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.

Analysis Seminar

Speaker: Eric Schippers (University of Manitoba)

"Dirichlet problem and jump decomposition on quasicircles"

Time: 15:30

Room: MC 108

Any complex harmonic function of finite Dirichlet energy on a Jordan domain has boundary values on the Jordan curve in a sense due to Osborn. For which Jordan curves must there be a harmonic function of finite Dirichlet energy on the complement with these same boundary values?

The Plemelj-Sokhotski jump formula says that a reasonably regular complex function on a reasonably regular Jordan curve can be written as the difference of boundary values of holomorphic functions on the domain and its complement. For which Jordan curves does the jump formula hold in the Dirichlet space setting?

The answer to both of these questions (once they are made suitably precise) is: for those Jordan curves which are quasicircles.

Speaker's web page: http://server.math.umanitoba.ca/~schippers/

PhD Thesis Defence

Speaker: Nadia: Public Lecture (Western)

"On Vector-Valued Automorphic Forms On Bounded Symmetric Domains"

Time: 14:00

Room: MC 107

Abstract: The main points of this talk are as follows:

- constructions of vector-valued automorphic forms on bounded symmetric domains via Poincare series

- vector-valued automorphic forms associated to submanifolds of the complex unit ball - studying the behavior of asymptotics of the inner product of two Poincare series associated to submanifolds of the complex unit ball, for large weights, with examples.

Algebra Seminar

Speaker: Tristan Freiberg (Waterloo)

"Distribution of sums of two squares"

Time: 14:30

Room: MC 107

We formulate a conjecture, analogous to the Hardy--Littlewood prime tuples conjecture, for tuples of sums of two squares. Assuming this conjecture holds, we show that sums of two squares are distributed among the natural numbers as if according to a Poisson process. We discuss numerical evidence, and unconditional results, in support of our conjecture. This is joint work with Pär Kurlberg and Lior Rosenzweig.

Algebra Seminar

Speaker: Vaidehee Thatte (Queen's University)

"Ramification theory for arbitrary valuation rings in positive characteristic"

Time: 14:30

Room: MC 108

Our goal is to develop ramification theory for arbitrary valuation fields, that is compatible with the classical theory of complete discrete valuation fields with perfect residue fields. We consider fields with more general (possibly non-discrete) valuations and arbitrary (possibly imperfect) residue fields. The defect case, i.e., the case where there is no extension of either the residue field or the value group, gives rise to many interesting complications. We present some new results for Artin- Schreier extensions of valuation fields in positive characteristic. These results relate the "higher ramification ideal" of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We also introduce a generalization and further refinement of Kato's refined Swan conductor in this case. Similar results are true in the mixed characteristic case.

Geometry and Topology

Speaker: Vic Snaith (Sheffield)

"The Bernstein centre of smooth representations"

Time: 15:30

Room: MC 107

In the 1980's Bernstein-Zelevinski calculated the centre of the abelian category of smooth representations on $GL_{n}K$ when $K$ is a local field. Soon after Deligne generalised this to all reductive algebraic groups $G$ over $K$. The centre of a category consists of all families $z_{A} \in End(A)$ as $A$ varies through all objects such that for any morphism in the category $f:A \longrightarrow B$ we have $fz_{A} = z_{B}f$. Deligne's answer comes in terms of distributions on $G$.

Over the last decade or so, I developed the notion of monomial resolutions for such representations. This amounts to an embedding of the representation category into a derived category of monomial objects. Using Bruhat's thesis I shall explain how to interpret the monomial morphisms in terms of spaces of distributions and thereby to re-derive Deligne's result.

I know to my cost how technical this stuff can get - so I shall try to navigate by means of conceptual insights. For example, for us topologists, I shall explain how sheaves of distributions behave in a manner precisely analogous to a famous result of Swan and Serre about sections of topological vector bundles.

Homotopy Theory

Speaker:

"Organizational meeting"

Time: 13:30

Room: MC 108

Algebra Seminar

Speaker: Marine Rougnant (Université de Franche-Comté)

"On the propagation of the mildness property along some imaginary quadratic extension of ℚ"

Time: 14:30

Room: MC 107

Let $p>2$ be a prime number and $K$ be a number field. Let $S$ be a finite set of primes of $K$ and let $K_S$ be the maximal pro-$p$ extension of $K$ unramified outside $S$; put $G_S=$ Gal $({K_S}{K})$. If $S$ contains the primes above $p$, we know that $cd(G_S)$ less than or equal to $2$, but what is going on if this is not the case?

Thanks to a criteria of Labute, Mináč and Schmidt, we can exhibit mild pro-$p$ groups $G_S$ (and then of cohomological dimension $2$). In this talk I will explain the question of the propagation of the mildness property along some quadratic extensions of $\mathbb{Q}$. In particular, I will give some statistics and some theoretical results.