Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"Morse inequalities through spectral geometry I"

Time: 14:30

Room: MC 108

Study of the topological and geometric properties of a (Riemannian) manifold by investigating the spectral properties of the geometric elliptic operators, or in general elliptic complexes, is the approach of the spectral geometry. Witten, in his famous paper "Supersymmetry and Morse theory", used the spectral properties of the perturbed de Rham complex, so called Witten complex, to prove the Morse inequalities. In this talk we shall cover his proof. The idea of the proof is to use the approximations of the eigenvalues of the corresponding laplacian. In the next step, we will have an overview on Bismut's proof. Bismut puts Witten's idea in another format. He proves the inequalities by studying the long term behavior of the heat kernel.

Geometry and Topology

Speaker: Cihan Okay (UBC)

"Homotopy Colimits of Classifying Spaces of Abelian Groups"

Time: 15:30

Room: MC 107

Homotopy colimit of classifying spaces of abelian subgroups of a finite group $G$ capture information about the commutativity of the group. For the class of extraspecial p-groups of rank at least 2 these colimits are not of the homotopy type of a $K(\pi,1)$ space. The main ingredient in the proof is the calculation of the fundamental group. Another natural question is the complex $K$-theory of these homotopy colimits which can be computed modulo torsion. In contrast to the classifying space $BG$, torsion groups may appear in $K^1$.

Analysis Seminar

Speaker: Roman Dwilewicz (Missouri University of Science and Technology)

"Recent Results on Holomorphic Extension of Functions for Unbounded Domains in $\mathbb{C}^n$"

Time: 14:30

Room: MC 108

In the talk there will be presented recent results on global holomorphic extensions for generalized tubes in $\mathbb{C}^n$ and tube-like domains in $\mathbb{C}^2$. There is an interesting geometry behind the extension problem for unbounded domains, namely (in some cases) it depends on the position of a complex variety in the closure of the domain. The extension problem appeared non-trivial and the work is in progress. However the talk will be illustrated by many figures and pictures and should be accessible also to graduate students. This is a common work with Al Boggess (Arizona State Univ.) and Zbigniew Slodkowski (Univ. Illinois at Chicago).

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"Dixmier Trace"

Time: 14:30

Room: MC 108

It is well known that a normal trace on $B(H)_{+}$ is proportional to the usual trace. On the other hand, it has been an open question whether or not a trace is proportional to the usual one on the set where that trace is finite. In 1966 Dixmier gave negative examples to this problem. Traces $f$ constructed by him have the following properties: $f(a) = 0$ for every operator a of finite rank, but $f(b) = 1$ for some compact operator $b$. Such traces are called Dixmier trace. In this talk I am going to talk about constructing Dixmier traces and deal with some examples.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"Morse inequalities through spectral geometry II"

Time: 14:30

Room: MC 108

Study of the topological and geometric properties of a (Riemannian) manifold by investigating the spectral properties of the geometric elliptic operators, or in general elliptic complexes, is the approach of the spectral geometry. Witten, in his famous paper "Supersymmetry and Morse theory", used the spectral properties of the perturbed de Rham complex, so called Witten complex, to prove the Morse inequalities. In this talk we shall cover his proof. The idea of the proof is to use the approximations of the eigenvalues of the corresponding laplacian. In the next step, we will have an overview on Bismut's proof. Bismut puts Witten's idea in another format. He proves the inequalities by studying the long term behavior of the heat kernel.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"The Mathai-Quillen superconnection construction"

Time: 13:30

Room: MC 108

I will review the paper "Superconnections, Thom classes, and equivariant differential forms" by Mathai and Quillen, and explain how their results can be used to simplify the cohomological formula for the index of an elliptic operator on a compact manifold.

Geometry and Topology

Speaker: Craig Westerland (Univ. of Minnesota)

"An analogue of K-theory for higher chromatic homotopy theory"

Time: 15:30

Room: MC 107

We introduce a new cohomology theory constructed using homotopy theoretic methods that bears some formal resemblance to K-theory -- it is equipped with Adams operations, a form of periodicity, and an analogue of the J-homomorphism. However, the information that it encodes is of a higher "chromatic level" than K-theory, and so may be suitable for studying higher chromatic phenomena in stable homotopy theory. Unfortunately, its geometric meaning is far from apparent, although there are some hints of a relationship with n-gerbes.

