Graduate Seminar

Speaker: Tyson Davis (Western)

"Essential Dimension of Moduli stacks"

Time: 11:20

Room: MC 106

Geometry and Topology

Speaker: Karol Szumilo (Western)

"Cofibration categories and quasicategories"

Time: 15:30

Room: MC 107

Approaches to abstract homotopy theory fall roughly into two types: classical and higher categorical. Classical models of homotopy theories are some structured categories equipped with weak equivalences, e.g. model categories or (co)fibration categories. From the perspective of higher category theory homotopy theories are the same as (infinity,1)-categories, e.g. quasicategories or complete Segal spaces. The higher categorical point of view allows us to consider the homotopy theory of homotopy theories and to use homotopy theoretic methods to compare various notions of homotopy theory. Most of the known notions of (infinity,1)-categories are equivalent to each other. This raises a question: are the classical approaches equivalent to the higher categorical ones? I will provide a positive answer by constructing the homotopy theory of cofibration categories and explaining how it is equivalent to the homotopy theory of (finitely) cocomplete quasicategories. This is achieved by encoding both these homotopy theories as fibration categories and exhibiting an explicit equivalence between them.

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"Zero sets of real analytic functions and the fine topology"

Time: 14:30

Room: MC 107

In this talk we will discuss some results concerning the zero sets of real analytic functions on open sets in $\mathbb{R}^n$. We will consider the related notion of analytic uniqueness sequences and, as an application, we will show that the zero set of every non-constant real analytic function on a domain has always empty interior with respect to the fine topology (which strictly contains the Euclidean one). Further, we will see that for a certain category of sets $E$ (containing the finely open sets), a function is real analytic on some open neighbourhood of $E$ if and only if it is real analytic ''at each point'' of $E$. (Joint work with Andre Boivin and Paul Gauthier.)

Homotopy Theory

Speaker: Karol Szumilo (Western)

"Univalence Axiom"

Time: 13:00

Room: MC 107

We will introduce the Univalence Axiom and discuss a few of its immediate consequences such as existence of types that are not sets, function extensionality or preservation of n-types by dependent products.

Graduate Seminar

Speaker: Ali Fathi (Western)

"Quantum anomalies"

Time: 11:20

Room: MC 106

Quantum anomalies can arise when regularized determinants and traces of infinite dimensional operators in quantum field theories lose their multiplicative and tracial property. I will explain the basic ideas and methods and if time allows, will go over the celebrated Polyakov conformal anomaly formula and how the anomaly cancellation dictates the critical dimension D=26 for bosonic string theory.

Geometry and Topology

Speaker: Martin Frankland (Western)

"Two-track algebras and the Adams spectral sequence"

Time: 15:30

Room: MC 107

The classical Adams spectral sequence can be computed via higher order operations in mod p cohomology. Baues and Jibladze carried out computations of the differential $d_2$ using the algebra of secondary operations. Baues and Blanc described an algebro-combinatorial structure which encodes enough information about $n^{th}$ order operations to compute the differential $d_n$. In joint work with Baues, we specialize that work to the case $n=3$ and describe a more concrete algebraic structure which suffices to compute the differential $d_3$.

Analysis Seminar

Speaker: Ilya Kossovskiy (University of Vienna)

"Dynamical Approach in CR-geometry and Applications"

Time: 14:30

Room: MC 107

Study of equivalences and symmetries of real submanifolds in complex space goes back to the classical work of Poincare and Cartan and was deeply developed in later work of Tanaka and Chern and Moser. This work initiated far going research in the area (since 1970's till present), which is dedicated to questions of regularity of mappings between real submanifolds in complex space, unique jet determination of mappings, solution of the equivalence problem, and study of automorphism groups of real submanifolds.

Current state of the art and methods involved provide satisfactory (and sometimes complete) solution for the above mentioned problems in nondegenerate settings. However, very little is known for more degenerate situations, i.e., when real submanifolds under consideration admit certain singularities of the CR-structure (such as non-constancy of the CR-dimension or that of the CR-orbit dimension).

The recent CR (Cauchey-Riemann Manifolds) -- DS (Dynamical Systems) technique, developed in our joint work with Shafikov and Lamel, suggests to replace a real submanifold with a CR-singularity by appropriate complex dynamical systems. This technique has recently hepled to solve a number of long-standing problems in CR-geometry, related to regularity of CR-mappings.

In this talk, we give an overview of the technique and the results obtained recently by using it. We also discuss a possible development in this direction, in particular, new sectorial extension phenomena for CR-mappings.

