Geometry and Topology

Speaker: Virginie Charette (Sherbrooke)

"Stretching three-holed spheres and the Margulis invariant"

Time: 15:30

Room: MC 107

A complete flat Lorentz 3-manifold M is a quotient of Minkowski (2+1)- spacetime by a discrete group G of affine isometries acting freely and properly. The study of such discrete groups relates to the deformation theory of hyperbolic structures on a hyperbolic surface S corresponding to M. Properness of G's action relates to lengthening (or shortening) of geodesics on S. Determining criteria for a proper action is, in general, a difficult problem. When S is a three-holed sphere, the sign of the "Margulis invariant" on each boundary components of S determines whether G acts properly or not -- this is a result we have shown with Drumm and Goldman. We will discuss this theorem and how it applies to deformations of hyperbolic structures on S.

Stable Homotopy

Speaker: Peter Oman (Western)

"Enriched Category Theory and Homotopy Theory 2"

Time: 14:00

Room: MC 107

Analysis Seminar

Speaker: Anna Valette (Jagiellonian University, Krakow)

"Geometry of polynomial mappings 1"

Time: 15:30

Room: MC 108

In the first lecture we will introduce some basic notions to study the behaviour of polynomial mappings, we will see what can happened at infinity and how the bifurcation set is related to the set of asymptotic critical values. Then, in the next two lectures we will focus on the asymptotic variety of polynomial mappings, i.e. on the set of points at which such a map fails to be proper. The geometry of this set and how we can get explicit description of it will be discussed.

Colloquium

Speaker: Gregory Pearlstein (Michigan State University)

"Normal functions and the Hodge conjecture"

Time: 15:30

Room: MC 108

The Hodge conjecture has its origins in the work of Lefschetz regarding which 2 dimensional homology classes on an algebraic surface could be represented via algebraic curves on the surface. Lefschetz's solution involved the study of a class of "Poincare normal functions" on the Riemann sphere minus a finite number of points. In this talk, I will outline Lefschetz's proof and discuss some recent work of Griffiths and Green towards studying the Hodge conjecture for higher codimension cycles using normal functions on higher dimensional parameter spaces.

Analysis Seminar

Speaker: Javad Mashreghi (Université Laval)

"Zero Sets of the Dirichlet Space"

Time: 14:30

Room: MC 108

There is a complete characterization of the zeros sets of the Hardy space Hp. However, at the present, we have just some partial characterizations for most of the relatives of Hp, e.g. the Bergman space and the Dirichlet space. For the latter space, we will discuss Carleson’s condition and its generalization by Shapiro-Shields. Then we give some new families of zero sets which are not covered by the preceding classical results.

Algebra Seminar

Speaker: Mehdi Garrousian (Western)

"Structured resolutions of monomial and binomial algebras"

Time: 15:30

Room: MC 106

The plan of the talk is to look at some explicit recipes for constructing minimal free resolutions of certain 'nice' classes of monomial and binomial ideals in a polynomial ring.

The general idea is to introduce appropriate simplicial or polytopal complexes which encode the information of a free resolution and read off the resolution from the complex.

Geometry and Topology

Speaker:

"No seminar"

Time: 15:30

Room: MC 107

Pizza Seminar

Speaker: Yuri Boykov (Western)

"Can computers see?"

Time: 16:30

Room: MC 108

Nowdays computers solve so many difficult tasks for us that we take them for granted. They can automatically land an airplane, detect a problem in your car, create special effects for movies, write music, and understand your questions over the telephone. Ask yourself, why is it that computers can predict global weather patterns yet, if you plug in a camera, they cannot tell a dog from a cat? The list of seemingly trivial visual tasks that computers cannot do goes on and on. For example, humans easily recognize their friends at a distance of 5-10 meters. Most people have no problems locating objects around them. We effortlessly percieve 3D shapes from just looking at photos or paintings. We can do all of this because our brain analyzes images projected onto our eyes' retinas in real-time. "Computer vision" is an area of computer science devoted to making computers understand images, much as humans do. Images come from a digital photo/video cameras connected to a desktop PC, laptop, or cellphone. They also come from MRI/CT scanners, electronic microscopes, ultrasound sensors, and many other sources. This talk will present some of the challenges facing computer vision and some of its recent advances. We will also discuss some mathematical models widely used in computer vision. In particular, discrete models based on Markov Random Fields and related models from based on deferential and integral geometry. We will also discuss some related optimization issues.

Stable Homotopy

Speaker: Peter Oman (Western)

"Model structures in stable homotopy theory"

Time: 14:00

Room: MC 107

Analysis Seminar

Speaker: Anna Valette (Jagiellonian University, Krakow)

"Geometry of polynomial mappings 2"

Time: 15:30

Room: MC 108

In the first lecture we will introduce some basic notions to study the behaviour of polynomial mappings, we will see what can happened at infinity and how the bifurcation set is related to the set of asymptotic critical values. Then, in the next two lectures we will focus on the asymptotic variety of polynomial mappings, i.e. on the set of points at which such a map fails to be proper. The geometry of this set and how we can get explicit description of it will be discussed.

