Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"NCG Learning Seminar: Introduction to Operator Algebra III: Gelfand-Naimark Theorems"

Time: 14:30

Room: MC 107

In this lecture we first give some examples of noncommutative C*-algebras. Then we introduce group C*-algebra of a discrete group. Finally we present one of the most important theorems in the theory of C*-algebras that is called GNS construction, by which we can see every C*-algebra as a norm closed subalgebra of the algebra of bounded linear operators on a Hilbert space.

Geometry and Topology

Speaker: Parker Lowrey (Western)

"Grothendieck-Riemann-Roch for derived schemes"

Time: 15:30

Room: MC 108

I will discuss an extension of Chow groups to derived schemes and discuss how to recover Kontsevich's formulas relating virtual structure sheafs and virtual fundamental classes as a byproduct of the use of quasi-smooth morphisms between derived schemes.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"The Gauss-Bonnet theorem and scalar curvature for noncommutative two-tori (1)"

Time: 14:30

Room: MC 107

I will survey on a recent joint work with M. Khalkhali and a paper by Connes and Moscovici on scalar curvature for noncommutative two-tori. The scalar curvature is computed by considering small time heat kernel expansions of the perturbed Laplacian which encodes the metric information of a general translation invariant conformal structure and a Weyl conformal factor on the noncommutative two-torus. There is an equivalent formulation for the scalar curvature in terms of special values of spectral zeta functions. I will also talk about our result on the Gauss-Bonnet theorem for noncommutative two-tori which extends the work of Connes and Tretkoff to the general conformal structures.

Analysis Seminar

Speaker: Burglind Joricke (Indiana University)

"On CR manifolds and the geometry of decomposition into orbits"

Time: 15:30

Room: MC 108

We will state some results and formulate some global problems related to the geometry of decomposition of CR manifolds into CR orbits.

Colloquium

Speaker: Burglind Joricke (Indiana University)

"Braids, Conformal Module and Entropy"

Time: 15:30

Room: MC 108

After a brief introduction to braids I will discuss a conformal invariant and a dynamical invariant, the relation between them and some applications.

Algebra Seminar

Speaker: Francois-Xavier Machu (Western)

"Monodromy of a class of logarithmic connections over an elliptic curve and local structure of the moduli space of connections"

Time: 14:30

Room: MC 108

The study of the relations between various moduli spaces arising from connections on vector bundles over algebraic varieties is an interesting topic. The most intriguing question is the relation between the moduli space of connections and that of the underlying vector bundles. We illustrate this concept in considering a familly of rank 2 logarithmic connections over an elliptic curve. These rank 2 logarithmic connections are obtained as direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We provide an explicit parameterization of all such connections and determine their monodromy and differential Galois group. We address the local structure of the moduli space of connections and give an example of a moduli space of connections having a singularity. This singularity is solved by means of the toric geometry. I will finish this talk in saying a few words on what happens while one considers the genus-1 singular curves.

Noncommutative Geometry

Speaker: Sajad Sadeghi (Western)

"NCG Learning Seminar: Hecke C*-algebras"

Time: 14:30

Room: MC 107

In this lecture we first give some examples of states and GNS constructions. Then we introduce the reduced group $C^*$-algebra of a discrete group. Moreover, the Hecke pair $(G,H)$ consisting of a group $G$ and an almost normal subgroup $H$ will be introduced. Indeed, a subgroup $H$ of a group $G$ is called almost normal if every double coset $HgH$ can be written as a union of finite number of left cosets. Using Hecke pairs we introduce Hecke $C^*$-algebras as a generalization of group $C^*$-algebras. Finally we piont to the Bost-Connes pair and its Hecke $C^*$-algebra.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"Fun with Feynman, part I: Wick`s Theorem"

Time: 16:30

Room: MC 108

This is the first of a series of two talks on finite dimensional Feynman Calculus. The second talk will be in the next week and will be given by Travis Ens. For the background on these series please check the Pizza Seminar blog http://pizzaseminaruwo.blogspot.ca/

Wick's theorem allows us to compute any exponential integral by repeatedly differentiating a Gaussian function. It is a bridge that will take us from Gaussian integrals to Feynman integrals and their evaluations in terms of graph sums and of course to a delicious Pizza at the end! These talks are designed for our undergrads, but good grad students and genius faculty can also attend!