Homotopy Theory

Speaker: Dan Christensen (Western)

"Higher inductive types"

Time: 13:00

Room: MC 107

Higher inductive types are a generalization of inductive types. While an inductive type is generated by certain terms, a higher inductive type may be generated by terms, paths between terms, paths between paths between terms, etc. In the homotopy theoretic model of type theory, this corresponds to constructing cell complexes. We will see many other uses of higher inductive types, and will sketch the argument that $\pi_1(S^1)$ is isomorphic to the integers.

Algebra Seminar

Speaker: Caroline Junkins (Western)

"Decomposability of algebras with involution"

Time: 14:30

Room: MC 107

Over many fields, including finite fields and imaginary number fields, any central simple algebra of exponent 2 can be decomposed as a product of quaternion algebras. However, over arbitrary fields there exist examples of indecomposable algebras of exponent 2. For division algebras, indecomposability can be detected by non-trivial torsion in the Chow group of the associated Severi-Brauer variety. In this talk, we consider the analogous problem of decomposability for algebras with involution. We replace the Severi-Brauer variety with G/B, the variety of Borel subgroups of an algebraic group G of inner type Dn, and estimate the Chow group of G/B via the gamma-filtration on its Grothendieck group. In particular, we look at groups of type D4 and their associated trialitarian triples.

Analysis Seminar

Speaker: Gord Sinnamon (Western)

"Products of Quadratic Forms / Angular Equivalence of Banach spaces"

Time: 14:30

Room: MC 107

I will present a sufficient condition, expressed in terms of the condition numbers of underlying matrices, for a product of positive definite quadratic forms to be convex. The condition is weaker than previously known sufficient conditions, and is also necessary in the case of a product of two forms.

I introduce an equivalence of norms on Banach spaces that is finer than the usual one. In addition to generating the same topology, angularly equivalent norms share certain geometric properties. Equivalent Hilbert space norms are always angularly equivalent, with the constants of equivalence related to the condition number of the matrix relating their inner products

Homotopy Theory

Speaker: Chris Kapulkin (Western)

"Models of type theory"

Time: 13:00

Room: MC 107

I'll discuss the notion of a model of dependent type theory. After outlining the general algebraic semantics, I'll show how such models can be constructed from categories that we encounter in the so-called "mathematical practice".

Algebra Seminar

Speaker: Johannes Middeke (Western University)

"49 years of Gr$\mathrm{\ddot{o}}$bner bases"

Time: 14:30

Room: MC 107

Ever since their first description in the 1965 PhD thesis of Bruno Buchberger, Gr$\mathrm{\ddot{o}}$bner bases have been an important tool for computational algebra. We can view Gr$\mathrm{\ddot{o}}$bner bases as a nonlinear version of Gaussian Elimination or a multivariate version of Euclid's Algorithm. They allow to answer problems in ideal theory, polynomial system solving, algebraic geometry, homological algebra, graph theory, diophantine equations and many other areas.

In this talk we will discuss the mathematical definition of Gr$\mathrm{\ddot{o}}$bner bases of polynomial ideals, computation of Gr$\mathrm{\ddot{o}}$bner bases with Buchberger's algorithm, conversion of Gr$\mathrm{\ddot{o}}$bner bases using the FGLM algorithm and the Gr$\mathrm{\ddot{o}}$bner walk, generalisations of Gr$\mathrm{\ddot{o}}$bner bases beyond commutative polynomials, and a selected number of applications including ideal comparison as well as symbolic summation.

Graduate Seminar

Speaker: Chandra Rajamani (Western)

"Torus actions on Manifolds"

Time: 11:20

Room: MC 106

Let $M$ be a $2n$ dimensional manifold with a symplectic form $\omega$. This symplectic form determines a Lie subgroup, $Symp_{\omega}(M)$, of $Diff(M)$ called the symplectomorphism group. There is yet another subgroup of interest called the Hamiltonian group. $Ham_{\omega}(M)$ is an infinite dimensional Lie group yet it has some properties of compact finite dimensional Lie groups. The presence of finite dimensional tori $T^k$, k≤n, inside $Ham_{\omega}(M)$ determines $M$ completely when $n = 2, 4$. I will sketch the proof of the case $n = 4$, a result of my supervisor and his colleagues.

Geometry and Topology

Speaker: Francesco Sala (Western)

"Sheaves on root stacks and Nakajima quiver varieties"

Time: 15:30

Room: MC 107

In the present talk I describe a (conjectural) relation between moduli spaces of (framed) sheaves on some two-dimensional root toric stacks and Nakajima quiver varieties of type the affine Dynkin diagram $\hat{A}_{n}$. If time permits, I will discuss an application of this relation to representation theory of Kac-Moody algebras (and vertex algebras).

