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

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"Connes' trace theorem"

Time: 14:30

Room: MC 108

I will give a proof of the following result known as Connes' trace theorem: Any pseudodifferential operator of order $-n$ acting on the smooth sections of a vector bundle over a compact $n$-dimensional manifold is in the domain of the Dixmier trace and its Dixmier trace coincides with its Wodzicki residue.

Analysis Seminar

Speaker: Nathaniel Johnston (University of Waterloo)

"The Separability Problem and its Variants in Quantum Entanglement Theory"

Time: 15:30

Room: MC 108

The separability problem, which is one of the central problems in the theory of quantum entanglement, asks for simple methods to determine whether a given quantum state is entangled (i.e., contains useful "quantumness") or separable (i.e., not entangled). We discuss how norms and eigenvector perturbation results can be used to approach problems like the separability problem, and we present some recent progress on these problems.

Homotopy Theory

Speaker: Martin Frankland (Western)

"Introduction to dg-categories II"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: Baran Serajelahi (Western)

"Morse Homology"

Time: 14:30

Room: MC 108

Let $f:M^n\rightarrow \mathbb{R}$ be a function with only nondegenerate critical points. Denote by $Crit_kf$ those critical points of f that have index k, let $c_k$ denote their total number. Consider the free abelian groups $C_k=\mathbb{Z}^{c_k}$, $C_k$ has one generator for each critical point of index k that f has. It is well known that that the strong Morse inequalities $c_k-c_{k-1}+\dots\pm c_0\geq b_k-b_{k-1}+\dots\pm b_0$ for $k=0,\dots,n-1$ and $c_n-c_{n-1}+\dots\pm c_0=b_n-b_{n-1}+\dots\pm b_0$, are equivalent to the existence of boundary homomorphisms $\partial_k:C_k\rightarrow C_{k-1}$ whose homology groups have rank, $b_k=Rank(H_k(M;\mathbb{Z}))$.There are several ways of getting to a boundary operator that will work. In this talk we will discuss one approach to constructing such a chain complex for a manifold M, given a metric g on M and a Morse function f on M. All approaches of which I am aware are based on the following observation. Associated to every Morse function f on M is a dynamical system given by the negative gradient flow of f. To define $\partial_k:C_k\rightarrow C_{k-1}$ we will investigate this dynamical system.

Index Theory Seminar

Speaker: Masoud Khalkhali (Western)

"The $\Gamma$ index theorem of Atiyah"

Time: 12:30

Room: MC 108

Abstract: The $\Gamma$ index theorem is one of the first index theorems on non-compact spaces. It plays an important role in extensions of index theory in noncommutative geometry and in representation theory of non compact Lie groups. I shall give a proof of this theorem and indicate some applications.

Algebra Seminar

Speaker: Johannes Middeke (Western)

"(postponed until Jan recovers!)"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"The Weitzenbock Formula"

Time: 14:30

Room: MC 108

Given a compact Riemannian manifold $M$ and $D^{2}$ the Dirac Laplacian on the Clifford bundle, Bochner discovered the existence of a self-adjoint, non-negative Laplacian $\Delta$ such that the difference $D^{2} - \Delta$ is a zero-order operator that can be expressed in terms of the curvature tensor of $M$. In fact, combined with some harmonic theory, these operators allowed Bochner to obtain fundamental vanishing theorems involving the Betti numbers of $M$. In this talk, I will recall the Dirac and connection Laplacian operators and prove the general Bochner identity. Using this, I will prove the Weitzenbock formula and a vanishing theorem of the first Betti number $b_{1}(M) = dim (H^{1}(M, \mathbb{R}))$.

Geometry and Topology

Speaker: Martin Brandenburg (Muenster)

"Algebraic geometry of tensor categories"

Time: 15:30

Room: MC 107

Various results by Tannaka, Krein, Deligne, Lurie, Hall, Schaeppi, Chirvasitu and B. show that a scheme / algebraic stack can be recovered from its tensor category of quasi-coherent sheaves. This motivates to generalize several constructions from algebraic geometry to tensor category theory. I would like to illustrate this process for affine and projective morphisms, tangent bundles, and fiber products.

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"Universal Taylor series"

Time: 15:30

Room: MC 108

A holomorphic function on a planar domain $\Omega$ is said to possess a universal Taylor series about a point in $\Omega$ if the partial sums of the Taylor series have the following surprising property: they can approximate arbitrary polynomials on arbitrary compact sets $K$ outside $\Omega$ (provided only that $K$ has connected complement). In the last few years, central questions about universal Taylor series have been addressed using potential theory. In this talk we will discuss some of these results and in particular we will focus on the boundary behaviour of such functions.

Noncommutative Geometry

Speaker: Matthias Franz (Western)

"Maximal syzygies in equivariant cohomology"

Time: 14:30

Room: MC 108

This is the last part of the syzygy saga (but independent of the previous talks).

