Colloquium

Speaker: Paul Baum (Penn State)

"What is K-theory and what is it good for?"

Time: 11:30 - 12:30

Room: MC 106

This talk will consist of four points:

1. The basic definition of K-theory
2. A brief history of K-theory
3. Algebraic versus topological K-theory
4. The unity of K-theory

Geometry and Topology

Speaker: Kyle Ormsby (Michigan)

"The motivic alpha family over p-adic fields"

Time: 15:30 - 16:30

Room: MC 108

Using a splitting of the algebraic Brown-Peterson spectrum (at the prime 2), I describe the E_2-term of the motivic Adams-Novikov spectral sequence over a p-adic field (p > 2) and identify an analogue of the alpha family within it. Inspired by classical computations from topology and previous work over algebraically closed fields, I determine the behavior of this family, discovering new phenomena (like the existence of nontrivial d_2-differentials) along the way. This produces an ``infinite result" in the stable motivic homotopy groups of the 2-complete sphere spectrum over p-adic fields.

Noncommutative Geometry

Speaker: Paul Baum

"WHAT IS K-HOMOLOGY ?"

Time: 14:00 - 15:30

Room: MC 106

K-homology is the dual theory to K-theory. This talk will give the basic definition (following Atiyah, Brown-Douglas-Fillmore, and Kasparov) of K-homology as abstract elliptic operators. A different approach ( due to Baum-Douglas) will also be indicated. This second definition of K-homology is closely connected to the D-branes of string theory. K-homology will then be used to state the BC (Baum-Connes) conjecture.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Holomorphic mappings in $\mathbb C^n$ : II. The Reflection Principle."

Time: 15:30 - 16:30

Room: MC 108

After a brief review on the Schwarz Reflection Principle in one variable, I will discuss the general situation in higher dimensions using the language of the so-called Segre varieties associated with real analytic hypersurfaces in $\mathbb C^n$. I will then explain how to use it for proving boundary regularity results for holomorphic mappings.

Noncommutative Geometry

Speaker: Paul Baum

"THREE CONJECTURES IN THE REPRESENTATION THEORY OF REDUCTIVE P-ADIC GROUPS (PART 1)"

Time: 11:30 - 13:00

Room: MC 106

The three conjectures are : Local Langlands, Baum-Connes, Aubert-Baum-Plymen. All three conjectures will be carefully stated. The point of view will then be developed that Aubert-Baum-Plymen provides a link between Local Langlands and Baum-Connes. This talk will include an introduction to p-adic numbers and to the representation theory of reductive p-adic groups.

Colloquium

Speaker: Doug Ravenel (Rochester)

"A solution to the Arf-Kervaire invariant problem"

Time: 14:30 - 15:30

Room: MC 107

Mike Hill, Mike Hopkins and I recently solved one of the oldest problems in algebraic topology. The theorem we proved is the opposite of what many people tried to prove about it in the 1970s. This talk will give some history and background of the problem and say a little about our method of proof.

Note: The talk starts at 2:30pm, and coffee will be served at 2pm.

Noncommutative Geometry

Speaker: Paul Baum

"THREE CONJECTURES IN THE REPRESENTATION THEORY OF REDUCTIVE P-ADIC GROUPS (PART 2)"

Time: 15:30 - 17:00

Room: MC 108

This talk will continue to develop the three conjectures (Local Langlands, Baum-Connes, Aubert-Baum-Plymen) and the interactions among them.

Stacks Seminar

Speaker: Jose Malagon-Lopez (Western)

"Some Characterizations of Stacks"

Time: 11:30 - 12:30

Room: MC 107

We will review the concept of a stack as a category where descent works. Starting with a stack as a sheaf of groupoids satisfying effective descent, we will review characterizations of stacks in terms of sieves, and in terms of fibrant models.

Algebra Seminar

Speaker: Letitia Banu (Western)

"Betti numbers of a rationally smooth toric variety"

Time: 12:30 - 13:30

Room: MC 108

Consider an irreducible representation of a semisimple algebraic group with $\lambda$ its highest weight and look at the action of the Weyl group $W$ on the rational vector space spanned by the roots. Take the convex hull of the $W$-orbit of $\lambda$ and obtain the polytope $P_{\lambda}= {\textrm{Conv}}(W.\lambda)$. We are interested in describing the Betti numbers of the toric variety $X(J)$ associated to the polytope $P_{\lambda}$ when the Weyl group is the $n$ symmetric group and $X(J)$ is a rationally smooth variety which doesn't depend on the highest weight $\lambda$ but on the set of reflections that fix $\lambda$ called $J$. The main result is a recursion formula for the Betti numbers of $X(J)$ in terms of Eulerian polynomials. The theory of algebraic monoids developed by Renner and Putcha is effectively used in our computations.

