Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Habemus Higgs!"

Time: 14:30 - 16:00

Room: MC 107

On 4 July 2012 at a conference in CERN, Geneva, the latest findings of the LHC experiment were unveiled: ``We observe in our data clear signs of a new particle, at the level of 5 sigma, in the mass region around 126 GeV. The outstanding performance of the LHC and ATLAS and the huge efforts of many people have brought us to this exciting stage", said ATLAS experiment spokesperson Fabiola Gianotti.

It is now believed beyond any doubts that the long sought after Higgs particle, the last missing part of the Standard Model of elementary particles, has been detected. In this talk I shall sketch the so called Higgs mechanism (due independently to Englert-Brout; Kibble-Guralnik-Hagen; and Higgs) and will show how symmetry breaking creates the much needed massive gauge bosons of the standard model. In a follow up talk I shall explain what noncommutative geometry adds to this picture.

Note: The 2013 Nobel prizes in physics will be announced tomorrow, Oct. 8. The odds for a prize going to a Higgs related work is quite high. We shall see!

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Structure of the polynomially convex hulls of singular real manifolds in $\mathbb C^n$."

Time: 14:30 - 15:30

Room: MC 108

I will discuss possible descriptions of the structure of polynomially convex hulls of singular totally real and Lagrangian submanifolds of $\mathbb C^n$. This is joint work with A. Sukhov.

Homotopy Theory

Speaker: Mike Misamore (Western)

"Localization of spectra with respect to homology"

Time: 14:30 - 15:30

Room: MC 108

Algebra Seminar

Speaker: Sean Fitzpatrick (Western)

"Clifford algebras and Bott periodicity in K-theory"

Time: 15:30 - 16:30

Room: MC 108

I will review the classic paper "Clifford modules", by Atiyah, Bott, and Shapiro, which had its 50th birthday last month. This paper explores both Clifford algebras and K-theory, and the connections between them. In particular, we see that Bott periodicity arises as a purely algebraic phenomenon in the study of Clifford modules. Although this paper did not give a proof of Bott periodicity in K-theory via Clifford algebras, it certainly provided strong evidence in that direction. (This problem was tackled by other authors at the same time.) Time permitting, I will discuss some of the topological applications that arise from applying the algebraic machinery of Clifford modules to K-theory, such as the Thom isomorphism and vector fields on spheres.

Analysis Seminar

Speaker: Petr V. Paramonov (Moscow State University)

"$C^m$ approximation by bianalytic functions on plane compact sets (Part I)"

Time: 14:30 - 15:30

Room: MC 108

It is planned to present certain necessary and sufficient conditions for approximability of functions on an arbitrary plane compact set by functions, bianalytic on neighborhoods of the set, in the norms of Whitney spaces $C^m$ for all $m \geq 0$. The formulated criteria are given in the so-called individual form and for each $m$ they are formulated in terms of geometric measure theory.

Homotopy Theory

Speaker: -

"No meeting today"

Time: 14:30 - 15:30

Room:

See you next week.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Introduction to gauge theory and Higgs mechanism"

Time: 14:30 - 16:00

Room: MC 107

This is a quick introduction to gauge theory as a classical field theory of elementary particles. Ideas of symmetry breaking and Higgs mechanism will be emphasized.

Index Theory Seminar

Speaker: Masoud Khalkhali (Western)

"Elliptic complexes and analytic index map"

Time: 14:00 - 15:30

Room: MC 107

I shall define the Fredholm index (resp. Euler characteristic) of an elliptic PDE (res. elliptic complex) acting on smooth sections of bundles over a closed manifold. This is the analytic index map. I shall also sketch a proof of the fact, due to Atiyah and Jaenich, that the space of Fredholm operators is a classifying space for K-theory. Time permitting, I shall indicate the classifying space of odd K-theory (space of self adjoint Fredholm operators and spectral flow). Basic tools of functional analysis like compact operators and pseudodifferential calculus will be recalled.

