Geometry and Topology

Speaker: Dan Isaksen (Wayne State)

"Computations in stable motivic homotopy groups"

Time: 15:30

Room: MC 108

The goal of the talk is to describe explicit generators and relations in the stable motivic homotopy groups. Methods include geometric constructions, Toda brackets, and the Adams spectral sequence.

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Properties and uniqueness of Chern classes for vector bundles"

Time: 14:00

Room: MC 106

In this talk, we will explore some properties of Chern classes. In the axiomatic way, these properties uniquely determine these Chern classes.

Analysis Seminar

Speaker: Andre Boivin (Western)

"Sets of approximation on Riemann surfaces"

Time: 15:30

Room: MC 108

Examples will be given to convince you that I do not know when a closed subset of a Riemann surface is a set of uniform approximation by holomorphic or meromorphic functions.

Pizza Seminar

Speaker: Zack Wolske (Western)

"Wallpaper groups"

Time: 17:30

Room: MC 107

A planar tiling is a repeating symmetric pattern in the plane. Because of their common everyday appearances such patterns are called "wallpaper groups." We follow Conway's orbifold notation, which describes the 17 wallpaper groups as certain topological spaces: quotients of the plane by some finite group. Completeness is given by computing the Euler characteristic of such spaces. No knowledge of groups, topology, orbifolds, or how to hang wallpaper required.

Colloquium

Speaker: Oliver Röndigs (Osnabrück)

"Homotopy types of curves"

Time: 15:30

Room: MC 106

Let X be a smooth projective curve over the complex numbers. The topological space of complex points of X is fairly simple: It is a one-point union of spheres, at least up to stable homotopy equivalence. If X is a smooth projective curve over an arbitrary field, one may consider it within the motivic homotopy theory of Morel and Voevodsky. Under the assumption that X has a rational point, it is possible to split off a top-dimensional sphere if and only if the tangent bundle of X admits a square root.

Stacks Seminar

Speaker: Tom Prince (Western)

"Homotopy Theory and Stacks"

Time: 11:00

Room: MC 107

TBA

Algebra Seminar

Speaker: David Doty (Western)

"Molecular algorithmic self-assembly: theoretical foundations and open problems"

Time: 14:30

Room: MC 108

We review a formal model of molecular self-assembly known as the abstract Tile Assembly Model (aTAM). The aTAM which models the interaction of artificial biochemical macromoleclues known as "DNA tiles", which are capable of binding to each other in specific and surprising ways. The goal of this and other models of self-assembly is to study the feasibility of engineering nanoscale structures through a bottom-up approach, through the "programming" of molecules to automatically assemble themselves, in contrast to top-down approaches such as lithography.

After presenting the aTAM and a few basic results that illustrate its power and its limitations, we survey some theoretical conjectures. These conjectures share the properties of being easy to state, easy to understand, "obviously true", and unresolved. A primary goal is to frustrate the audience with the simplicity of these problems, in the hopes that one of them will step in and solve them.

Geometry and Topology

Speaker: Gereon Quick (Muenster)

"Torsion algebraic cycles and étale cobordism"

Time: 15:30

Room: MC 108

We show that the classical integral cycle class map from algebraic cycles to étale cohomology factors through a quotient of l-adic étale cobordism over an algebraically closed field of positive characteristic. This implies that there is a strong topological obstruction for cohomology classes to be algebraic and that examples of Atiyah and Hirzebruch for non-algebraic integral cohomology classes and of Totaro for non-trivial elements in the Griffiths group also work in positive characteristic.

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Properties and uniqueness of Chern classes for vector bundles"

Time: 14:00

Room: MC 106

In this talk, we will explore some properties of Chern classes. In the axiomatic way, these properties uniquely determine these Chern classes.

Pizza Seminar

Speaker: Rasul Shafikov (Western)

"The mathematics of music: from the wave equation to equal temperament."

Time: 17:30

Room: MC 107

In this talk I will explain how the solution of the wave equation can be used to explain music scales, temperament (i.e., music tuning) and harmony. We will also do a few experiments on a guitar.

Algebra Seminar

Speaker: Martin Bendersky (CUNY)

"The unstable chromatic spectral sequence"

Time: 14:30

Room: MC 107

The (stable) chromatic spectral sequence has had a significant impact on our understanding of the stable homotopy groups of the spheres. I will talk about preliminary attempts to construct an unstable version. I will try to describe a filtration of the stable chromatic spectral sequence induced by the Hopf rings for the odd spheres. There are natural questions that arise in the unstable world (e.g. an unstable version of the Morava stabilizer algebra) and a chromatic interpretation of the Hopf invariant.

Analysis Seminar

Speaker: Martin Bendersky (Hunter College, CUNY)

"A spectral sequence approach to normal forms"

Time: 15:30

Room: MC 107

Normal forms have been used since Poincare. The problem of converting an element in a Lie algebra into its normal form can be a difficult calculation. In joint work with Rick Churchill we have applied the method of spectral sequences to this problem. The talk will be both an introduction to normal forms and (to a lesser degree) an introduction to spectral sequences.

