Geometry and Topology

Speaker: Tatyana Foth (Western)

"Holomorphic k-differentials on Riemann surfaces"

Time: 15:30 - 16:30

Room: MC 107

Let k be a positive integer. A k-differential on a Riemann surface C is a section of the k-th tensor power of the canonical bundle of C. I will review what is known about the space of holomorphic k-differentials in the case when C is compact. I will state some new results for the case when C is non-compact.

Operads Seminar

Speaker: Zackary Wolske (Western)

"Introduction to $A_{\infty}$-Algebras"

Time: 14:30 - 15:30

Room: MC 107

An $A_{\infty}$-algebra generalizes an associative algebra, by requiring the binary operation to only be associative up to the derivative of a ternary operation. There is then a 4-ary operation satisfying some relation between these two, and we can continue to get the structure of an operad. Beginning with an augmented differential graded associative (dga) algebra, we use the Eilenberg-MacLane bar construction to get a dga coalgebra with some nice properties. We use this to define a general $A_{\infty}$-algebra, and then manipulate it as a chain complex to find the homotopy.

Colloquium

Speaker: Patrick Iglesias-Zemmour (Marseille)

"A guided tour of diffeology"

Time: 15:30 - 16:30

Room: MC 108

After introducing rapidly the context of diffeology, I'll try to review some of the main constructions and show, through examples, how this differential geometry treat objects which do not belong to the classical category of manifolds. I will show also how using diffeology some classical theorems are dramatically simplified, and extended, in particular in de Rham calculus. I may also, if I have the time, introduce the program of symplectic diffeology: results, questions and perspectives.

Algebra Seminar

Speaker: Richard Gonzales (Western)

"Equivariant Euler classes and rational cells"

Time: 14:30 - 15:30

Room: MC 107

Let $X$ be a complex affine variety with an action of a torus $T$, and an attractive fixed point $x_0$. We say that $X$ is a rational cell if $H^{2n}(X,X-\{x_0\})=\mathbb{Q}$ and $H^{i}(X,X-\{x_0\})=0$ for $i\neq 2n$, where $n={\rm dim\,}_{\mathbb{C}}(X)$. These objects appear naturally in the study of group embeddings. A fundamental result in equivariant cohomology asserts that the transgression ${\bf Eu}_T \in H^{2n}(BT)$ of a generator of $H^{2n}(X,X-\{x_0\})$ splits into a product of singular characters, ${\bf Eu}_T={\chi_1}^{k_1}\ldots {\chi_m}^{k_m}$. This characteristic class is by definition the Equivariant Euler class of $X$ at $x_0$. Loosely speaking, one could think of $X$ as a sort of $T$-vector bundle over a point. My goal in this talk is to make this claim precise, and to show why one could hope to build similar elements in equivariant $K$-theory, i.e. Bott classes, by using localization and completion techniques. This is work in progress.

Geometry and Topology

Speaker: Christopher Allday (Hawaii)

"A new look at Duflot's results on the equivariant cohomology of the orbit filtration"

Time: 15:30 - 16:30

Room: MC 107

In 1983 Jeanne Duflot published some results showing that certain long exact sequences were, in fact, short exact. Her methods confined her to smooth actions. Recently, Matthias Franz, Volker Puppe and I have needed to reconsider Duflot's results, and, in the process, we arrived at new proofs that also work for continuous actions on topological manifolds. The main tool in the proof is a spectral sequence that is an equivariant version of Poincare - Alexander - Lefschetz duality. The proof also makes use of some basic properties of Cohen - Macaulay modules and a little bit of local cohomology.

Analysis Seminar

Speaker: Serge Randriambololona (Western)

"Eliminating quantifiers"

Time: 15:30 - 16:30

Room: MC 107

I will discuss about quantifiers, why one may want to eliminate them and about a joint result with S. Starchenko (University of Notre Dame). This talk should contain very little analysis.

Pizza Seminar

Speaker: (Western)

"N is a Number: A Portrait of Paul Erdos"

Time: 16:30 - 17:30

Room: MC 105B

This one hour documentary follows Paul Erdos as he travels the world solving math problems. It features a number of famous mathematicians talking about his influence, and the way that mathematics is done in reality. As usual, pizza follows in the Grad Club.

Operads Seminar

Speaker: Enxin Wu (Western)

"Koszul Duality I"

Time: 14:30 - 15:30

Room: MC 107

This talk focuses on Koszul duality of quadratic algebras. We will quickly review the basic definition and important properies of twisted morphism from a (differential graded) coalgebra to a (differential graded) algebra, equivalent characterization of Koszul morphisms, and the adjunction between cobar and bar constructions.Then we apply this to quadratic algebras, and some classical examples will be introduced

Symplectic Learning Seminar

Speaker: Martin VanHoof (Western)

"Building symplectic toric orbifolds from weighted polytopes."

