Symplectic Learning Seminar

Speaker: M. Pinsonnault (Western)

"The Linear Nonsqueezing Theorem"

Time: 13:30

Room: MC 108

Gromov's celebrated "Nonsqueezing Theorem" states that one cannot "squeeze" a symplectic ball into a thin cylinder using a symplectic transformation. This shows that symplectic diffeomorphisms are more rigid that volume preserving ones, and that there exists a notion of "size" peculiar to symplectic object. In this talk, I will explain the linear version of this theorem.

Everyone interested in symplectic geometry / topology is welcome.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Gelfand's theory of commutative Banach algebras 1"

Time: 14:00

Room: MC 108

Abstract: To start this year's NCG seminar series, I shall give a quick survey of one of the jewel's of functional analysis, namely Gelfand's theory of commutative Banach algebras. This will naturally lead to Gelfand-Naimark's characterization of commutative C*-algebras and the notion of noncommutative space. I shall give several key examples and applications illustrating the power of Gelfand's old theory going back to the years 1939-1943!

Noncommutative Geometry

Speaker: Farzad Fathizadeh (Western)

"Introduction to Dixmier trace 2"

Time: 14:00

Room: MC 108

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Gelfand's theory of commutative Banach algebras 2"

Time: 14:00

Room: MC 108

Symplectic Learning Seminar

Speaker: M. Pinsonnault (Western)

"Linear Symplectic Width and Ellipsoids"

Time: 13:30

Room: MC 108

I will continue the survey on linear aspects of symplectic geometry by discussing the relations between linear symplectic capacities and ellipsoids.

Algebra Seminar

Speaker: Emre Coskun (Western)

"The Fine Moduli Space of Representations of Clifford Algebras, Part 1"

Time: 14:30

Room: MC108

Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated Clifford algebra is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the $rd$-dimensional representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$, and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1) $, where $g$ denotes the genus of $C$.

Math Scholars

Speaker: (Western)

"Discussion Group"

Time: 16:30

Room: MC 108

First meeting.

Analysis Seminar

Speaker: Serge Randriambololona (Western)

"A non-superposition result for global subanalytic functions I"

Time: 15:40

Room: MC 108

O-minimal structures are categories of sets and mapping having nice geometrical properties. To each o-minimal expansion of a real closed field, one can associate the set of germs at infinity of its unary functions, which form a Hardy field. Valuational properties of these Hardy fields give good information about the initial structure. After a lengthy introduction of all the previously named objets and motivated by a conjecture of L. van den Dries and a result of F.-V. and S. Kuhlmann, I will discuss whether an o-minimal expansions of the field of the reals is, in general, fully determined by its associated Hardy field. I will also relate this question to the Hilbert's 13th Problem.

Colloquium

Speaker: Alejandro Uribe (U Michigan)

"On Donaldson's complexification of the group of automorphisms of a symplectic manifold"

Time: 15:30

Room: MC 108

I will review the notion in the title (which, as it turns out, is not a group), and show how to construct in certain cases an "exponential" in the complexification. The construction is motivated by quantum mechanics.

Algebra Seminar

Speaker: Gregory Chaitin (IBM Research)

"Mathematics, Biology and Metabiology"

Time: 14:30

Room: MC 105b

CS Department colloquium

It would be nice to have a mathematical understanding of basic biological concepts and to be able to prove that life must evolve in very general circumstances. At present we are far from being able to do this. But I'll discuss some partial steps in this direction plus what I regard as a possible future line of attack.

Algebra Seminar

Speaker: Emre Coskun (Western)

"postponed"

Time: 14:30

Room: MC108

Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated Clifford algebra is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the $rd$-dimensional representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$, and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1) $, where $g$ denotes the genus of $C$.

Geometry and Topology

Speaker: Spiro Karigiannis (Waterloo)

"Curvature of the moduli space of $G_2$ metrics"

Time: 15:30

Room: MC 108

I will talk about the geometry of the moduli space $\mathcal M$ of holonomy $G_2$ metrics. In particular I will discuss the Hessian metric structure, the Yukawa coupling, and the sectional curvature of this moduli space. This is a combination of past work with Conan Leung and new work in progress with Christopher Lin.

Math Scholars

Speaker:

"Discussion Group"

Time: 16:30

Room: MC 104

Algebra Seminar

Speaker: Mark Hovey (Wesleyan University)

"Watts' theorems in homological algebra and algebraic topology"

Time: 15:00

Room: MC 107

The classical Watts' theorems identify functors which are tensor products or Hom functors by internal properties. We extend these theorems to homological algebra and algebraic topology. So, in the easiest case, we characterize all functors from the unbounded derived category $D(R)$ of a ring $R$ to $D(S)$ which are given by the derived tensor product with a complex of bimodules (recovering a result of Keller's in this case). We draw conclusions about Brown representability of homology and cohomology functors.

Note room change: MC107.