Analysis Seminar

Speaker: Petr V. Paramonov (Moscow State University)

"$C^m$ approximation by bianalytic functions on plane compact sets (Part II)"

Time: 11:30 - 12:30

Room: MC 107

It is planned to present certain necessary and sufficient conditions for approximability of functions on an arbitrary plane compact set by functions, bianalytic on neighborhoods of the set, in the norms of Whitney spaces $C^m$ for all $m \geq 0$. The formulated criteria are given in the so-called individual form and for each $m$ they are formulated in terms of geometric measure theory.

Algebra Seminar

Speaker: Nguyen Duy Tan (Western)

"Triple Massey products in Galois cohomology"

Time: 14:30 - 15:30

Room: MC 108

This is joint work with J$\mathrm{\acute{a}}$n Min$\mathrm{\acute{a}\check{c}}$. Let $F$ be a field. We show that any triple Massey product in Galois cohomology of $F$ with coefficient $\mathbb{F}_2$ contains 0 whenever it is defined. This extends a recent theorem of M. J. Hopkins and K. G. Wickelgren, from a global field to an arbitrary field.

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western)

"Deformation Quantization"

Time: 14:30 - 16:00

Room: MC 107

The idea of making spacetime into a noncommutative space goes back to the late 60's, and deformation quantization was first introduced in 1978 by Bayen, Flato, Fronsdal, Lichnerowicz, and Sternheimer. In my talk, I will define a star product on a spacetime $M$, which is a noncommutative deformation of $C^\infty(M)$ and (essentially uniquely) quantize the star product. My goal is to use this to construct a noncommutative field theory on the principal $G$-bundle $P$, considered as a finitely generated projective module in the quantized star product over the star product algebra $C^\infty(M)$.

Geometry and Topology

Speaker: Daniel Schaeppi (UWO)

"A Tannakian characterization of categories of coherent sheaves"

Time: 15:30 - 16:30

Room: MC 108

Classical Tannaka duality is a duality between groups and their categories of representations. The two basic questions it answers are the reconstruction problem (when can a group be reconstructed from its category of representations?) and the recognition problem (can we characterize categories of representations abstractly?).

I will outline how the notion of a Tannakian category can be weakend in order to solve the recognition problem for categories of coherent sheaves of algebraic stacks (if you are an algebraic geometer), respectively categories of comodules of Hopf algebroids (if you are an algebraic topologist). I will end with an application that illustrates one of the differences between these two perspectives.

Analysis Seminar

Speaker: Ryan Berndt (Otterbein University)

"Weight problem for the Fourier transform"

Time: 14:30 - 15:30

Room: MC 108

We will discuss conditions, necessary or sufficient, that the Fourier transform maps one weighted Lebesgue space into another weighted Lebesgue space.

Noncommutative Geometry

Speaker: Yanli Song (University of Toronto)

"Geometric K-Homology and [Q, R]=0 problem"

Time: 14:30 - 16:00

Room: MC 107

The quantization commutes with reduction problem for Hamiltonian actions of compact Lie groups was solved by Meinrenken in the mid-1990s, and solved again afterwards by many other people using different methods. In this talk, I will consider a generalization of [Q, R]=0 theorem when the manifold is non-compact. In this case, the main issue is that how to quantize a non-compact manifold. I will adopt some ideas from geometric K-homology introduced by Baum and Douglas in 1980s and examine this problem from a topological perspective. One of the applications is that it provides a geometric model for the Kasparov KK group KK(C*(G, X), C).

Homotopy Theory

Speaker: Martin Frankland (Western)

"The R-nilpotent completion"

Time: 14:30 - 15:30

Room: MC 108

Index Theory Seminar

Speaker: Mitsuru Wilson (Western)

"Equivariant K-theory"

Time: 14:00 - 15:30

Room: MC 107

K theory is a study of the Abelian group constructed from vector bundles $E\to X$ over some topological space $X$ with respect to the direct sum of the vector bundles then taking the Grothendieck closure. KG group is the K theory of $E\to X$ and a space X equipped both with continuous G actions. Moreover, it forms a commutative ring with multiplication $\otimes$ . Of our par- ticular interest to the index seminar I will define equivariant K theory and allude to its applications to index of elliptic operators. The goal of this talk is to develop sufficient tools to understand the index theory of elliptic operators.