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"The Gauss-Bonnet theorem and scalar curvature for noncommutative two-tori (2)"

Time: 14:30

Room: MC 107

I will survey on a recent joint work with M. Khalkhali and a paper by Connes and Moscovici on scalar curvature for noncommutative two-tori. The scalar curvature is computed by considering small time heat kernel expansions of the perturbed Laplacian which encodes the metric information of a general translation invariant conformal structure and a Weyl conformal factor on the noncommutative two-torus. There is an equivalent formulation for the scalar curvature in terms of special values of spectral zeta functions. I will also talk about our result on the Gauss-Bonnet theorem for noncommutative two-tori which extends the work of Connes and Tretkoff to the general conformal structures.

Analysis Seminar

Speaker: Roman Dwilewicz (Missouri University of S&T)

"Hartogs Type Holomorphic Extensions"

Time: 14:30

Room: 108

In the talk there will be given a short review of holomorphic extension problems starting with the famous Hartogs theorem (1906), via Severi-Kneser-Fichera-Martinelli theorems, up to some recent results on global holomorphic extensions for unbounded domains obtained together with Al Boggess (Arizona State Univ.) and Zbigniew Slodkowski (Univ. Illinois at Chicago). The classical Hartogs theorem solves the extension problem for bounded domains in C^n and clearly shows the difference between one and many-variables cases. The theorem is considered as an informal beginning of Complex Analysis in Several Variables. Surprisingly, the unbounded case was missed by analysts for more than a hundred years, even though it is important not only in Complex Analysis, but also in Partial Differential Equations and Algebraic Geometry. The problem appeared highly 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.

Colloquium

Speaker: Vestislav Apostolov (UQAM)

"Calabi extremal metrics on polarized projective varieties"

Time: 14:30

Room: MC 108

After a brief introduction to the Calabi program concerning the search of canonical riemannian metrics on a smooth compact polarized projective variety, I will discuss some recent developments relating analytical with algebro-geometric aspects of the problem. I will then provide some non-trivial examples of projective varieties on which the Calabi problem can be solved.

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"The Gauss-Bonnet theorem and scalar curvature for noncommutative two-tori (3)"

Time: 14:30

Room: MC 107

I will survey on a recent joint work with M. Khalkhali and a paper by Connes and Moscovici on scalar curvature for noncommutative two-tori. The scalar curvature is computed by considering small time heat kernel expansions of the perturbed Laplacian which encodes the metric information of a general translation invariant conformal structure and a Weyl conformal factor on the noncommutative two-torus. There is an equivalent formulation for the scalar curvature in terms of special values of spectral zeta functions. I will also talk about our result on the Gauss-Bonnet theorem for noncommutative two-tori which extends the work of Connes and Tretkoff to the general conformal structures.

Geometry and Topology

Speaker: Enxin Wu (Western)

"What is a diffeological moment map?"

Time: 15:30

Room: MC 108

I will try to discuss how to extend the classical moment maps to the diffeological case, and hopefully some applications to some orbifolds, etc.

Analysis Seminar

Speaker: Masoud Khalkhali (Western)

"A new evaluation of zeta values at even integers, I"

Time: 15:30

Room: MC 108

TBA

Pizza Seminar

Speaker: Travis Ens (Western)

"Fun with Feynman, part II: Graph Sums"

Time: 16:30

Room: MC 108

Abstract: Feynman's theorem, a fundamental mathematical tool in quantum field theory, provides a way to evaluate complicated integrals by summing over finite graphs. After exploring how to obtain this sum from an integral, we will reverse the correspondence and prove a famous result of Cayley, that the number of labelled trees with n vertices is nn−2, using known values of integrals. No physics will be needed to follow this lecture. For some background material please check out the Pizza Seminar blog http://pizzaseminaruwo.blogspot.ca/

Noncommutative Geometry

Speaker: Alimjon Eshmatov (Western)

"On Dixmier Groups"

Time: 14:30

Room: MC 107

I will discuss about my recent joint work with Yu. Berest and F. Eshmatov on the structure of automorphism groups of class of algebras closely related to the first Weyl algebra $A_1(k)$. In particular, we give a geometric presentation for these groups using the Bass-Serre theory, thus answering question posed by T. Stafford in 80's. A key role in our approach is played by a transitive action of the automorphism group $Aut(A_1)$ on the Calogero-Moser varieties, which are certain variation of Hilbert schemes of points on the plane. In particular, using this geometric description of these groups we have been able to classify these groups up to isomorphism. The results obtained generalize some classic theorems of J. Dixmier and L. Makar-Limanov. In the end, we discuss a few open problems related to these groups such as the Dixmier Conjecture.