Time: 15:00 - 16:30

Room: MC 105C

In 1988, Delzant showed how to construct a toric manifold from a special type of polytope (now called a Delzant polytope). This result, combined with the Atiyah, Guillemin-Sternberg convexity theorem, establishes a 1-1 correspondence between symplectic toric manifolds and Delzant polytopes. A few years later, Lerman and Tolman did the same with toric orbifolds, thus establishing an analogous correspondence in this more general case, though the polytopes are now more general and come with weights attached to each facet. In this talk, we will focus on a very specific case of the Lerman-Tolman construction; that of constructing a weighted projective space starting with a weighted (non isosceles) triangle.

Colloquium

Speaker: Volker Puppe (Konstanz)

"Binary codes and involutions on manifolds"

Time: 15:30 - 16:30

Room: MC 107

There is a strong relationship between involutions on compact manifolds and binary codes. Constructions on the side of codes have their counterparts for involutions. Every binary, self-dual code can be obtained from an involution on a 3-dimenional compact manifold with 'maximal' number of isolated fixed points. The code in turn determines the cohomology algebra structure of the manifold. Certain properties of the code are reflected in the geometry of the manifold.

The talk is based on joint work with M. Kreck.

Algebra Seminar

Speaker: Sanghoon Baek (Ottawa)

"Essential dimensions of algebraic groups of types $A_n$"

Time: 14:30 - 15:30

Room: MC 107

In this talk, we introduce the notion of the essential dimension of an algebraic structure and discuss some recent results on the essential dimension of certain classes of central simple algebras. We also relate these results to the essential dimension of split simple groups of type $A_n$.

Geometry and Topology

Speaker: Graham Denham (Western)

"Topological aspects of partial product spaces"

Time: 15:30 - 16:30

Room: MC 107

The notion of a partial product space is a relatively recent unification of various combinatorial constructions in topology. This construction is variously known as the generalized moment-angle complex, or (more euphoniously) as the polyhedral product functor. Some instances of it are closely related to Davis and Januszkiewicz's quasitoric manifolds: these include the moment-angle complexes (Buchstaber and Panov) and homotopy orbit spaces for quasitoric manifolds. By making suitable choices, one also obtains classifying spaces for right-angled Artin groups and Coxeter groups, as well as certain real and complex subspace arrangements.

One advantage to this generality is that some topological information about such spaces can sometimes be expressed directly in combinatorial terms: presentations of cohomology rings; a homotopy-theoretic decomposition of the suspension of a partial product space; descriptions of rational homotopy Lie algebras and the Pontryagin algebra. I will give an introductory overview of some remarkable results along these lines.

Analysis Seminar

Speaker: Gord Sinnamon (Western)

"Positive Integral Operators"

Time: 15:30 - 16:30

Room: MC 107

Norm inequalities determine whether or not an operator acts as a bounded map between two Banach spaces. For a large range of indices an explicit parameterization gives, with best constant, all possible Lebesgue norm inequalities for positive integral operators. This result is outlined and extended to a class of nonlinear integral operators.

Colloquium

Speaker: Noriko Yui (Queen's)

"The modularity (automorphy) of Calabi-Yau varieties over the rationals"

Time: 15:30 - 16:30

Room: MC 107

According to the Langlands Philosophy, every algebraic variety defined over the rationals or a number field should be modular (automorphic).

In this talk, I will concentrate on a special class of algebraic varieties, called Calabi-Yau varieties (of dimension at most three), defined over the rationals, and report on the current status of their modularity (automorphy).

Geometry and Topology

Speaker: Christian Haesemeyer (UCLA)

"Rational points, zero cycles of degree one, and $A^1$-homotopy theory"

Time: 15:30 - 16:30

Room: MC 107

A smooth proper variety with a zero cycle of degree one (that is, closed points of relatively prime degrees) need not have a rational point. In this talk we aim to explain how this phenomenon relates to the difference between unstable and stable $A^1$-homtopy theory.

Geometry and Topology

Speaker: Thomas Huttemann (Belfast)

"Algebraic K-theory of projective toric schemes"

Time: 15:30 - 16:30

Room: MC 107

A projective toric scheme is specified by combinatorial data, viz., a polytope with integral vertex coordinates. I will show how the geometry of the polytope leads to a simple splitting result in the algebraic K-theory of the scheme. In the special case of projective space (given by a standard simplex) this reduces to the well-known splitting of K(P^n) into n+1 copies of the K-theory of the ground ring. - The combinatorial approach is flexible enough to include the case of schemes defined over an arbitrary (possibly non-commutative) ring.

Pizza Seminar

Speaker: Ali Moatadelro (Western)

"Untying Knots with the Jones Polynomial"

Time: 16:30 - 17:30

Room: MC 107

A knot is a smooth embedding of a circle in \(R^3\). A major problem in knot theory is to classify knots. Two knots are identified if one can transform one knot to the other one without tearing.

The discovery of the Jones polynomial in 1980's has been considered as a great achievement in the classification of knots. The Jones polynomial is a sensitive knot invariant which is easy to compute. On the other hand, theories including geometry of low dimensional spaces, quantum groups and statistical mechanics meet each other because of this polynomial.

In this talk we introduce the Jones polynomial in an elementary way from two different point of views. This talk will be accessible to undergraduate students.