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1 Noncommutative Geometry
Noncommutative Geometry Speaker: Travis Ens (Western) "NCG Learning Seminar: Path Integrals in Quantum Mechanics (4)" Time: 14:30 - 16:00 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. |
2 Analysis Seminar
Analysis Seminar Speaker: Dayal Dharmasena (Syracuse University) "Holomorphic Fundamental Semigroup of Riemann Domains" Time: 15:30 - 16:30 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.
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3 Noncommutative Geometry
Noncommutative Geometry Speaker: Masoud Khalkhali (Western) "Localization in equivariant cohomology and index formula" Time: 14:30 - 16:00 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. |
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5 Noncommutative Geometry
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 - 12:00 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
Algebra Seminar Speaker: David Riley (Western) "On the behaviour of the Frobenius map in a noncommutative world" Time: 14:30 - 15:30 Room: MC 108 |
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8 Noncommutative Geometry
Noncommutative Geometry Speaker: Travis Ens (Western) "Matrix integrals and a theorem of t'Hooft" Time: 14:30 - 16:00 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
Geometry and Topology Speaker: Matthias Franz (Western) "Equivariant (co)homology and syzygies" Time: 15:30 - 16:30 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. |
9 Analysis Seminar
Analysis Seminar Speaker: Damir Kinzebulatov (Fields Institute) "Kohn decomposition for forms taking in values in holomorphic Banach vector bundles" Time: 15:30 - 16:30 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
Pizza Seminar Speaker: Mitsuru Wilson and Masoud Khalkhali (Western) "A stroll on Strange Spaces/ First Steps in Quantum Computing" Time: 16:30 - 18:30 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! |
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11 Analysis Seminar
Analysis Seminar Speaker: Rasul Shafikov (Western) "On Alexander's proof of Gromov's Theorem" Time: 15:30 - 16:30 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. |
12 Noncommutative Geometry
Noncommutative Geometry Speaker: Masoud Khalkhali (Western) "Feynman-Kac Formula" Time: 10:30 - 12:00 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
Algebra Seminar Speaker: Ilya Shapiro (Windsor) "Bitorsors, gerbes, and duality" Time: 14:30 - 15:30 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
Dept Oral Exam Speaker: Masoud Ataei Jaliseh (Western) "On the Tower of Function Fields" Time: 15:30 - 16:30 Room: MC 108 |
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15 Geometry and Topology
Geometry and Topology Speaker: Nima Rasekh (Western) "The Chromatic Spectral Sequence" Time: 15:30 - 16:30 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. |
16 Analysis Seminar
Analysis Seminar Speaker: Hristo Sendov (Western) "Spectral Manifolds" Time: 15:30 - 16:30 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. |
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24 Noncommutative Geometry
Noncommutative Geometry Speaker: Mohammad Hassanzadeh (University of Windsor) "Galois extensions for Hopf algebras" Time: 14:30 - 16:00 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. |
25 Colloquium
Colloquium Speaker: Alex Suciu (Northeastern University) "Automorphism groups, Lie algebras, and resonance varieties" Time: 15:30 - 16:30 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).
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26 Dept Oral Exam
Dept Oral Exam Speaker: Baran Serejelahi (Western) "Geometric Quantization" Time: 13:00 - 15:00 Room: MC 106 |
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30 Algebra Seminar
Algebra Seminar Speaker: Brian Pike (Toronto) "Milnor fibers of non-isolated singularities" Time: 14:30 - 15:30 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. |
1 Comprehensive Exam Presentation
Comprehensive Exam Presentation Speaker: Allen O'Hara (Western) "The Bruhat Decomposition For Rational Monoids" Time: 13:30 - 15:00 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. |
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