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.

Noncommutative Geometry

Speaker: Magdalena Georgescu (University of Victoria)

"Spectral flow: An introduction I"

Time: 14:30

Room: MC 108

In the context of B(H) (the set of bounded operators on a separable Hilbert space), the spectral flow counts the net number of eigenvalues which change sign as one travels along a path of self-adjoint Fredholm operators. The ability to calculate the spectral flow allows one to calculate the Fredholm index of some operators, making it of interest in the study of non-commutative geometry. It is possible to generalize the concept of spectral flow to a semifinite von Neumann algebra, as we can use a trace on the algebra to measure the amount of spectrum which changes sign. During the course of the two talks, I will start by giving a detailed introduction to spectral flow (for both bounded and unbounded operators), followed by an overview of some important results for the B(H) case, including a characterization of spectral flow due to Lesch, integral formulas for spectral flow, and geometric interpretations (e.g. spectral flow as an intersection number). I will give sketches of some of the more illuminating proofs, and conclude by discussing some of the changes required for the generalization to semifinite von Neumann algebras.

Geometry and Topology

Speaker: Kirill Zaynullin (Ottawa)

"Oriented cohomology of projective homogeneous spaces"

Time: 15:30

Room: MC 107

Oriented cohomology theories and the associated formal groups laws have been a subject of intensive investigations since 60's, mostly inspired by the theory of complex cobordism. In the present talk we discuss several recent developments in the study of algebraic analogues of such theories, e.g. algebraic cobordism of Levine-Morel or algebraic elliptic cohomology, of projective homogeneous spaces.

Analysis Seminar

Speaker: Tatyana Barron (Western)

"Line bundles and automorphic forms"

Time: 15:30

Room: MC 108

I will explain how automorphic forms appear in quantization of compact Riemann surfaces, or, in higher dimensions, quotients of bounded symmetric domains (e.g. ball quotients). I will mention some of my results on explicit construction of automorphic forms as Poincare series. After that I will briefly mention some results from my two papers with N. Askaripour and will pose a few related questions, hoping that maybe someone in the audience will have comments or suggestions.

Noncommutative Geometry

Speaker: Magdalena Georgescu (University of Victoria)

"Spectral flow: An introduction II"

Time: 14:30

Room: MC 108

In the context of B(H) (the set of bounded operators on a separable Hilbert space), the spectral flow counts the net number of eigenvalues which change sign as one travels along a path of self-adjoint Fredholm operators. The ability to calculate the spectral flow allows one to calculate the Fredholm index of some operators, making it of interest in the study of non-commutative geometry. It is possible to generalize the concept of spectral flow to a semifinite von Neumann algebra, as we can use a trace on the algebra to measure the amount of spectrum which changes sign. During the course of the two talks, I will start by giving a detailed introduction to spectral flow (for both bounded and unbounded operators), followed by an overview of some important results for the B(H) case, including a characterization of spectral flow due to Lesch, integral formulas for spectral flow, and geometric interpretations (e.g. spectral flow as an intersection number). I will give sketches of some of the more illuminating proofs, and conclude by discussing some of the changes required for the generalization to semifinite von Neumann algebras.

Homotopy Theory

Speaker:

"Talk CANCELED"

Time: 14:30

Room:

We will resume next week.

Colloquium

Speaker: Alex Buchel (Western)

"Localization and holography in N=2 gauge theories"

Time: 15:30

Room: MC 107

Gauge theory/string theory correspondence maps dynamics of strongly coupled gauge theories to that of the classical supergravity. We describe a highly non-trivial check of the correspondence in the context of Seiberg-Witten models. Specifically, using localization techniques, the path-integral of N=2 supersymmetric SU(Nc) gauge theory can be computed exactly by reducing it to a certain matrix model. In the large-Nc limit the saddle point of the matrix integral picks a particular point on the Coulomb branch of the moduli space. We show that precisely the same point is picked out by the dual gravitational description of the theory. We comment on supersymmetric Wilson loops and the free energy of the theory.

Algebra Seminar

Speaker: Johannes Middeke (Western)

"TBA"

Time: 14:30

Room: MC 107