numbers.

Algebra Seminar

Speaker: Andrew Schultz (Wellesley)

"Counting solutions to Galois embedding problems"

Time: 14:30 - 15:30

Room: MC 107

For any given field $F$ there is a well known parametrizing space for elementary p-abelian Galois extensions of $F$; for example, if $K$ contains a primitive pth root of unity, Kummer theory provides this parametrizing space for us. By putting additional structure on these parametrizing spaces, we are able to give a parametrizing space for solutions to any given embedding problem where the quotient is a cyclic $p$-group and the kernel is an elementary $p$-abelian group. This allows us to give an explicit count to the number of such solutions, and in particular we can make certain universal statements about the number of solutions to such embedding problems. For instance, we use our results to show that $p$-groups have unbounded realization multiplicity.

Geometry and Topology

Speaker: Tomoo Matsumura (Cornell)

"Hamiltonian Torus Actions on Orbifolds"

Time: 15:30 - 16:30

Room: MC 108

When a symplectic manifold M carries a Hamiltonian torus R action, the injectivity theorem states that the R-equivariant cohomology of M is a subring of the one of the fixed points and the GKM theorem allows us to compute this subring by only using the data of 1-dimensional orbits. The results in the first part of this talk are a generalization of this technique to Hamiltonian R actions on orbifolds and an application to the computation of the equivariant cohomology of compact toric orbifolds. In the second part, we will introduce the equivariant Chen-Ruan cohomology ring which is a symplectic invariant of the action on the orbifold and explain the injectivity/GKM theorem for this ring.

Analysis Seminar

Speaker: Ekaterina Shemyakova (Western)

"Differential Transformations for Integrable PDEs"

Time: 15:30 - 16:30

Room: MC 107

Transformational Methods are known to be one of the most efficient methods for finding exact solutions of Partial Differential Equations. In this talk we shall be concentrated on the differential transformations introduced by Darboux (DT). DT can be defined by so-called (m,n)-transformations which are Linear Partial Differential Operators without mixed derivatives. The (m,n)-transformations have interesting algebraic structure. The (m,n)-transformations can help us to solve the problem of the generality of the Darboux Wronskian formulas. Namely, Darboux stated and different authors proved for different cases that given some number of partial solutions, a DT can be defined via some Wronskians. Darboux believed that the reverse statement will be true "generally speaking". In this talk we show several results on our way to prove this reverse statement and to decide what is "the general case" in this context. The second part of the talk will be devoted to an invariant description of the DT. We start with an idea that in view of the said above it would be more efficient to defined DT in terms of invariants of the pair (L,z), where L is a Linear Partial Differential Operator, and z is an element of its kernel. We show that such invariants is in correspondence with solutions of certain PDE, and that instead of a chain of DT we can consider mappings of invariants.

Colloquium

Speaker: Gene Freudenberg (University of Western Michigan)

"Locally Nilpotent Derivations of Rings with Roots Adjoined"

Time: 15:30 - 16:30

Room: MC 107

Working over a ground field k of characteristic zero, this talk will discuss locally nilpotent derivations of rings of the form $B = R[z ]$, where $R$ is a commutative $k$-domain, and $z^n\in R$ for some positive integer $n$. Such a ring has a natural grading by $Z_n$ . We give basic properties of locally nilpotent derivations $D$ of $B$ which are homogeneous relative to this grading. In particular, $D$ is always a quasi-extension of a locally nilpotent derivation $\delta$ of $R$, and $D^2 z = 0$. This approach yields strong sufficient conditions for a ring of this type to be rigid, using in particular the absolute degree $|f |_R$ of elements of $R$. For example, we show that if $R$ is $Z$-graded, $f ∈ R$ is $Z$-homogeneous of degree coprime to $n$, and $|f |_R\ge 2$, then the ring $B = R[f^{1/n}]$ is rigid. The main idea is to study the locally nilpotent derivations of $B$ by looking at those of $R$.

Several applications of our results will be discussed. For example, we study $G_a$ -actions of Pham- Brieskorn surfaces and threefolds, with particular interest in questions of rigidity and stable rigidity. Our methods permit us to show rigidity for many cases which were previously open. This talk represents the speaker’s joint work with L. Moser-Jauslin.

Algebra Seminar

Speaker:

"Algebra Seminar cancelled"

Time: 14:30 - 15:30

Room: MC 107

Geometry and Topology

Speaker: Brooke Shipley (Illinois/Chicago)

"An algebraic model for rational torus-equivariant stable homotopy theory"

Time: 13:30 - 14:30

Room: MC 107

Distinguished Lecture

Speaker: Gunnar Carlsson (Stanford)

"Topology and Data, I"

Time: 15:00 - 16:00

Room: MC 107

The need for effective methods of generating information and ultimately knowledge from experimental data is increasingly acute, due to the fact that such data is being generated at a rapidly growing rate and that the data now comes in much more varied forms than even a few years ago. One way of thinking about data is through its "shape", i.e. about the geometry it caries through the existence of a metric on it. In these talks, we will discuss how one can measure and represent the shape of data using adaptations of methods of topology to point cloud data, or finite metric spaces. We will discuss the computational methodology, and illustrate with various examples.

Distinguished Lecture

Speaker: Gunnar Carlsson (Stanford)

"Topology and Data, II"

Time: 11:00 - 12:00

Room: MC 108

Algebra Seminar

Speaker:

"Algebra Seminar cancelled"

Time: 14:30 - 15:30

Room: MC 107

Algebra Seminar

Speaker: Ali Moatadelro (Western)

"Noncommutative complex geometry of the quantum projective space"

Time: 14:30 - 15:30

Room: MC 107

We consider a natural holomorphic structure on the quantum projective space $\mathbb{C}P^l_q$ already presented in the literature and define holomorphic structures on canonical quantum line bundles on it. The space of holomorphic sections of these line bundles then will determine the quantum homogeneous coordinate ring of $\mathbb{C}P^l_q$. We define bimodule connections on canonical line bundles and this enables us to identify the quantum homogeneous coordinate ring of $\mathbb{C}P^l_q$ with the ring of twisted polynomials. We also introduce a twisted positive Hochschild $2l$-cocycle on $\mathbb{C}P^l_q$, by using the complex structure of $\mathbb{C}P^l_q$, and show that it is cohomologous to its fundamental class which is represented by a twisted cyclic cocycle. This certainly provides further evidence for the belief that holomorphic structures in noncommutative geometry should be represented by (extremal) positive Hochschild cocycles within the fundamental class. Finally we verify directly that the main statements of the Riemann-Roch formula and Serre duality theorem hold for $\mathbb{C}P^1_q$ and $\mathbb{C}P^2_q$. This is joint work with Masoud Khalkhali.