Geometry and Topology

Speaker: Dan Grayson (UIUC)

"Homotopy Type Theory and Univalent Foundations"

Time: 15:30

Room: MC 107

Homotopy type theory with the univalence axiom of Voevodsky provides both a new logical foundation for mathematics (Univalent Foundations) and a formal language usable with computers for checking the proofs mathematicians make daily. As a foundation, it replaces set theory with a framework where sets are defined in terms of a more primitive notion called "type". As a formal language, it encodes the axioms of mathematics and the rules of logic simultaneously, and promises to make the extraction of algorithms and values from constructive proofs easy. With a semantic interpretation in homotopy theory, it offers an alternative world where the proofs of basic theorems of mathematics can be formalized with minimal verbosity and verified by computer.

As a relative newcomer to the field, I will survey these recent developments and sketch the basic concepts for a general mathematical audience.

Noncommutative Geometry

Speaker: Alimjon Eshmatov (Western University (Assistant Professor and Postdoctoral Fellow))

"Noncommutative Poisson Structures II"

Time: 15:00

Room: MC 107

In this talk, we will discuss Noncommutative Poisson Structure which was introduced by Crawley-Boevey and how it fits nicely with Kontsevich-Rosenberg principle. We will also give some examples. If time allows, we will also discuss its relation to Van den Bergh's Double Poisson Algebras.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"Why $ \infty ! = \sqrt{2 \pi} $"

Time: 17:30

Room: MC 108

A few years ago I gave a Pizza Seminar talk where I showed how to regularize an infinite sum like \( 1+2+3+4+5+\cdots \) and show that it is equal to \( \frac{-1}{12} \). In this talk I shall discuss a multiplicative version and show how one can regularize infinite products like \( 1.2.3.4.\cdots \). This topic is intimately related to Riemann's zeta function and its analytic continuation and special values. Some tools of classical analysis like Euler-Maclaurin summation formula will be introduced and used extensively in my talk.

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Universal Toda brackets"

Time: 14:00

Room: MC 107

I will discuss universal Toda brackets due to Sagave. They are Mac Lane cohomology classes that determine Toda brackets in certain stable homotopy theories and provide an obstruction theory to the problem of realizing $\pi_* R$-modules as $R$-modules for a ring spectrum $R$.

Colloquium

Speaker: Dimitri Gurevich (Valenciennes University, France)

"From Quantum Groups to Noncommutative Geometry"

Time: 15:30

Room: MC 107

Since creation of quantum groups theory numerous attempts to elaborate an appropriate corresponding differential calculus were undertaken. Recently, a new type of noncommutative geometry has been obtained this way. Namely, we have succeeded in introducing the notions of partial derivatives on the enveloping algebras U(gl(m)) and constructing the corresponding de Rham complexes. All objects arising in our approach are deformations of their classical counterparts. In my talk I plan to introduce some basic notions of the Quantum Groups theory and to exhibit possible applications of this type Noncommutative Geometry to quantization of certain dynamical models.

Pizza Seminar

Speaker: Shiyamalen Thavandiran (Western)

"The NSA Back Door to NIST"

Time: 16:30

Room: MC 108

I will present an article on cryptography by Thomas Hales which appeared in the Notices of the American Mathematical Society journal. A familiarity with basic group theory and finite fields would be helpful.

Noncommutative Geometry

Speaker: Alimjon Eshmatov (Western University (Assistant Professor and Postdoctoral Fellow))

"Noncommutative Poisson Structures III"

Time: 15:00

Room: MC 107

In this talk, we will discuss Noncommutative Poisson Structure which was introduced by Crawley-Boevey and how it fits nicely with Kontsevich-Rosenberg principle. We will also give some examples. If time allows, we will also discuss its relation to Van den Bergh's Double Poisson Algebras.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

"Introduction to cyclic cohomology and its applications"

Time: 11:00

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Algebra Seminar

Speaker: Marc Moreno Maza (Western)

"(Postponed to later date)"

Time: 14:30

Room: MC 107

Noncommutative Geometry

Speaker: Ali Fathi (Western University)

"Canonical trace and spectral invariants of noncommutative tori"

Time: 15:00

Room: MC 107

I will explain some of our new results on spectral eta invariant of certain families of Dirac operators on NC tori.

PhD Thesis Defence

Speaker: Wayne Grey (Western)

"Inclusions Among Mixed-Norm Lp Spaces"

Time: 13:30

Room: MC 107

Mixed norm Lebesque spaces are Banach spaces of multi-variable functions for which a different Lp norm is used in each variable. In many cases, there are simple, easy-to-compute conditions that ensure one given mixed norm space is contained in another. In other cases, only complicated, hard-to-compute conditions will do.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

"Introduction to cyclic cohomology and its applications II"

Time: 11:00

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Analysis Seminar

Speaker: Javier Soria (Universidad de Barcelona)

"Recent results on optimal Hardy's inequalities and isometric averaging operators"

Time: 15:30

Room: MC 107

Classical Hardy's inequalities have been recently considered in a more general context than Lebesgue spaces (e.g., monotone functions, weighted inequalities, Hardy operator minus the identity, etc.) I will present some of the main results obtained in these new settings, as well as an extension to discrete averaging isometries. (Joint work with Santiago Boza.)

PhD Thesis Defence

Speaker: Ali Fathi (Western)

"On certain spectral invariants of noncommutative tori and curvature of Quillen's determinant line bundle for noncommutative two-torus"

Time: 13:30

Room: MC 108

By extending the canonical trace of Kontsevich-Vishik to Connes' pseudodifferential operators on noncommutative tori, we study various spectral invariants associated to elliptic operators in this setting. We also consider a family of Cauchy-Riemann operators over noncommutative 2-torus and using the machinery of canonical trace, we compute the curvature form of the associated Quillen determinant line bundle.

Dept Oral Exam

Speaker: Mike Rogelstad (Western)

"Combinatorial Techniques in the Galois Theory of p-Extensions"

Time: 15:30

Room: MC 108

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.

Noncommutative Geometry

Speaker: Masoud Khalkhali ((Western University))

"Introduction to cyclic cohomology and its applications III"

Time: 11:00

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Analysis Seminar

Speaker: Stephen Gardiner (University College Dublin)

"Universal Taylor series, conformal mappings and boundary behaviour"

Time: 11:30

Room: MC 108

In various mathematical contexts it is possible to find a single object which, when subjected to a countable process, yields approximations to the whole universe under study. Such an object is termed "universal" and, contrary to expectations, such objects often turn out to be generic rather than exceptional. This talk will focus on this phenomenon in respect of the Taylor series of a holomorphic function, and how the partial sums behave outside the domain of the function. It will discuss how potential theory reveals much about the boundary behaviour of such functions, and their relationship with conformal mappings.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

"Introduction to cyclic cohomology and its applications IV"

Time: 14:30

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Colloquium

Speaker: Tobias Fritz (Perimeter)

"Ordered commutative monoids and their applications"

Time: 15:30

Room: MC 107

Ordered commutative monoids are mathematical structures that are simple to define, yet display a very diverse phenomenology. I will introduce these structures and explain how they formalize situations in which one deals with resource objects and how they can be combined with each other or converted into each other, such as the molecules in a chemical reaction like $2H_2 + O_2 \rightarrow 2H_2O$. Some standard theorems of functional analysis yield results on ordered commutative monoids, which in turn have applications to Shannon's theory of communication and the ordered commutative monoid of graphs.