Algebra Seminar

Speaker: Adam Chapman (Tel-Hai Academic College)

"Linked fields of characteristic 2 and their u-invariant"

Time: 14:30

Room: MC 107

The u-invariant of a field is the maximal dimension of a nonsingular anisotropic quadratic form over that field, whose order in the Witt group of the field is finite. By a classical theorem of Elman and Lam, the u-invariant of a linked field of characteristic different from 2 can be either 0, 1, 2, 4 or 8. The analogous question in the case of characteristic 2 remained open for a long time. We will discuss the proof of the equivalent statement in characteristic 2, recently obtained in joint work by Andrew Dolphin and the speaker.

Basic Notions Seminar

Speaker: Masoud Khalkhali (Western)

"What is Spectral Geometry?"

Time: 15:30

Room: MC 107

Spectral geometry, among other things, asks the question `can one hear the shape of a drum?' To a mathematical object, say a Riemannian manifold, one can attach its spectrum and one is interested to know to what extent the object can be recovered from its spectrum. The spectral information can be encoded in terms of zeta functions, heat trace, or wave trace. Isometry invariants like volume and total scalar curvature can be obtained as special values of the spectral zeta function (Weyl's law). I shall give a quick introduction to these ideas and will end by giving the first example of two isospectral manifolds which are not isometric. The example, due to Milnor (using some deep work of Ernst Witt based on the theory of modular forms), exhibits two 16 dimensional flat tori which are isospectral but not isometric. This talk will be accessible to all grad students.

Analysis Seminar

Speaker: Dilian Yang (University of Windsor)

"[Rescheduled]"

Time: 15:30

Room: MC 108

Moved to the next session of the seminar.

Algebra Seminar

Speaker: Matthias Franz (Western)

"Symmetric products of maximal varieties"

Time: 14:30

Room: MC 107

The fixed point set of an anti-holomorphic involution on a compact Riemann surface X, say of genus g, consists of at most g+1 circles. If this bound is attained, one calls X maximal. More generally, an algebraic variety X with an anti-holomorphic involution is called maximal if the sum of the mod-2 Betti numbers of X is equal to the corresponding sum for the fixed point set.

Recently, Biswas and D'Mello have shown that the n-th symmetric product of a maximal curve is again maximal provided that n <= 3 or n >= 2g-1. In this talk, we will explain how to generalize this result not only to all n and to other maximal varieties, but even beyond the realm of algebraic geometry.

Analysis Seminar

Speaker: Dilian Yang (University of Windsor)

"Boundary quotient C*-algebras of products of odometers"

Time: 15:30

Room: MC 108

There is a semigroup naturally associated to the standard product of $k$ odometers. In this talk, we will focus on the boundary quotient of the full C*-algebra of the semigroup. Among other things, we will provide some characterizations on the simplicity of the boundary quotient, and discuss its nuclearity and pure infiniteness. If time permits, some relations with the C*-algebra $\mathcal{Q}_{\mathbb{N}}$ introduced by Cuntz will also be given.

This is joint work with Hui Li.

Speaker's web page: http://www1.uwindsor.ca/math/dilian-yang

Analysis Seminar

Speaker: Maritza Branker (Niagara University)

"[Cancelled]"

Time: 15:30

Room: MC 108

Cancelled by the speaker.

Pizza Seminar

Speaker: Masoud Khalkhali (Western)

"Infinitely many primes and beyond"

Time: 18:30

Room: MC 107

I shall explain different proofs by Euclid and Euler which show that there are infinitely many primes. I will then indicate how attempts to refine and extend this result and understand the distribution of primes, led mathematicians to beautiful discoveries and open problems, including the celebrated Riemann hypothesis.

Since this is a pi day talk, the number pi will surely appear at some point through a beautiful formula of Euler relating primes to pi. I shall try to indicate a much more modern understanding of this formula as equality of volumes of some geometric objects.

With free pie!

Geometry and Topology

Speaker: Chris Hall (Western)

"Families of covers of graphs"

Time: 15:30

Room: MC 107

We will discuss the notion of a random cover of an undirected graph and averages one can calculate for such covers. While the definition we will give for a cover is likely to be familiar to anyone who has studied a bit of algebraic topology, we will not assume everyone in the audience has the background. The notion of random we will give is quite naive, but it leads to algebraic objects we regard as very interesting, e.g., d-matchings polynomials. I will explain some known properties of these geometric and algebraic, if time permits, point out open questions which I regard as interesting.

I intend for this talk to be accessible to graduate students and encourage anyone who thinks they might be interested to attend.

Analysis Seminar

Speaker: Eleonore Faber (University of Michigan)

"Reflection groups and the McKay correspondence"

Time: 15:30

Room: MC 108

Let $G$ be a finite subgroup of $GL(n,\mathbb{C})$. Then $G$ acts linearly on the polynomial ring $S$ in $n$ variables over $\mathbb{C}$. When $G$ is generated by reflections, then the discriminant $D$ of the group action of $G$ on $S$ is a hypersurface with singular locus of codimension 1. The classical McKay correspondence relates the geometry of the resolutions of singularities of so-called Kleinian surfaces with the representation theory of finite subgroups of $SL(2,\mathbb{C})$. In particular, there is an algebraic version of this correspondence, due to M. Auslander.

In this talk we present a version of the McKay correspondence when $G$ is a finite group generated by reflections: We give a natural construction of a so-called noncommutative resolution of the coordinate ring of $D$ as a quotient of the skew group ring $A=S*G$. We will explain this construction, which allows to extend Auslander's theorem to reflection groups. This is joint work with R.-O. Buchweitz and C. Ingalls.

Speaker's web page: http://www-personal.umich.edu/~emfaber/index.html

Analysis Seminar

Speaker: Luke Broemeling (Western)

"A generalization of Kallin's Lemma to Stein manifolds"

Time: 15:30

Room: MC 108

Kallin's Lemma is a technical tool useful for proving the polynomial convexity of certain unions of polynomially convex compacts (such as the union of 3-balls). We show that this result extends to holomorphically convex compacts in a Stein manifold. We will define Kallin's Lemma, introduce the necessary background from the theories of uniform algebras and Stein manifolds, and prove the generalization.

This is a PhD Comprehensive Examination.

Speaker's web page: http://www.math.uwo.ca/index.php/profile/view/208/

Colloquium

Speaker: Frank Sottile (Texas A&M)

"Galois groups in Enumerative Geometry and Applications"

Time: 15:30

Room: MC 107

In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.

I will describe this background and discuss some work in a long-term project to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry. A main focus is to understand Galois groups in the Schubert calculus, a well-understood class of geometric problems that has long served as a laboratory for testing new ideas in enumerative geometry.

Pizza Seminar

Speaker: Frank Sottile (Texas A&M)

"Shape of Space"

Time: 17:30

Room: MC 108

In mathematics and science, we often need to think about high (3 or more) dimensional objects, called spaces, which are hard or impossible to visualize. Besides the question of what such objects are or could be, is the problem of how can we make sense of such spaces.

The goal of this discussion is to give you an idea of how mathematicians manage to make sense of higher-dimensional spaces. We will do this by exploring the simplest spaces, and through our explorations, we will begin to see how we may tell different spaces apart.