Colloquium

Speaker: Michael Farber (Warwick)

"Large Random Spaces and Groups"

Time: 15:30 - 16:30

Room: MC 108

Large random spaces can be used to model large systems which arise in applications in computer science, biology and engineering. Large random spaces can be also used in pure mathematics to test probabilistically challenging mathematical problems. In my talk I will focus on large random 2-dimensional simplicial complexes which are generalisations of random graphs of Erdös and Rényi. One wants to understand topological properties of random complexes, in particular those which are satisfied with probability close to 1. In the talk I will also describe properties of the fundamental groups of random 2-complexes (cohomological dimension, torsion and others). Aspherical subcomplexes of random 2-complex satisfy the Whitehead conjecture, i.e. all their subcomplexes are also aspherical. The proofs exploit strong hyperbolicity property of random 2-complexes and use inequalities for Cheeger constants and systoles of simplicial surfaces.

The talk is based on a joint work with A. E. Costa.  

Analysis Seminar

Speaker: Myrto Manolaki (Western)

"Ostrowski-type theorems for harmonic functions"

Time: 11:30 - 12:30

Room: MC 107

Ostrowski showed that there are intimate connections between the gap structure of a Taylor series and the phenomenon of "overconvergence"; that is convergence of subsequences of the partial sums outside the disk of convergence. In this talk we will discuss the corresponding problem for the homogeneous polynomial expansion of a harmonic function. As we will see, the results for harmonic functions display new features in the case of higher dimensions.

Noncommutative Geometry

Speaker: Asghar Ghorbanpour (Western)

"Quantum dynamical systems I: Geodesic flow in NCG"

Time: 14:30 - 16:00

Room: MC 107

This talk is an introduction to the classical dynamical systems and the way it can be generalized to the quantum setting. We shall start with a Hamiltonian on the phase space and show that how the flow can be lifted on the observables after quantization. The Heisenberg equation can be obtained from its classical counterpart, i.e. Hamilton's equations. It is a well known fact that the geodesic flow of a Riemannian manifold $M$ is the flow of the Hamiltonian given by $H(q,p)=g_q(p,p)/2$. This can be generalized to spectral triples. We also show that for a finitely summable (even) spectral triple $(A,H,D)$ the analogue of the geodesic flow on the bounded operators of H, is given by $$F_t(T)=e^{it|D|}T e^{-it|D|}.$$ Finally I shall recall Egorov's theorem.

This is the first talk in a series of talks on "quantum dynamical systems and their properties" which will be jointly delivered by Ali Fathi and Asghar Ghorbanpour.

Geometry and Topology

Speaker: Marcy Robertson (UWO)

"Schematic Homotopy Types of Operads"

Time: 15:30 - 16:30

Room: MC 108

The rational homotopy type $X_{\mathbb{Q}}$ of an arbitrary space $X$ has pro-nilpotent homotopy type. As a consequence, pro-algebraic homotopy invariants of the space $X$ are not accessible through the space $X_{\mathbb{Q}}$. In order to develop a substitute of rational homotopy theory for non-nilpotent spaces Toen introduced the notion of a pointed schematic homotopy type over a field $\mathbb{k}$, $(X\times k)^{sch}.$

In his recent study of the pro-nilpotent Grothendieck-Teichmuller group via operads, Fresse makes use of the rational homotopy type of the little $2$-disks operad $E_2$. As a first step in the extension of Fresse's program to the pro-algebraic case we discuss the existence of a schematization of the little $2$-disks operad.

Homotopy Theory

Speaker: Martin Frankland (Western)

"Relation between completion and localization"

Time: 14:30 - 15:30

Room: MC 108

Noncommutative Geometry

Speaker: Mitsuru Wilson (Western)

"Formal Deformation Theory of Poisson Manifolds"

Time: 14:30 - 15:30

Room: MC 107

Pioneer works in formal deformation theory goes back to the late 60's, where a noncommutative space from a classical algebra of smooth functions was constructed. In his '98 paper Kontsevich proved the existence of a deformation called star products, on $C^\infty(M)$ where the product is understood to be a formal power series in one variable, on a Poisson manifold. In this talk I will define and derive an explicit formula on $\mathbb{R}^n$ of the star product. Furthermore I will explain Kontsevich's proof inasmuch elementary words as possible.