Analysis Seminar

Speaker: Serge Randriambololona (Western)

"A non-superposition result for global subanalytic functions II"

Time: 15:40

Room: MC 108

O-minimal structures are categories of sets and mapping having nice geometrical properties. To each o-minimal expansion of a real closed field, one can associate the set of germs at infinity of its unary functions, which form a Hardy field. Valuational properties of these Hardy fields give good information about the initial structure. After a lengthy introduction of all the previously named objets and motivated by a conjecture of L. van den Dries and a result of F.-V. and S. Kuhlmann, I will discuss whether an o-minimal expansions of the field of the reals is, in general, fully determined by its associated Hardy field. I will also relate this question to the Hilbert's 13th Problem.

Algebra Seminar

Speaker: Jon Carlson (U. Georgia)

"Endotrivial modules"

Time: 15:00

Room: MC 108

This is a report on efforts to classify the endotrivial modules over the modular groups algebras of groups which are not $p$-groups. A classification of the endotrivial modules over $p$-groups was completed by the speaker and Th\'evenaz a few years ago, building on the work of many others, notably Dade and Alperin. The endotrivial modules form an important part of the Picard group of self equivalences of the stable category of modules over the group algebra. For groups which are not $p$-groups, the problem of determining the endotrivial modules often reduces to discovering when the Green correspondent of an endotrivial module is endotrivial. This investigation often involves a detailed study of the representation theory of the groups in question.

Colloquium

Speaker: Jon F. Carlson (U Georgia)

"Modules of constant Jordan type"

Time: 15:30

Room: MC 108

This talk will present an introduction to some continuing work being conducted with Eric Friedlander, Julia Pevtsova and Andrei Suslin. The work is concerned with some basic questions about sets of commuting nilpotent operators on vector spaces. The objects that we construct generalize the class of endotrivial modules that is important in the modular representation theory of finite groups. They can also be used to construct bundles on projective spaces and Grassmannians.

Symplectic Learning Seminar

Speaker: M. Pinsonnault

"Dynamics and Symplectic Capacities"

Time: 13:30

Room: MC 107

I will talk about the dynamical aspects of Linar Symplectic Widths, and then I will introduce nonlinear capacities.

Algebra Seminar

Speaker: Vikram Balaji (Chennai Math Institute)

"Vector bundles and non-abelian mathematics"

Time: 14:30

Room: MC108

The aim of the talk will be to look at non-abelian analogues of Kummer theory for function fields of Riemann surfaces and relate them to bundles. We will trace the ramifications of Weil's paper "Generalizations des fonctiones abeliennes".

Geometry and Topology

Speaker: Matthias Franz (Western)

"Describing toric varieties and their equivariant cohomology"

Time: 15:30

Room: MC 108

I will explain how complex and real toric varieties and their non-negative parts can easily be defined topologically. This gives in particular canonical cell decompositions of these spaces.

I will also discuss consequences to the ordinary and equivariant integral cohomology of toric varieties. For example, if the ordinary cohomology is concentrated in even degrees, then the equivariant cohomology can be described by piecewise polynomials. If the toric variety is in addition smooth or compact, then its ordinary cohomology is necessarily torsion-free.

Noncommutative Geometry

Speaker: A. Motadelro (Western)

"holomorphic structure on quantum projective line"

Time: 14:30

Room: MC 106

Abstract: The aim of this talk is to present the notion of holomorphic structure in noncommutative setting. Focusing on quantum projective line we will see that some of the classical structures have perfect analogues here. Also we shall explain a twisted positive Hochschild cocycle related to this complex structure.

Analysis Seminar

Speaker: Tatyana Foth (Western)

"On holomorphic k-differentials on some open Riemann surfaces"

Time: 15:30

Room: MC 108

Let X be a hyperbolic Riemann surface and A be a closed subset of X. We study spaces of integrable, square-integrable and bounded holomorphic k-differentials on X-A. Our main results provide a description of the kernel of the Poincare series map. This is joint work with N. Askaripour.

Pizza Seminar

Speaker: Sheldon Joyner (Western)

"Solving Rubik's cube using group theory"

Time: 17:00

Room: 108

Group theory is the mathematical language of symmetry, and as such has many real world applications, ranging from the study of crystals to fundamental ideas about the workings of the universe. In this talk, we will introduce group theory and see how it is used to create a wonderful algorithm to solve Rubik's cube.

Everyone welcome!

Algebra Seminar

Speaker: Sheldon Joyner (Western)

"The geometry of the functional equation of Riemann's zeta function"

Time: 14:30

Room: MC108

In a seminal 1859 paper, Riemann gave two proofs of the analytic continuation and functional equation of his zeta function. The ideas behind his theta function proof were later developed into a powerful theory of Fourier analysis on number fields, in work of Hecke, Tate and others. In this talk, I will focus instead on the contour integral proof, and based on the ideas therein, will present two infinite families of new proofs of the analytic continuation and functional equation. The proofs are facilitated by geometric data coming from the fact that the polylogarithm generating function is a flat section of the universal unipotent bundle with connection over $\mathbb{P}^{1} \backslash \{0,1,\infty\}$.