Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Feynman's diagrams and Feynman's theorem (2)"

Time: 14:30 - 14:00

Room: MC 107

Analysis Seminar

Speaker: Martin Pinsonnault (Western)

"Symplectomorphisms and Lagrangian $RP^2$ in the cotangent bunble $T^*RP^2$"

Time: 15:30 - 16:30

Room: MC 108

In this talk, we will investigate the symplectomorphism groups of the simplest open manifolds with both convex and concave ends, namely the symplectizations $sL(n,1)$ of Lens spaces $L(n,1)$. We will see that the compactly supported symplectomorphism group $Symp_c(sL(n,1))$ is homotopy equivalent to a loop space. As a corollary, we will show that the space of Lagrangian $RP^2$ in the cotangent bunble $T^*\RR P^2$ is weakly contractible. (Part of a joint work with R. Hind and W. Wu.)

Graduate Seminar

Speaker: Baran Serajelahi (Western)

"Nambu-mechanics"

Time: 16:30 - 18:00

Room: MC 108

We will discuss the basic formalism of Hamiltonian mechanics and of its generalization Nambu-mechanics. Notions from symplectic geometry will allow us to lay out this formalism in a coordinate independent way and will lead to the definition of Poisson manifolds, which serve as phase spaces for Hamiltonian mechanics. We will prove Liouville’s theorem for Hamiltonian mechanics which states that the volume of any region in phase space is preserved under the phase flow (time evolution) and we will see in a special case that the Liouville theorem itself is preserved in the generalization from Hamiltonian to Nambu-mechanics. Finally we will introduce the notion of a Nambu-Poisson manifold (Phase space for Nambu-mechanics) a natural generalization of the notion of a Poisson manifold. We will see for example that the theorem from Hamiltonian mechanics that the bracket of two integrals of motion is again an integral of motion holds also for Nambu’s dynamics by the very definition of Nambu-Poisson manifold. We will end with many examples; in particular, multi-symplectic manifolds will be introduced.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"The heat equation proof of the Atiyah-Singer index theorem (2)"

Time: 14:30 - 16:00

Room: MC 107

I shall survey the main steps in the heat equation proof of the Atiyah-Singer index theorem for Dirac operators on spin manifolds.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Feynman's diagrams and Feynman's theorem (3)"

Time: 14:30 - 16:00

Room: MC 107

Algebra Seminar

Speaker: Kristin Shaw (University of Toronto)

"Tropical intersection theory and approximating tropical curves"

Time: 15:30 - 16:30

Room: MC 108

One of the major successes of tropical geometry is Mikhalkin's correspondence theorem, which relates complex and tropical curves in toric surfaces. In non-toric surfaces Mikhalkin's correspondence does not hold; there are tropical curves not arising from complex curves or algebraic curves over any field. We will explain some local obstructions to approximating tropical curves coming from a tropical intersection product. This product is related to the intersection product defined by Kaveh and Khovanskii on divisors in more general spaces. We will also see examples of tropical curves which do not satisfy some classical theorems of geometry.

Analysis Seminar

Speaker: Franklin Vera Pacheco (University of Toronto)

"Desingularization preserving stable simple normal crossings"

Time: 15:30 - 16:30

Room: MC 108

Resolution of singularities consists in constructing a non-singular model of an algebraic variety. This is done by applying a proper birational map that is a local isomorphism at the smooth points. Often too much information is lost about the original variety if the smooth points are the only ones where the desingularization map is a local isomorphism. In these cases, a desingularization preserving some minimal singularities is necessary. This suggests the question of whether, given a class of singularity types S, it is possible to remove with a birational map all singularities not in S while still having a local isomorphism over the singularities of type S. We will study this problem when S consists of all stable simple normal crossings.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Applications of the Atiyah-Singer index theorem 1: Hirzebruch signature theorem"

Time: 14:30 - 16:00

Room: MC 107

After giving the final details of the heat equation proof, I shall give some application. Most notably Hirzebruch's Signature theorem and the Riemann-Roch theorem for compact complex manifolds.

