Noncommutative Geometry

Speaker: TBA (Western)

"Learning Seminar"

Time: 11:00

Room: MC 108

The NCG learning seminar's topic this year will be Geometric Analysis. One of our goals is to go through a heat equation proof of the celebrated Atiyah-Singer index theorem. The following topics will be covered: 1. Operators of Dirac type and its main examples, 2. Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula, 3. Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula, 4. The index problem for elliptic PDE's, characteristic classes via Chern-Weil theory, 5. Miraculous cancellations, Getzler's supersymmetric proof of the Atiyah-Singer index theorem, special cases: Gauss-Bonnet-Chern, Hirzebruch signature theorem, and Riemann-Roch. 6. Approach via path integrals and quantum mechanics, 7. Atiyah-Bott-Lefschetz fixed point formula.

Colloquium

Speaker: Alejandro Adem (UBC/Mitacs)

"Topology of Commuting Matrices"

Time: 14:30

Room: MC 107

In this talk we will describe basic topological properties of the space of commuting unitary matrices. In particular we will show that they can be assembled to form a space which classifies commutativity for vector bundles and which has very interesting homotopy-theoretic properties.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:00

Room: MC 106

We will continue with the following topics:

---Operators of Dirac type and its main examples, ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula.

Geometry and Topology

Speaker: Christin Bibby (Western)

"Rational homotopy theory of chordal arrangements"

Time: 15:30

Room: MC 107

To any graph, one may associate three flavors of hyperplane arrangements: linear (subspaces of a complex vector space), toric (subtori of a complex torus), or abelian (abelian subvarieties of a complex abelian variety). In the linear case, there is considerable literature on the rational homotopy theory of the complement, and the toric case is similar in flavor. The abelian case is more complicated due to lack of formality of the space. When the graph is chordal, we have a Koszul model and use quadratic-linear duality to compute the minimal model and show that the space is rationally $K(\pi,1)$. This is joint work with Justin Hilburn.

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western)

"Hilbert C*-modules: KSGNS construction"

Time: 12:30

Room: MC 107

In the first part, we continue with the following topics: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

In the second part, Mitsuru Wilson (Western) will continue developing the theory of Hilbert C*-modules and discuss the KSGNS construction.

Geometry and Topology

Speaker: Karol Szumilo (Western)

"Simplicial Localization of Cofibration Categories"

Time: 15:30

Room: MC 107

Simplicial localization is a localization of a category at a class of morphisms yielding a higher category that captures homotopy theory of the original category. I will begin with a review of motivations behind the simplicial localization and some of its classical constructions. In the main part of the talk, I will discuss a recent result, joint with Chris Kapulkin, about a new way of constructing simplicial localizations.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

In this lecture, we continue with the following topics: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Approximation on Real Surfaces"

Time: 15:30

Room: MC 107

I will discuss the following result, which is joint work with A. Sukhov: on any compact real surface $S$, except the sphere and $RP^2$, there exist two smooth functions with the property that their rational combinations are dense in the space of continuous functions on $S$. The proof relies on many different results in several complex variables, symplectic geometry, and approximation theory. I will give a general overview of this.

Colloquium

Speaker: Markus Mueller (Western)

"Majorization, thermodynamics, and a characterization of Shannon entropy"

Time: 15:30

Room: MC 107

In the last few years, there has been a wave of interest in thermodynamics of very small or strongly correlated systems. This research, rooted in quantum information theory, has given rise to several interesting mathematical questions on majorization and its behavior under tensor products. In the talk, I will present one theorem that concerns majorization of probability distributions under the creation of correlations in auxiliary systems. As a by-product, we obtain a characterization of Shannon entropy as the essentially unique real function on probability distributions which is Schur-concave, additive, and subadditive.

Algebra Seminar

Speaker: Cam McLeman (University of Michigan - Flint)

"Old and new invariants in the class field tower problem"

Time: 14:30

Room: MC 107

Given a number field, we form the Hilbert p-class field tower by repeatedly iterating the Hilbert p-class field construction. Work of Golod and Shafarevich demonstrated that this construction can fail to stabilize after finitely many steps, and the slew of natural questions that this observation unleashed remain largely unanswered. In particular, there is no known definitive criterion for deciding if a number field has an infinite such tower. In the case of quadratic imaginary number fields, however, we are close to honing in on a set of invariants which completely resolve the question. This talk will cover both historical such invariants and modern ones, giving the current status of our ability to determine the infinitude of such towers.

Geometry and Topology

Speaker: Pal Zsamboki (Western)

"QC(X)"

Time: 15:30

Room: MC 107

I will explain how to construct the symmetric monoidal infinity-category of complexes of quasi-coherent sheaves on a stack using higher algebra. Afterwards, I will talk about how this might be used to compactify moduli stacks of torsors.

Noncommutative Geometry

Speaker: (Western)

"Learning Seminar"

Time: 11:30

Room: MC 107

We continue the lectures with : ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula. ---Heat kernel and its asymptotic expansion, Gilkey's formula, McKean-Singer formula.

Analysis Seminar

Speaker: Lubos Pick (Charles University, Prague)

"Traces of Sobolev functions --- old and new"

Time: 15:30

Room: MC 107

The talk will focus on the classical problem of traces of functions from Sobolev spaces, which had originated in connection with some specific problems in PDEs and then mushroomed into a separate field of research in functional analysis and the function spaces theory. One important property enjoyed by functions from the Sobolev space $W^ {m,p}(\mathbb{R}^ n )$, where $m\in \mathbb{N}$ and $p\in[1,\infty]$, is that their restrictions, called traces, to lower dimensional spaces $\mathbb{R}^ d$ can be properly defined, provided that the dimension $d$ of the relevant subspaces is not too small, depending on the values of $n$, $m$ and $p$. In such case one can ask whether some properties such as a certain degree of integrability of a trace can be expected, and, naturally, which of these properties are the best possible. We shall survey both classical and recent results concerning traces of Sobolev functions. We shall consider basic questions concerning the very existence of trace as well as deeper problems such as optimal trace embeddings involving specific function spaces.

Algebra Seminar

Speaker: Detlev Hoffmann (Technische Universität Dortmund)

"Equivalence relations for quadratic forms"

Time: 14:30

Room: MC 107

We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.