Colloquium

Speaker: Eric Katz (University of Waterloo)

"Tropicalization and Combinatorial Abstraction"

Time: 15:30

Room: MC 108

Given a mathematical object, one may associate a combinatorial object that captures some of its properties.  Natural examples would be matroids as combinatorial abstractions of linear subspaces and Newton polytopes as combinatorial abstractions of hypersurfaces.  Then one has a class of combinatorial objects that behave somewhat like the original mathematical objects.  Two questions arise: which properties of the original objects do the combinatorial objects encode?; and how to characterize the combinatorial objects that are abstractions of an object in the original category. 

In this talk, we discuss tropical varieties which are combinatorial abstractions of algebraic varieties and contain as examples the theories of Newton polytopes and matroids.  We develop some examples and share the progress that has been made on those two questions.

Algebra Seminar

Speaker: Hugo Bacard (Western)

"Enrichment: past, present and ..."

Time: 14:30

Room: MC 108

The theory of $\it{enriched}$ categories generalizes naturally the classical theory of categories and has at least as much impact in mathematics as classical category theory. Following ideas of B$\mathrm{\acute{e}}$nabou, Grothendieck, Kelly, Mac Lane, Segal and others, we develop a theory of $\it{weakly}$ $\it{enriched}$ categories; these structures arise naturally when Algebra meets Homotopy and give an alternative approach to $\it{higher}$ $\it{categories}$. In this talk I will give an overview of classical enriched category theory and then will talk about the extension to $\it{weak}$ $\it{enrichment}$.

Noncommutative Geometry

Speaker: Jason Haradyn (Western)

"NCG Learning Seminar: Spin Geometry (1)"

Time: 14:30

Room: MC 107

In the first of a series of talks, I would like to introduce the notions of a Clifford algebra of a vector space $V$ over $\mathbb{R}$ and of a spin structure on a Riemannian manifold. I will discuss when a Riemannian manifold does in fact carry a spin structure, thus allowing it to admit spinors. This is not always possible because there may be topological obstructions on the manifold that inhibit it from carrying such a structure. Nevertheless, spin manifolds are useful for determining whether or not an orientable Riemannian manifold admits spinors. Once this is in place, we will look at the Dirac operator associated to a spin module and some of its properties, including how it operates on sections of the spinor bundle.

Analysis Seminar

Speaker: Masoud Khalkhali (Western)

"A new evaluation of zeta values at even integers, II"

Time: 15:30

Room: MC 108

TBA

Graduate Seminar

Speaker: Piers Lawrence (UWO Applied Math)

"Mandelbrot Polynomials and Matrices"

Time: 17:00

Room: MC 108

We explore a family of polynomials whose roots are related to the Mandelbrot set. The roots correspond to the $k$-periodic points of the iteration defining the Mandelbrot set. The Mandelbrot polynomials are defined by $p_0(\zeta)=0$ and $p_{j+1}(\zeta)=\zeta p^2_{j}(\zeta)+1$. These polynomials give rise to a novel family of recursively constructed zero-one matrices whose eigenvalues are the roots of $p_k(\zeta)$. The LU decomposition of the resolvent of these matrices is highly structured, and one linear solve can be done in $O(n)$ operations. Krylov based eigenvalue solvers can then be used to compute the eigenvalues of these matrices in an efficient manner.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Spin Groups and their Representation Theory"

Time: 14:30

Room: MC 107

The spin group, $Spin(n)$, for $n>2$, can be defined as the universal covering group of $SO(n)$. They can be explicitly constructed as a subgroup of the group of invertible elements of the Clifford algebra. One can easily see that any irreducible $SO(n)$ representation gives an irreducible representation of $Spin(n)$, however, some irreducible $Spin(n)$ representations cannot be constructed in this way. The main goal of this talk is to construct such representations using the representation theory of Clifford algebras.

Algebra Seminar

Speaker: Stefan Tohaneanu (Western)

"From Spline Approximation to Roth's Equation via Schur Functors"

Time: 14:30

Room: MC 108

Let $\Delta$ be a triangulation of a topological open disk in the real plane. Let $r$ and $d$ be two positive integers. On this region one defines a piecewise $C^r$ function, such that on each triangle the function is given by a polynomial in two variables of degree $\leq d$. The set of these functions forms a finite dimensional vector space, and one of the major questions in Approximation Theory is to find the dimension of this space. It was conjectured that for $d\geq 2r+1$, this dimension is given by a precise formula that depends on the combinatorial information of the simplicial complex $\Delta$, and on the local geometric data. The conjecture is very difficult, and trying to prove it for the simplest nontrivial example has been a challenge for about 10 years. Jan Minac and myself answered this question by the means of Commutative Algebra, showing also that a direct approach to solve this conjecture for this particular example leads to difficult questions in Matrix Theory, such as the LU-decomposition of an invertible matrix. In this talk I am presenting an overview of these problems. The talk is accessible to graduate students.