Colloquium

Speaker: Farzad Fathizadeh (Western)

"Spectral Geometry of Noncommutative Tori"

Time: 15:30 - 16:30

Room: MC 108

SPECIAL FIELDS POSTDOCTORAL FELLOW TALK

I will first give a brief introduction to the metric aspects of noncommutative geometry and ideas from spectral geometry that have played an important role in their development. Noncommutative tori $\mathbb{T}_\theta^n$ are important $C^*$-algebras that have been studied vastly in noncommutative geometry due to their importance, among which is their role in the study of foliated manifolds. In a recent seminal paper, A. Connes and P. Tretkoff proved the Gauss-Bonnet theorem for the noncommutative two torus $\mathbb{T}_\theta^2$ equipped with its canonical conformal structure. In a series of joint works with M. Khalkhali, we extended this result to general translation invariant conformal structures, computed the scalar curvature, and proved the analog of Weyl's law and Connes' trace theorem for $\mathbb{T}_\theta^2$. Our final formula for the curvature of $\mathbb{T}_\theta^2$ precisely matches with the one computed independently by A. Connes and H. Moscovici. A purely noncommutative feature is the appearance of the modular automorphism from Tomita-Takesaki theory in the computations and the final formula for the curvature. In this talk I will review these results and will then turn to part of our recent work on the curved geometry of noncommutative four tori $\mathbb{T}_\theta^4$. That is, I will explain the computation of scalar curvature and the analog of the Einstein-Hilbert action for $\mathbb{T}_\theta^4$, and show that metrics with constant curvature are critical points of this action.

Noncommutative Geometry

Speaker: Travis Ens (Western)

"NCG Learning Seminar: Feynman's diagrams and Feynman's theorem (4)"

Time: 14:30 - 16:00

Room: MC 107

Geometry and Topology

Speaker: Steven Rayan (University of Toronto)

"Generalized holomorphic bundles on ordinary complex surfaces"

Time: 15:30 - 16:30

Room: MC 108

Generalized holomorphic bundles are a feature of Hitchin's generalized geometry. On an ordinary complex manifold, a generalized holomorphic bundle is not necessarily a holomorphic bundle. More generally, it is a kind of Higgs bundle -- sometimes called a co-Higgs bundle. I will discuss issues regarding stability and integrability for generalized holomorphic bundles over ordinary complex surfaces. In particular, I will construct a sequence of families of stable, integrable rank-2 generalized holomorphic bundles on CP^2, using the classical Schwarzenberger construction of ordinary holomorphic bundles.

Dept Oral Exam

Speaker: Claudio Quadrelli (Western)

"p-rigid fileds - a high cliff on the p-Galois see"

Time: 11:00 - 13:00

Room: MC 106

I plan to discuss my recent joint work with S. Chebolu and J. Minac. Let p be an odd prime and assume that a primitive p-th root of unity is in a field F. Then F is said to be p-rigid if only those cyclic algebras are split which are split for trivial reasons. I will present new characterizations of such fields and their Galois groups, which come from a more group-theoretical and cohomological approach. Our work extends, illustrates and simplifies some previous results and provides a new direct foundation of rigid fields which does not rely on valuation techniques. This work shows in fact how this new cohomological approach on maximal p-extensions of fields can be powerful, especially after the proof of the Milnor-Bloch-Kato conjecture.

Analysis Seminar

Speaker: Blagovest Sendov (Bulgarian Academy of Sciences)

"Hausdorff Approximations"

Time: 15:30 - 16:30

Room: MC 108

Let $A$ be a functional space of high or infinite dimension, $r(f,g);\; f,g\in A$ be a metric defined on $A$ and $\PP_n\subset A$ be an $n$-dimensional subset of $A$. The main goal of Approximation Theory, which is a theoretical basis for Numerical analysis and Numerical methods, is for given $f\in A$ to find a $p\in \PP_n$, such that $r(f,p)$ is as small as possible. Hausdorff Approximation (see \cite{BS}) is a part of Approximation Theory, in which to every function $f\in A$ corresponds a closed and bounded point set $\bar{f}$, and the distance between two functions $f,g\in A$ is defined as the Hausdorff distance between $\bar{f}$ and $\bar{g}$. An important fact is that the Hausdorff distance is not derived from a norm.

In this lecture, we underline the specifics of Hausdorff Approximation and formulate the most interesting results.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Applications of the Atiyah-Singer index theorem 2: Hirzebruch signature theorem (continued)"

Time: 14:30 - 16:00

Room: MC 107

After giving the final details of the heat equation proof, I shall give some applications. Most notably Hirzebruch's Signature theorem and the Riemann-Roch theorem for compact complex manifolds.