Colloquium

Speaker: Martin Bendersky (CUNY)

"Structure of some toric spaces"

Time: 15:30

Room: MC 108

There is some interest in the structure of toric spaces (e.g. moment angle complexes, toric manifolds, subspace arrangements, etc.) In recent work with Tony Bahri, Fred Cohen and Sam Gitler we have shown that many of these spaces stably split into canonical pieces. I shall describe how this splitting determines the product structure in the cohomology of a generalized moment angle complex and allows us to compute the KO theory of Davis-Januskiewicz spaces (joint with BBCG, Don Davis and Nigel Ray).

Stacks Seminar

Speaker: Tom Prince (Western)

"Homotopy Theory and Stacks II"

Time: 11:30

Room: MC 108

TBA

Algebra Seminar

Speaker: Ivan Dimitrov (Queen's)

"A geometric realization of extreme components of the tensor product of modules over algebraic groups"

Time: 14:30

Room: MC 108

In this talk I will explain how the celebrated theorem of Borel-Weil-Bott provides a natural realization of some extreme components of the tensor product of two irreducible modules of simple algebraic groups. I will also discuss a number of connections of our construction with problems coming from Representation Theory, Combinatorics, and Geometry, including questions about the Littlewood-Richardson cone related to Horn's conjecture, settled by Knutson and Tao in the late 1990's.

The talk is based on joint work with Mike Roth.

Geometry and Topology

Speaker: Igor Kriz (Michigan)

"Equivariant and real motivic stable homotopy theory"

Time: 15:30

Room: MC 108

I will discuss G-equivariant motivic stable homotopy theory for G finite, and some applications, mostly for G=Z/2. In this case, I will construct a motivic (=algebraic) analogue of Atiyah's real K-theory, and related periodicity theorems (Karoubi-Hornbostel theorems, and a new theorem), and also a solution (in some sense) of the completion problem for Hermitian K-theory. I will also discuss an algebraic (=motivic) version of Landweber's Real cobordism.

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern Classes for Hermitian Holomorphic Vector Bundles"

Time: 14:00

Room: MC 106

This is Shiing-Shen Chern's original work on Chern classes. In this talk, we will discuss three things: 1. Any complex vector bundle has a Hermitian structure. 2. Due to the Hermitian holomorphic structure, the formulas for the canonical connection and its curvature are very simple and easy to calculate. 3. More examples of Chern classes will be computed through Chern-Weil's theory.

Analysis Seminar

Speaker: Rasul Shafikov (Western)

"Holomorphic mappings in $\mathbb C^n$: I. Boundary regularity."

Time: 15:30

Room: MC 108

In this survey talk I will give an overview of various results and approaches to the boundary regularity problem of holomorphic mappings between domains in $\mathbb C^n$.

Pizza Seminar

Speaker: Ajnit Dhillon (Western)

"Remote coin tossing"

Time: 17:00

Room: MC 107

Judy and Andrew are going through a bitter divorce. They live thousands of miles apart. They wish to toss a coin over the phone to see who will keep the car. How can they do this without anyone cheating?

Colloquium

Speaker: Adrian Wadsworth (UCSD)

"Computing SK_1 of Division Algebras"

Time: 15:30

Room: MC 108

If D is a division algebra finite dimensional over its center, then SK_1(D) = { a in D | Nrd(a) = 1 }/[D*,D*]. This is a subtle arithmetic invariant of D which has had ongoing interest and applications in the theory of algebraic groups. We will discuss how valuation theory on D is used to compute SK_1(D) for specific division algebras.

Stacks Seminar

Speaker: Enxin Wu (Western)

"Diffeological Bundles and the Irrational Torus"

Time: 11:30

Room: MC 107

This talk will describe some beautiful work of Patrick Iglesias-Zemmour on diffeology. We will start from the definition of diffeological spaces, with some motivations, give some nice properties of diffeological spaces, and introduce diffeological bundles from the point of view of groupoids. Finally, we will discuss the example of irrational torus from a diffeological viewpoint.

Algebra Seminar

Speaker: Adrian Wadsworth (UC San Diego)

"Valued and graded division algebras"

Time: 14:30

Room: MC 108

Given a valuation $v$ on a division algebra $D$, there is an associated graded division algebra $\textrm{gr}(D)$, which is often much easier to work with than $D$ itself, but which retains surprisingly much of the structure of $D$. We illustrate this with $SK_1(D)$. If $v$ on the center of $D$ is Henselian and $D$ is tame, then $SK_1(D)$ is isomorphic to $SK_1(\textrm{gr}(D)).$

Noncommutative Geometry

Speaker: Ajnit Dhillon (Western)

"The Riemann-Roch theorem from Riemann to Hirzebruch and Grothendieck"

Time: 11:30

Room: MC 106

We will begin by discussing the relationship between divisors, line bundle and maps to projective space on a compact Riemann surface. This will motivate the main theorem of the talk, the Riemann-Roch theorem on a compact Riemann surface. Using a result of Chow, we show that this theorem implies that every compact Riemann surface comes from a projective algebraic curve.