I will recall how syzygies interpolate between torsion-free and free modules and why certain syzygies are ruled out in the torus-equivariant cohomology of compact orientable manifolds. More precisely, by a result of Allday, Puppe and myself there is a bound on the syzygy order unless the equivariant cohomology is free. The main point of this talk is to show that this bound is sharp. I will do so by exhibiting a new class compact orientable manifolds with torus action. These manifolds are related to polygon spaces as studied by Hausmann, Farber and many others.

Homotopy Theory

Speaker: Hugo Bacard (Western)

"Model categories and dg-categories"

Time: 14:30

Room: MC 107

Index Theory Seminar

Speaker: Masoud Khalkhali (Western)

"Index theorem for homegenous differential operators"

Time: 12:00

Room: MC 107

By an old result of Raoul Bott, the index of homogeneous Dirac operators can be computed using Weyl character formula. Thus at least in this case one can in principle bypass a substantial amount of analysis and reduce the Atiyah-Singer index theorem to representation theory of compact Lie groups. In my talk I shall recall these results and discuss their impact on equivariant index theorems.

Colloquium

Speaker: Manfred Kolster (McMaster)

"Special values of zeta-functions and motivic cohomology"

Time: 15:30

Room: MC 107

In the 1970's Lichtenbaum conjectured a formula for special values of zeta-functions of number fields at negative integers in terms of algebraic K-groups. I will give an overview of the results on this and related conjectures and show how Voevodsky's proof of the Bloch-Kato Conjecture not only allowed to prove the Lichtenbaum conjecture for abelian number fields, but suggests a more complete motivic cohomology version, which includes the prime 2.

Geometry and Topology

Speaker: Reading Week (Western)

"No Seminar"

Time: 15:30

Room: MC 107

Homotopy Theory

Speaker:

"Reading Week. No meeting today."

Time: 14:30

Room:

We will resume next week as usual.

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"The analytic index of transversally elliptic operators"

Time: 12:00

Room: MC 107

Given the action of a compact Lie group on a smooth manifold, it is possible to define a class of (pseudo-)differential operators known as the transversally elliptic operators. The principal symbol of such an operator is only guaranteed to be invertible for nonzero covectors that are conormal to the group orbits. I will give the precise definition of these operators, and explain why they have a well-defined equivariant index, in spite of the fact that in general, they are not Fredholm operators.

Colloquium

Speaker: Johanes Walcher (Dept. of Physics, McGill)

"Mirror symmetry at the math-physics interface"

Time: 15:30

Room: MC 107

Mirror symmetry proposes a rather general correspondence of algebraic structures in complex and symplectic geometry, and illustrates the interdependence of mathematics and theoretical physics. I will review selected aspects of the story, from the origins in Calabi-Yau compactifications of the superstring to recent calculations relating to arithmetic of algebraic cycles.

Geometry and Topology

Speaker: Philip Hackney (Stockholm)

"Infinity Properads"

Time: 15:30

Room: MC 107

Properads are an extension of the notion of operad which allow one to model structures with many-to-many operations, such as various kinds of bialgebras. In this talk we will discuss up-to-homotopy versions of properads, as well as potential applications.

Algebra Seminar

Speaker: Thomas WEigel (Milan-Bicocca)

"Necklaces, finite fields, and Lie algebras"

Time: 15:30

Room: MC 108

Necklace polynomials arise in different areas of mathematics: combinatorics, arithmetic and Lie theory. In my talk, I will discuss their significance in each of these areas with special emphasis on a generalised Witt formula that one may deduce for graded Lie algebras. This formula can be used to prove a Gromov-like theorem for graded Lie algebras of type FP. Necklaces, finite fields, and Lie algebras

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Introduction to Harish-Chandra characters of semi-simple Lie groups"

Time: 14:30

Room: MC 108

The index of a transversally elliptic operator is not an integer. It is a character of an infinite dimensional representation. Such characters, when they can be defined, make sense only as distribution but under some mild conditions they can be shown to be representable by locally integrable functions. This talk is a general introduction to such characters mostly from a purely representation theoretic view and can be followed independently of index theory seminar talks.

Homotopy Theory

Speaker: Daniel Schaeppi (Western)

"The homotopy category of dg-categories"

Time: 14:30

Room: MC 107

Index Theory Seminar

Speaker: Sean Fitzpatrick (Western)

"Properties of the index of transversally elliptic operators"

Time: 12:00

Room: MC 107

Continuing from last week's lecture, I will discuss some of the functorial properties of the index of transversally elliptic operators, and give some basic examples.

Colloquium

Speaker: André Joyal (UQAM)

"What is Homotopy Type Theory?"

Time: 15:30

Room: MC 107

HOTT is a new branch of mathematics arising from the unexpected encounter of logic with homotopy theory. It provides a new foundation of mathematics which can be implemented in a computerised proof assistant like Coq or Agda. I will briefly describe the history of the subject, from Martin-Löf, to Awodey, Warren and Voevodsky. I will describe HOTT in the language of category theory and discuss the geometric meaning of Voevodsky's univalence axiom.