Analysis Seminar

Speaker: Stephen Gardiner (University College Dublin)

"Universal Taylor series, conformal mappings and boundary behaviour"

Time: 11:30

Room: MC 108

In various mathematical contexts it is possible to find a single object which, when subjected to a countable process, yields approximations to the whole universe under study. Such an object is termed "universal" and, contrary to expectations, such objects often turn out to be generic rather than exceptional. This talk will focus on this phenomenon in respect of the Taylor series of a holomorphic function, and how the partial sums behave outside the domain of the function. It will discuss how potential theory reveals much about the boundary behaviour of such functions, and their relationship with conformal mappings.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

"Introduction to cyclic cohomology and its applications IV"

Time: 14:30

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Colloquium

Speaker: Tobias Fritz (Perimeter)

"Ordered commutative monoids and their applications"

Time: 15:30

Room: MC 107

Ordered commutative monoids are mathematical structures that are simple to define, yet display a very diverse phenomenology. I will introduce these structures and explain how they formalize situations in which one deals with resource objects and how they can be combined with each other or converted into each other, such as the molecules in a chemical reaction like $2H_2 + O_2 \rightarrow 2H_2O$. Some standard theorems of functional analysis yield results on ordered commutative monoids, which in turn have applications to Shannon's theory of communication and the ordered commutative monoid of graphs.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western University)

"Introduction to cyclic cohomology and its applications V"

Time: 11:00

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Noncommutative Geometry

Speaker: Masoud Khalkhali (Western)

"Introduction to cyclic cohomology and its applications VI"

Time: 11:00

Room: MC 106

In these series of lectures I shall cover some of the main ideas and results in cyclic cohomology and its application to problems in index theory.

Algebra Seminar

Speaker: Michael Bush (Washington and Lee University)

"Hilbert class towers and their Galois groups"

Time: 14:30

Room: MC 107

Given a number field K, the Hilbert class field of K is the maximal unramified abelian extension of K. Iterating this construction, one obtains a tower of fields known as the Hilbert class tower of K. The question of whether such towers can be infinite or must always stabilize after a finite number of steps is connected with a certain embedding problem and was finally resolved in the 1960s by Golod and Shafarevich. In the first part of the talk, I'll discuss this history in more detail. In the second part, I'll explain how one can sometimes obtain information indirectly about such towers using methods from computational group theory.

Dept Oral Exam

Speaker: Javad Rastegari Koopaei (Western)

"Fourier inequalities in Lorentz and Lebesgue spaces"

Time: 14:30

Room: MC 107

This talk is on the mapping properties of the Fourier transform between Banach function spaces. These are generalizations of Hausdorff-Young and Pitt's inequalities.

We provide several relations between weight functions, that guarantee the boundedness of the Fourier series coefficients, viewed as a map between weighted Lorentz spaces. As a useful machinery, we briefly introduce the quasi concave functions and generalize a number of known inequalities. Finally, we apply our results to Fourier inequalities in weighted Lebesgue spaces and Lorentz-Zygmund spaces.

PhD Thesis Defence

Speaker: Mike Rogelstad (Western)

"Combinatorial Techniques in the Galois Theory of p-Extensions"

Time: 13:30

Room: MC 107

A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of $p$-extensions is an important step toward a solution. We illustrate several techniques for counting Galois $p$-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the $\mathbb{F}_p$-dimension of an $n$-th graded piece of the Zassenhaus filtration for various finitely generated pro-$p$ groups, including free pro-$p$ groups, Demushkin groups and their free pro-$p$ products. Several examples are provided to illustrate the importance of these dimensions in characterizing pro-$p$ Galois groups. We also show that knowledge of small quotients of pro-$p$ Galois groups can provide information regarding the form of relations among the group generators.

Algebra Seminar

Speaker: Sunil K. Chebolu (Illinois State University)

"Recent progress on a problem posed by Laszlo Fuchs"

Time: 14:30

Room: MC 107

More than 50 years ago, Laszlo Fuchs asked which abelian groups can be the group of units of a commutative ring. Though progress has been made, the question remains open. We answer this question in the case of indecomposable abelian groups by classifying the indecomposable abelian groups that are realizable as the group of units of a ring of any given characteristic. This is joint work (arXiv:1505.03508) with Keir Lockridge

Comprehensive Exam Presentation

Speaker: Nicholas Meadows (Western)

"The Local Joyal Model Structure"

Time: 13:00

Room: MC 107

Higher category theory is an emerging area of modern mathematics with applications to areas as diverse as algebraic geometry and theoretical computer science. However, there are multiple proposed foundations for this subject. Jacob Lurie believes that higher category theory should be founded on the theory of quasi-categories, which are studied using the Joyal model structure on simplicial sets. On the other hand, Carlos Simpson believes that higher category theory should be founded on simplicial presheaves. This talk will be the beginnings of trying to reconcile these two views, by developing a model structure for simplicial presheaves on a topological space where the weak equivalences are maps which are Joyal weak equivalences on stalks.

Geometry and Topology

Speaker: Alex Suciu (Northeastern University)

"Hyperplane arrangements and Milnor fibrations"

Time: 15:30

Room: MC 107

Much of the fascination with arrangements of complex hyperplanes comes from the rich interplay between the combinatorics of the intersection lattice and the algebraic topology of the complement. A key bridge between the two is provided by the resonance varieties of the cohomology ring and by the characteristic varieties of the fundamental group. This approach helps explain how certain homological invariants associated to the Milnor fibration of the complement can be computed in terms of multinets supported by the underlying matroid.