Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Path Integrals in Quantum Mechanics (4)"

Time: 14:30

Room: MC 107

By transforming to momentum space, the integrals used to compute the Feynman weight of a graph can be simplified. After carrying out this process, I will compute the partition function of two simple systems, quantum mechanics on a circle and circle-valued quantum mechanics. Finally I will discuss how these methods easily generalize to quantum field theory in the case of a free scalar bosonic field.

Analysis Seminar

Speaker: Dayal Dharmasena (Syracuse University)

"Holomorphic Fundamental Semigroup of Riemann Domains"

Time: 15:30

Room: MC 108

Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\partial\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\overline {\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $\Pi\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ are called {\it holomorphically homotopic or $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into ${\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $\eta_1(W,M,w_0)$. \par There is a natural mapping $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $\eta_1(W,M,w_0)$ a binary operation $\star$ which induces on $\eta_1(W,M,w_0)$ a structure of a semigroup with unity and show that $\eta_1(W,M,w_0)$ is an algebraic biholomorphic invariant of Riemann domains. Moreover, $\iota_1([f_1]\star[f_2])=\iota_1([f_1])\cdot\iota_1([f_2])$, where $\cdot$ is the standard operation on $\pi_1(W,w_0)$. Then we establish standard properties of $\eta_1(W,M,w_0)$ and provide some examples. When $W$ is a finitely connected domain in $M=\mathbb C$ and $\Pi$ is the identity, we show that $\iota_1$ is an isomorphism of $\eta_1(W,M,w_0)$ onto the minimal subsemigroup of $\pi_1(W,w_0)$ containing holomorphic generators and invariant with respect to the inner automorphisms. In particular, we show for a general domain $W\subset\mathbb C$ that $[f_1]=[f_2]$ if and only if $\iota_1([f_1])=\iota_1([f_2])$. This is a joint work with Evgeny Poletsky.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Localization in equivariant cohomology and index formula"

Time: 14:30

Room: MC 107

The path integral formula for the index of the Dirac operator can be interpreted as a localization formula for U(1)-equivariant cohomology of the free loop space of the manifold. In this lecture I shall first recall the Cartan model of equivariant differential forms of a finite dimensional manifold and the localization formula of Berline-Vergne. We shall then see that the loop space analogue of this result will give the A hat genus. This can be regarded as the bosonic component of the index formula. The corresponding localization formula in the supersymmetric case gives the full index formula.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"NCG Learning Seminar: Applications of the Atiyah-Singer Index theorem 4: the Hirzebruch-Riemann-Roch Theorem"

Time: 10:30

Room: MC 107

Following the previous talks on the Atiyah-Singer index theorem by Masoud, we will prove another important special case, namely the Hirzebruch-Riemann-Roch theorem. This theorem gives the holomorphic Euler characteristic of a holomorphic vector bundle over a compact Kähler manifold in terms of the Todd class of the manifold and the Chern character of the vector bundle. It will be shown how in the case of a holomorphic line bundle over a Riemann surface this reduces to the classical Riemann-Roch theorem.

Algebra Seminar

Speaker: David Riley (Western)

"On the behaviour of the Frobenius map in a noncommutative world"

Time: 14:30

Room: MC 108

Noncommutative Geometry

Speaker: Travis Ens (Western)

"Matrix integrals and a theorem of t'Hooft"

Time: 14:30

Room: MC 107

Regarding the space of Hermitian matrices as an N^2 dimensional real vector space with nondegenerate bilinear form given by the trace, we may apply Feynman's theorem to compute matrix integrals. First I will show how to evaluate such integrals by a sum over compact oriented surfaces with boundary, and then I will use this expansion to prove a theorem of t'Hooft which states that in the limit for large N of such integrals the sum only depends on the contribution of planar connected fat graphs.

Geometry and Topology

Speaker: Matthias Franz (Western)

"Equivariant (co)homology and syzygies"

Time: 15:30

Room: MC 108

After defining equivariant (co)homology for torus actions, I will present an equivariant version of Poincaré-Alexander-Lefschetz duality and relate it to an old result of Duflot.