Geometry and Topology

Speaker: Sam Isaacson (Western)

"Minimal model structures"

Time: 15:30

Room: MC 108

In a 2002 paper, D.-C. Cisinski completely characterized the accessible model structures on a Grothendieck topos in which the cofibrations are the monomorphisms. All such model structures are Bousfield localizations of a "minimal model structure." I'll discuss some properties of these model structures and two extreme examples: model structures on presheaf topoi and the minimal model structure on the category of simplicial sets. This latter example sheds some light on the weak equivalences in Rezk's category of complete Segal spaces.

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern Classes for Hermitian Holomorphic Vector Bundles II"

Time: 14:00

Room: MC 106

This is Shiing-Shen Chern's original work on Chern classes. In this talk, we will discuss three things: 1. Any complex vector bundle has a Hermitian structure. 2. Due to the Hermitian holomorphic structure, the formulas for the canonical connection and its curvature are very simple and easy to calculate. 3. More examples of Chern classes will be computed through Chern-Weil's theory.

Colloquium

Speaker: Bruce Gilligan (U Regina)

"Holomorphic reductions of homogeneous complex manifolds"

Time: 15:30

Room: MC 108

The Maximum Principle in one complex variable implies that every holomorphic function on any compact complex manifold is constant. One can then ask the question: which non-compact complex manifolds have no non-constant holomorphic functions? In full generality this is difficult to answer. However, if one ask this of complex Lie groups, then one is considering Cousin groups (also called toroidal groups). It turns out that these groups play a central role in the structure theory of some non-compact homogeneous complex manifolds - a subtitle of the talk could be: "How I came to know and love Cousin groups".

In our talk we recall the notions of Lie groups, Lie algebras, and the exponential maps between them. We will show how to construct one particular Cousin group and prove all holomorphic functions on it are constant by using Liouville's theorem, the density of one set in another, and the Identity Principle. From this construction one sees what the structure of all Cousin groups must be and can then classify Abelian complex Lie groups (they are direct products of copies of $\mathbb C$ and $\mathbb C^*$ with Cousin groups). We also define and investigate properties of holomorphic reductions of complex Lie groups and complex homogeneous spaces (with examples, particularly in the nilpotent and solvable cases). In an analogous way one can get reductions relative to other analytic objects on our manifolds, e.g., bounded holomorphic functions and analytic hypersurfaces.

Analysis Seminar

Speaker: Bruce Gilligan (Regina)

"Homogeneous Kahler manifolds"

Time: 14:30

Room: MC 108

We will present some conditions for the existence of Kahler structures on (non-compact) manifolds that are homogeneous under the holomorphic actions of complex Lie groups, particularly when the groups are either solvable or reductive. This includes recent joint work with Karl Oeljeklaus and Christian Miebach of the Universite de Provence in Marseille, France.

Noncommutative Geometry

Speaker: Ajnit Dhillon (Western)

"The Riemann-Roch theorem from Riemann to Hirzebruch and Grothendieck"

Time: 11:30

Room: MC 106

This talk will take place entirely in the algebraic world. We will start with a quick introduction to intersection theory and recall the relevant results from K-theory. The main result is the Grothendieck - Riemann - Roch theorem. Although the theorem is profound the proof is not too difficult so we indicate it. We close by showing that other Riemann-Roch theorems are special cases of this one.

Geometry and Topology

Speaker: Jose Malagon-Lopez (Western)

"The Descent Problem for Presheaves of Spectra"

Time: 15:30

Room: MC 108

Given a presheaf of spectra F, the problem of descent for F can be divided in two. First, to show that any stably fibrant replacement GF of F is sectionwise stable equivalent to F. Second, to obtain a spectral sequence that compute the sheaf \pi_* (GF) by means of cohomology groups with coefficients in the sheafification of \pi_* F. We will review these notions and some known cases.

Noncommutative Geometry

Speaker: Enxin Wu (Western)

"Chern-Simons Forms for Vector Bundles"

Time: 14:00

Room: MC 106

In this talk, we will review the proof of the independence of the choice of connections for Chern classes from Chern-Weil theory, which will lead a way to Chern-Simons theory.

Noncommutative Geometry

Speaker: Ajnit Dhillon (Western)

"Riemann-Roch theorem from Riemann to Hirzebruch and Grothendieck"

Time: 14:00

Room: MC 106

This talk will take place entirely in the algebraic world. We will start with a quick introduction to intersection theory and recall the relevant results from K-theory. The main result is the Grothendieck - Riemann - Roch theorem. Although the theorem is profound the proof is not too difficult so we indicate it. We close by showing that other Riemann-Roch theorems are special cases of this one.