Analysis Seminar

Speaker: Patrick Speissegger (McMaster University)

"A quasianalytic algebra based on the Hardy field of log-exp-analytic functions"

Time: 14:30

Room: MC 107

In his work on Dulac's problem, Ilyashenko uses a quasianalytic class of functions that is a group under composition, but not closed under addition or multiplication. When trying to extend Ilyashenko's ideas to understand certain cases of Hilbert's 16th problem, it seems desirable to be able to define corresponding quasianalytic classes in several variables that are also closed under various algebraic operations, such as addition, multiplication, blow-ups, etc. One possible way to achieve this requires us to first extend the one-variable class into a quasianalytic algebra whose functions have unique asymptotic expansions based on monomials definable in $R_{an,exp}$. I will explain some of the difficulties that arise in constructing such an algebra and how far (or close) we are to obtaining it. (This is joint work with Tobias Kaiser.)

Homotopy Theory

Speaker: Sina Hazratpour (Western)

"Category Theory in HoTT"

Time: 13:00

Room: MC 107

In this talk, we will show a way to develop category theory in Univalent foundations. As it turns out, the naive reformulation of the standard axioms from set theory leads to a rather ill-behaved notion. We show how it can be refined. We also observe that for these redefined categories, two concepts of equivalence and isomorphism are the same.

Colloquium

Speaker: Zeljko Cuckovic (University of Toledo)

"Operator theory in several complex variables"

Time: 15:30

Room: MC 107

Operator theory on spaces of holomorphic functions has undergone a rapid development in the last several decades. It started with spaces of functions holomorphic on the unit disk in the complex plane and it kept developing into higher dimensions. Particularly well studied cases are operators acting on Bergman spaces on the ball and polydisk. We use $\overline\partial$-techniques to study compactness of Hankel and Toeplitz operators on Bergman spaces on pseudoconvex domains in $\mathbb{C}^n$. This is joint work with Sonmez Sahutoglu.

Algebra Seminar

Speaker:

"Postponed"

Time: 14:30

Room: MC 107

Graduate Seminar

Speaker: Masoud Ataei (Western)

"Carlitz extension"

Time: 11:20

Room: MC 106

In this talk, I'll start with definition of Carlitz polynomial and discuss about some analogy of that with polynomial $X^m -1$ over rational numbers . After that, we will see the module structure of $\bar{F_p(T)}$ as $F_p(T)$-module using Carlitz polynomial. So, that leads us to the definition of Carlitz extension which is analogue of cyclotomic extension over rational numbers. At the end, we will see the analogue of Quadratic Reciprocity over finite fields.

Geometry and Topology

Speaker: Cihan Okay (Western)

"Filtrations of Classifying Spaces"

Time: 15:30

Room: MC 107

The classifying space $BG$ of a group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. The smallest subspace in this filtration is $B(2,G)$ which is obtained from commuting elements in the group. When $G$ is finite describing these subspaces as homotopy colimits is convenient to study the cohomology, and also generalized cohomology theories. I will describe the complex $K$-theory of $B(2,G)$ modulo torsion, and discuss examples where non-trivial torsion part appears.

Analysis Seminar

Speaker: Javad Mashreghi (U. Laval)

"Carleson measures for analytic function spaces"

Time: 14:30

Room: MC 107

Let $\mathcal{H} \subset \mbox{Hol}(\mathbb{D})$ be a Hilbert space of analytic functions. A finite positive Borel measure $\mu$ on $\mathbb{D}$ is a Carleson measure for $\mathcal{H}$ if \[ \|f\|_{L^2(\mu)} \leq C \|f\|_{\mathcal{H}}, \qquad f \in \mathcal{H}. \] Equivalently, we can say that $\mathcal{H}$ embeds in $L^2(\mu)$. In 1962, Carleson solved the corona problem. But, besides solving this difficult problem, he opened many other venues of research. For example, he characterized such measures (now called Carleson measures) for the Hardy-Hilbert space $H^2$. However, the same question perfectly makes sense for any other Hilbert space of functions. We will discuss Carleson measures for the classical Dirichlet space $\mathcal{D}$.

Homotopy Theory

Speaker: Fall study break (Western)

"No meeting today"

Time: 13:00

Room: MC 107

We resume next week.

Algebra Seminar

Speaker:

"Fall study break (Western) No meeting today"

Time: 14:30

Room: MC 107