Then I will turn to syzygies in equivariant cohomology. Syzygies are modules (over a polynomial ring) that interpolate between torsion-free and free modules. I will recall how syzygies are related to equivariant homology, the Atiyah-Bredon sequence and the equivariant Poincaré pairing. For actions on manifolds I will then give a "geometric criterion" that characterizes such syzygies in terms of the orbit space together with its stratification by orbit dimension. At the end I will discuss the existence of "maximal" syzygies for compact orientable manifolds. Here an interesting connection with singularities of real algebraic varieties appears.

This is joint work with Chris Allday and Volker Puppe.

Analysis Seminar

Speaker: Damir Kinzebulatov (Fields Institute)

"Kohn decomposition for forms taking in values in holomorphic Banach vector bundles"

Time: 15:30

Room: MC 108

Abstract: The fundamental Kohn's Decomposition Theorem relates cohomology groups of forms on compact subdomains of complex manifolds (e.g. pseudoconvex), to finite-dimensional spaces of harmonic forms on these subdomains. In my talk I will introduce a variant of Kohn's theorem for forms defined on non-compact subdomains, and satisfying additional constraints on their growth along discrete subsets (joint work with Alex Brudnyi). Its proof is based on a quite useful technique for dealing with infinite-dimensional holomorphic Banach vector bundles, which I will also describe. Finally, I will demonstrate how infinite-dimensionality of vector bundle, combined with Oka principle, can lead to better results than in the finite-dimensional case.

Pizza Seminar

Speaker: Mitsuru Wilson and Masoud Khalkhali (Western)

"A stroll on Strange Spaces/ First Steps in Quantum Computing"

Time: 16:30

Room: MC 108

To celebrate the year's end, we shall have two talks this Tuesday. The first talk is in our Pizza Seminar series and will be given by Mitsuru Wilson and the second will be the last lecture in our Discovery Cafe weekly meetings by Masoud Khalkhali. We shall then all go to the grad club for Pizza, courtesy of Math Department! Tea will be served in the lounge between the two talks.

The first talk is a gentle and friendly introduction to evolution of geometric thought through history of mathematics, culminating in some current ideas on noncommutative spaces.

The second talk is an introduction to Shor's fast factorization quantum computing algorithm and some of its physics and mathematics background. Please check the one and only Pizza Seminar blog http://pizzaseminaruwo.blogspot.ca/ for more details!

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"On Alexander's proof of Gromov's Theorem"

Time: 15:30

Room: MC 107

In a seminal paper of 1985 Gromov proved that any compact Lagrangian submanifold of $C^n$ admits a nonconstant analytic disc attached to it. I will outline Alexander's proof of this result and discuss possible generalizations for immersed Lagrangian manifolds.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Feynman-Kac Formula"

Time: 10:30

Room: MC 107

This is the Euclidean Wick rotated analogue of Feynman's formula for the propagator. Unlike the latter, it can be rigorously proved. I shall first define the Wiener measure on the space of continuous paths and then prove the formula. I end with a few examples and applications.

Algebra Seminar

Speaker: Ilya Shapiro (Windsor)

"Bitorsors, gerbes, and duality"

Time: 14:30

Room: MC 108

This talk is based on ongoing work with X. Tang and H. Tseng that grew out of my attempt to understand a certain duality for gerbes on orbifolds that Tang and Tseng studied in their paper "Duality theorems of etale gerbes on orbifolds". Our new approach is more conceptual, allowing the definition of duality to be extended in greater generality. In the talk I will explain gerbes from the point of view of bitorsors and sketch the constructions involved in duality, both original and twisted.

Dept Oral Exam

Speaker: Masoud Ataei Jaliseh (Western)

"On the Tower of Function Fields"

Time: 15:30

Room: MC 108

Geometry and Topology

Speaker: Nima Rasekh (Western)

"The Chromatic Spectral Sequence"

Time: 15:30

Room: MC 108

The chromatic spectral sequence is an algebraic spectral sequence constructed from the Brown-Peterson spectrum, which converges to the second sheet of the Adams-Novikov Spectral sequence, the primary tool to understand stable homotopy. In this talk we build this spectral sequence and show how it helps us to understand the stable homotopy ring.