Stacks Seminar

Speaker: Emre Coskun (Western)

"Gerbes from a Stack-Theoretic Perspective"

Time: 13:30 - 14:30

Room: MC 107

In this talk, we will define a gerbe and examine the relation to the second cohomology with coefficients in an abelian group scheme. As examples, we will discuss relations to Azumaya algebras and Brauer-Severi schemes

Stacks Seminar

Speaker: Emre Coskun (Western)

"Gerbes from a Stack-Theoretic Perspective - Part II"

Time: 13:00 - 14:00

Room: MC 107

In this talk, we will discuss some of the properties of gerbes from a stack-theoretic perspective. We will talk about the universal property of the structure morphism of a gerbe, define a neutral gerbe and prove that a neutral gerbe is of the form BG, where G is a sheaf of groups.

Algebra Seminar

Speaker: Bahram Rangipour (New Brunswick)

"Hopf algebras in Geometry without groups"

Time: 14:00 - 14:00

Room: MC 107

It is now more than a decade that Hopf algebras established themselves as an integral part of Noncommutative Geometry via the work of Connes and Moscovici on the computation of the index of hypoelliptic operators on manifolds. The latest Hopf algebras constructed were those associated to Cartan-Lie pseudogroups. In this talk we canonically associate a Hopf algebra to any bicrossed sum Lie algebras. This construction covers all known cases in type II and also type III. The constructed Hopf algebra is naturally equipped with a modular pair in involution which is the coefficients for the Hopf cyclic cohomology of the Hopf algebra. At the end we show how to compute the Hopf cyclic cohomology of these Hopf algebras.

Distinguished Lecture

Speaker: Ivan Fesenko (Nottingham)

"Higher fields and adeles associated to arithmetic surfaces, and a translation invariant integration on higher local fields and adeles"

Time: 15:30 - 16:30

Room: MC 107

This is part 1 of the series "A generalization of the adelic analysis theory of Tate and Iwasawa to arithmetic surfaces"

On arithmetic surfaces one can work with refined structures which the classical algebraic geometry does not really see. Some of them come as higher global, local-global and local fields. The latter fields include power series over usual local fields. Their topology is very unusual. Remarkably, one can integrate over them.

Unlike the classical case of dimension one, arithmetic surfaces have two quite different adelic spaces associated to them: one which is good for geometric applications and another which is good for zeta functions and integration.

The apparent separation between geometry and analysis is a typical phenomenon in dimension two; the BSD conjecture shows that they are not unrelated.

Distinguished Lecture

Speaker: Ivan Fesenko (Nottingham)

"K-delic structures on arithmetic surfaces and two-dimensional class field theory"

Time: 15:30 - 16:30

Room: MC 107

This is part 2 of the series "A generalization of the adelic analysis theory of Tate and Iwasawa to arithmetic surfaces"

Abelian extensions of a two-dimensional local field can be described by open subgroups of the topological Milnor K₂-group of the field. Abelian extensions of the field of rational functions of an arithmetic surface can be described using a Milnor K₂-deles which generalize the K₁-group of adeles in dimension one, and its appropriate quotients. The K₁-groups of the two adelic spaces on the arithmetic surface are related via the K₂-delic groups.

Distinguished Lecture

Speaker: Ivan Fesenko (Nottingham)

"Zeta integral on a regular model of elliptic curve over global field and applications to three fundamental properties of its zeta function"

Time: 15:30 - 16:30

Room: MC 107

This is part 3 of the series "A generalization of the adelic analysis theory of Tate and Iwasawa to arithmetic surfaces"

Using the integration on the two-dimensional objects we define the zeta integral on a relative arithmetic surface whose generic fibre is a smooth geometrically irreducible curve. The theory is the simplest when the genus of the curve is 1.

In dimension two mathematicians have been working with the L-function rather than with the zeta function; however, the existing methods are very restrictive: the base number field cannot be too far away from totally real fields. The zeta integral allows to study the zeta function of the surface directly, for the first time.

The additive duality in dimension two leads to a new theta formula and a two-dimensional version of the Tate thesis. Two-dimensional adelic analysis reduces the analytic properties of the zeta function to those of a so called boundary term given in its integral adelic representation. The boundary term related the geometric and analytic structures. In particular, two-dimensional adelic analysis includes a new powerful method to settle the BSD conjecture. Aspects of the meromorphic continuation and functional equation, location of poles and behavior at the central point become very closely interrelated with each other in the two-dimensional theory.

Colloquium

Speaker: Ján Mináč (Western)

"Galois theory of p-extensions and what I would like to do when I become 90 years old"

Time: 15:30 - 16:30

Room: MC 108

At that time I would like to invent yet another deep Hilbert 90-like theorem and apply it to the structure of Galois groups and arithmetic algebraic geometry. I will try to motivate my cheerful dream about youthful old age by historical remarks about Hilbert 90 and its applications in recent work with Sunil Chebolu and Ido Efrat.