Noncommutative Geometry

Speaker: Sean Fitzpatrick (Western)

"Geometric quantization of Kaehler manifolds"

Time: 14:30

Room: MC 108

This will be a survey talk on the method of geometric quantization in symplectic geometry, which attempts to associate to each symplectic manifold, viewed as a model for classical mechanics, a "quantum" Hilbert space in a way that is consistent with quantum mechanics. For simplicity I will stick to the case of Kaehler manifolds. In particular I will discuss the Kostant-Souriau approach via prequantum line/circle bundles and polarizations; the Poisson algebra of functions and Hamiltonian group actions; and the role of index theory for Dirac operators. Time permitting, I will also mention the relationship between geometric quantization and Berezin-Toeplitz quantization.

Colloquium

Speaker: Jean-François Lafont (Ohio State University)

"Constructing closed aspherical manifolds"

Time: 15:30

Room: MC 107

A manifold is aspherical if its universal cover is contractible. There are only a few known techniques for constructing closed aspherical manifolds. I will give an overview of these techniques, and explain how they can be used to produce some interesting examples.

Homotopy Theory

Speaker: Mike Misamore (Western)

"Introduction to dg-categories"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: Ali Fathi (Western)

"The Eta Invariant "

Time: 14:30

Room: MC 108

The Eta invariant appeared for the first time as a correction term in Atiyah-Patodi-Singer Index formula for manifolds with boundary. It can be thought of as regularized signature for infinite matrices which measures the amount of spectral asymmetry. I will first introduce the eta function of a self adjoint differential operator and explain the meromorphic structure of it, then I will give a variational proof for the conformal invariance of the eta invariant for the class of conformally covariant differential operators.

Geometry and Combinatorics

Speaker: Lior Bary-Soroker (Tel Aviv University)

"Number theory in function fields"

Time: 09:30

Room: MC 107

There is a deep and fascinating connection between the ring of integer numbers and the ring of univariate polynomials over a finite field.

In this talk I will discuss the classical theory, and I will present a new approach based on Galois theory and Field Arithmetic.

I will demonstrate the method by solving a function field version of the classic problem on primes in short intervals.

Geometry and Combinatorics

Speaker: Danny Neftin ((?))

"The absolute Galois group of $Q$ and its Sylow subgroups"

Time: 10:30

Room: MC 107

Understanding the rich structure of the absolute Galois group of the field $Q$ of rational numbers is a central goal in number theory.

Following Serre's question, the Sylow subgroups of the absolute Galois group of the fields $Q_p$ of $p$-adic numbers were studied and completely understood by Labute. However, the structure of the $p$-Sylow subgroups of the absolute Galois group of $Q$ is much more subtle and mysterious.

We shall discuss the first steps towards its determination via a surprisingly simple decomposition.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Mathai-Quillen meets Chern-Weil"

Time: 13:30

Room: MC 108

Last week, I outlined an approach due to Mathai and Quillen which uses Clifford algebras to give an explicit computation of the Chern character of a superconnection on the level of differential forms, in which we see the emergence of a Gaussian-shaped Thom form. This week, I will explain how one can simplify the construction using equivariant differential forms, and obtain a 'universal' Thom form as the result. Time permitting, I'll explain how this construction, which is also due to Mathai and Quillen, can easily be extended to define the equivariant characteristic classes that appear in the cohomological equivariant index.

Geometry and Topology

Speaker: Piotr Maciak (EPFL)

"Bounds for the Euclidean minima of algebraic number and function fields"

Time: 15:30

Room: MC 107

The Euclidean division is a basic tool when dealing with the ordinary integers. It does not extend to rings of integers of algebraic number fields in general. It is natural to ask how to measure the "deviation" from the Euclidean property, and this leads to the notion of Euclidean minimum. The case of totally real number fields is of special interest, in particular because of a conjectured upper bound (conjecture attributed to Minkowski). The talk will present some recent results concerning abelian fields of prime power conductor. We will also define Euclidean minima for function fields and give some bounds for this invariant. We furthermore show that the results are analogous to those obtained in the number field case.

Colloquium

Speaker: Spiro Karigiannis (University of Waterloo)

"An introduction to $G_2$ manifolds and $G_2$ conifolds"

Time: 15:30

Room: MC 108

The exceptional properties of the octonion algebra allow us to define the notion of a $G_2$ structure on an oriented spin 7-manifold, which is a certain ``nondegenerate'' 3-form that induces a Riemannian metric in a nonlinear way. The manifold is called a $G_2$ manifold if the 3-form is parallel. Such manifolds are always Ricci-flat, and are of interest in physics. More recently, however, there has been interest in G2 ``conifolds'', which have a finite number of isolated ``cone-like'' singularities. We will begin with an introduction to $G_2$ manifolds for a general audience, paying particular attention to the similarities and differences of $G_2$ geometry with respect to the geometries of K\"ahler manifolds and of 3-manifolds. Then we will define $G_2$ conifolds, and discuss some results about them, including their desingularization and their deformation theory.