Analysis Seminar

Speaker: Hristo Sendov (Western)

"Spectral Manifolds"

Time: 15:30

Room: MC 108

It is well known that the set of all $n \times n$ symmetric matrices of rank $k$ is a smooth manifold. This set can be described as those symmetric matrices whose ordered vector of eigenvalues has exactly $n-k$ zeros. The set of all vectors in $\mathbb{R}^n$ with exactly $n-k$ zero entries is itself an analytic manifold.

In this work, we characterize the manifolds $M$ in $\mathbb{R}^n$ with the property that the set of all $n \times n$ symmetric matrices whose ordered vector of eigenvalues belongs to $M$ is a manifold. In particular, we show that if $M$ is a $C^k$ manifold then so is the corresponding matrix set for all $k \in \{2,3,\ldots, \infty, \omega\}$. We give a formula for the dimension of the matrix manifold in terms of the dimension of $M$.

This is a joint work with A. Daniilidis and J. Malick.

Noncommutative Geometry

Speaker: Mohammad Hassanzadeh (University of Windsor)

"Galois extensions for Hopf algebras"

Time: 14:30

Room: MC 107

Galois theory for Hopf algebras has its roots in works of Chase, Harrison and Rosenberg in 1965 where they wanted to extend the classical Galois theory of fields. Fifteen years later, Kreimer and Takeuchi defined Hopf Galois extensions which are noncommutative analogue of torsors and principal bundles.

In this talk we explain the basics of Hopf Galois extensions and introduce several examples. Furthermore we explain the homology theories related to a Hopf Galois extension and explain some new results.

Colloquium

Speaker: Alex Suciu (Northeastern University)

"Automorphism groups, Lie algebras, and resonance varieties"

Time: 15:30

Room: MC 108

The automorphism group of a group $G$ comes endowed with a natural filtration: an automorphism belongs to the $k$-th term of this ``Johnson filtration" if it has the same $k$-jet as the identity, with respect to the lower central series of $G$. In this talk, I will discuss the Johnson filtration of the automorphism group of a finitely generated free group, and that of the mapping class group of a surface, with emphasis on the homological finiteness properties of the first few terms in these filtrations.

A key ingredient in this approach is a rather surprising relationship between the classical representation theory of a complex, semisimple Lie algebra $\mathfrak{g}$ and the resonance varieties $R(V,K)\subset V^*$ attached to irreducible $\mathfrak{g}$-modules $V$ and submodules $K\subset V\wedge V$. In the case when $\mathfrak{g}= \mathfrak{sl}_2(\mathbb{C})$, this relationship sheds new light on certain modules studied by Weyman and Eisenbud in the context of Green's conjecture on free resolutions of canonical curves.

This is joint work with Stefan Papadima (arXiv:1011.5292, 1207.2038).

Dept Oral Exam

Speaker: Baran Serejelahi (Western)

"Geometric Quantization"

Time: 13:00

Room: MC 106

Algebra Seminar

Speaker: Brian Pike (Toronto)

"Milnor fibers of non-isolated singularities"

Time: 14:30

Room: MC 108

The Milnor fiber is a topological space which can be associated to certain germs of singular complex analytic or algebraic varieties. For isolated hypersurface singularities and isolated complete intersection singularities, the Milnor fiber can be described by a single "Milnor number." Beyond these cases, the topology of the Milnor fiber is often much more complicated. An alternative generalization of the classical isolated situations is the "singular Milnor fiber," which often has a simple topology even for nonisolated singularities. We'll discuss joint work with James Damon on finding a way to compute the corresponding "singular Milnor number." In a few cases, this work has led to ways of computing information about the classical Milnor fiber.

Comprehensive Exam Presentation

Speaker: Allen O'Hara (Western)

"The Bruhat Decomposition For Rational Monoids"

Time: 13:30

Room: MC 107

In this talk we'll be covering some work by Sandeep Varma concerning rational submonoids of algebraic monoids. We begin by reviewing the Bruhat decomposition for algebraic groups, as discovered by Francois Bruhat. From there we'll take two different paths, the first will generalize the Bruhat decomposition to groups with BN-pairs. The second will deal with the Bruhat decomposition for algebraic monoids. Finally we will cover the results from Varma's paper which culminate in extending the Bruhat decomposition to rational submonoids.