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Jérémy Champagne

"Weyl's equidistribution theorem in function fields"

Finding a proper function field analogue to Weyl's theorem on the equidistribution of polynomial sequences is a problem that was originally considered by Carlitz in 1952. As noted by Carlitz, Weyl's classical differencing methods can only handle polynomials with degree less than the characteristic of the field. In this talk, we discuss some recent methods which avoid this "characteristic barrier", and we show the existence of polynomials with extremal equidstributive behaviour. 

This is joint work with Yu-Ru Liu, Thái Hoàng Lê and Trevor D. Wooley.

MC 5403

Speaker: Thomas Bray

"The Birkhoff-Kakutani Theorem"

I will introduce the basic concepts pertaining to topological groups. After this, I will show one way of proving the Birkhoff-Kakutani theorem. Time permitting, I will demonstrate how one can use Birkhoff-Kakutani to build a complete metric on a Polish group.

MC 5403

Speaker: William Gollinger

Topology Learning Seminar: The Adams Spectral Sequence

The Adams Spectral Sequence was introduced by Frank Adams in his paper "On The Structure and Application of the Streenrod Algebra" (1958) with applications to the stable homotopy groups of spheres and the Hopf-Invariant One problem. In the context of the stable homotopy category it was soon upgraded to the general problem of computing the coefficients of extraordinary cohomology theories.  In this series of lectures we will outline a construction of the Adams Spectral Sequence following Ravenel's "Green Book", and give applications including computations of some stable homotopy groups of spheres as well as certain Madsen-Tillmann bordism groups which have recently been of interest in the theory of TQFTs. 

The seminar assumes some basic knowledge of algebraic topology (in particular homotopy theory and ordinary homology theory) but is aimed to be expository, introducing the audience to important topological concepts such as stable homotopy theory and cohomology operations. The topics presented will be roughly in the following order: examples of the Leray-Serre spectral sequence; the Stable Homotopy Category; construction of the Adams Spectral Sequence; the Steenrod Algebra and its dual; computations.

MC 5417

Sascha Troscheit, University of Oulu

A classical problem in dynamical systems is known as the shrinking target problem: given a sequence of 'target' subsets A_n \subset X and a dynamic T: X \to X we ask how 'large' the set of all points R \subset X is whose n-th iterate hits the target, T^n (x) \in A_n, infinitely often. Much progress has been made on understanding this type of 'recurrent' set and I will highlight some recent results on this and the related 'dynamical covering problem' which is a dynamical generalisation of the Dvoretzky covering problem. The talk is based on joint results with Balázs Bárány, and Henna Koivusalo and Balázs Bárány.

MC5417

Alex Chirvasitu, University at Buffalo

Pr¨ufer surfaces are non-metrizable separable 2-manifolds originally defined by Calabi and Rosenlicht by doubling the upper half-plane along a continuum’s worth of real-line boundary components. The construction and variations on it have since been studied by Gabard, Baillif and many others for the purpose of probing the pathologies of non-paracompact manifolds. The fundamental groups of such surfaces and higher-dimensional cousins are known to be (essentially) free on the sets S of connected boundary components, so their first cohomotopy groups (i.e. sets of homotopy classes of continuous maps to rather than from the circle) are identifiable with maps from S to the integers. Which functions S → Z arise in this manner is a natural question, with (perhaps) a surprising answer. The goal will be to discuss that problem, but the manifolds themselves might provide some entertainment value on their own.

MC5417

Benoit Charbonneau, Department of Pure Mathematics, University of Waterloo

“Coxeter groups and Clifford Algebras”

If one wants to understand representation theory of the rotation group of the icosahedron, or of its lift to Sp(1), it is extremely useful to be able to compute things intelligently. It turns out that instead of using matrices, it is much better to play with Clifford Algebras. I’ll explain those concepts and illustrate them.

MC 5417

Spiro Karigiannis, Department of Pure Mathematics, University of Waterloo

“The linear algebra of 2-forms in 4-dimensions”

I will present some important facts about the linear algebra of 2-forms in 4 dimensions, which everyone should know. We start with classical results about self-dual and anti-self dual 2-forms, and then proceed to discuss "hypersymplectic" structures in 4d à la Donaldson. Then we put all this on an oriented Riemannian 4-manifold.

MC 5417

Andy Zucker, Department of Pure Mathematics, University of Waterloo

“Preliminaries on Polish Spaces”

This inaugural talk will introduce some of the background on Polish spaces that we will need in our study of Polish groups. We will mostly draw from the early chapters of Kechris's book ‘Classical Descriptive Set Theory’.

MC 5403

Aleksandar Milivojevic, Department of Pure Mathematics, University of Waterloo

"Obstructions to almost complex structures following Massey"

I will report on work in progress with Michael Albanese, in which we prove statements claimed by Massey in 1961 concerning the obstructions to finding an almost complex structure on an orientable manifold (or more generally, reducing the structure group of a real vector bundle over a CW complex to the unitary group). These obstructions involve the integral Stiefel-Whitney classes – which detect the existence of integral lifts of the mod 2 Stiefel-Whitney classes, namely putative Chern classes – and relations between the Pontryagin and Chern classes. A somewhat surprising aspect of these obstructions is that they are in fact generally proper fractional parts of what one might at first expect. For example, the obstruction in degree eleven is 1/24 of the eleventh integral Stiefel-Whitney class.

MC 5403

Anne Johnson, Department of Pure Mathematics, University of Waterloo

"Attributes and morphisms of schemes"

We start Chapter 3 of Eisenbud and Harris, discussing finiteness conditions, properness and separation. We discuss the construction of Proj S as time allows.

MC 5417