Algebra Seminar

Speaker: Richard Kane (Western)

"Topological K-theory"

Time: 15:30

Room: MC 106

Topological K-theory was the first "extraordinary" cohomology theory introduced into algebraic topology. It is extraordinary not only in that it generalized ordinary cohomology theory but also because of its immediate success in solving important problems and in contributing to the development of new mathematical theory. Topological K-theory was developed in the late 1950's by Atiya and Hirzebruch, building on the work of Grothendieck. During the 1960's in particular it found a number of important applications by Atiyah and collaborators. The goal of my talk is to explain the basics of this classical theory and to outline some of its important applications, namely those concerning how it can be used to study Lie groups and their classifying spaces. The usefulness of equivariant K-theory is a particular theme of these applications.

Geometry and Topology

Speaker: Jose Malagon Lopez (Western)

"Equivariant Algebraic Cobordism"

Time: 15:30

Room: MC 107

In a joint work (in progress) with J. Heller, following Edinin-Graham and Totaro's construction for equivariant Chow groups we construct a Borel-style G-equivariant algebraic cobordism for G-schemes, where G is a lineal algebraic group over a field of characteristic zero. We will discuss some basic properties and some computations for some groups G.

Analysis Seminar

Speaker: Anna Valette (Jagiellonian University, Krakow)

"Geometry of polynomial mappings 3"

Time: 15:30

Room: MC 108

In the first lecture we will introduce some basic notions to study the behaviour of polynomial mappings, we will see what can happened at infinity and how the bifurcation set is related to the set of asymptotic critical values. Then, in the next two lectures we will focus on the asymptotic variety of polynomial mappings, i.e. on the set of points at which such a map fails to be proper. The geometry of this set and how we can get explicit description of it will be discussed.

Colloquium

Speaker: Mark Spivakovsky (Université Paul Sabatier (Toulouse III))

"The Pierce-Birkhoff conjecture and the real spectrum of a ring"

Time: 15:30

Room: MC 108

A function f : Rⁿ -> R is said to be piecewise polynomial if there exist finitely many polynomials f_i in n variables such that for every point a in Rⁿ we have f(a) = f_i(a) for at least one f_i. The celebrated Pierce-Birkhoff conjecture asserts that every piecewise polynomial function f on Rⁿ can be obtained from a finite collection of polynomials by iterating the operations of maximum and minimum. This is equivalent to saying that there exists a finite collection f_{ij} of polynomials such that f = max limits_i (min limits_j f_{ij}). In this lecture, I will describe an approach to proving this conjecture proposed by J. Madden in the nineteen eighties and which we continue to  develop more recently with F. Lucas, D. Schaub and J. Madden. Our key tool is  the real spectrum of a ring; a large part of the lecture will be devoted to introducing the real spectrum.

Algebra Seminar

Speaker: Tony Bahri (Rider University)

"Algebras related to Borel constructions in toric geometry and topology"

Time: 15:30

Room: MC 106

Toric spaces have associated to them Borel constructions with respect to the actions of various tori. The cohomology (corresponding to complex-oriented theories) can be related to Stanley-Reisner rings, rings of piecewise polynomials and associated quotients. Examples exist for which the distinction between a ring of piecewise polynomials and the Stanley-Reisner ring mirrors that between the true orbit space and the Borel construction. The discussion will include also a short overview of work in progress on the KO-theory of these spaces.

The material is based on joint work with Matthias Franz and Nigel Ray and touches on additional joint work with Martin Bendersky, Fred Cohen and Sam Gitler.

Geometry and Topology

Speaker: Adam Sikora (SUNY/Buffalo)

"The relations between Jones polynomial of knots and the topology of their complements"

Time: 15:30

Room: MC 107

Coffee

Speaker:

"Coffee will be served"

Time: 15:00

Room: MC 109A

Distinguished Lecture

Speaker: Fred Cohen (Rochester)

"On natural subspaces of products, and their applications"

Time: 15:30

Room: MC 108

One basic example is the configuration space of unordered k-tuples of distinct points in a space M. When specialized to the case where M is given by the complex numbers, these spaces can be identified as the space of complex, monic polynomials of degree k which have exactly k distinct roots. Features of these spaces as well as their connections to knots, links, and homotopy groups will be addressed.

Coffee

Speaker:

"Coffee will be served"

Time: 15:00

Room: MC 109A

Distinguished Lecture

Speaker: Fred Cohen (Rochester)

"Generalized moment-angle complexes"

Time: 15:30

Room: MC 108

A second subspace of a product is the generalized moment-angle complex with notable cases given by subspaces of products of infinite dimensional complex projective space 'indexed by a finite simplicial complex'. These spaces appearing in work of Goresky-MacPherson, Buchstaber-Panov-Ray, Denham-Suciu, Franz as well as many others encode information ranging from the structure of toric varieties in one guise as well as 'motions of certain types of robotic legs' in another guise.

Further elementary features of these spaces are developed within the context of classical homotopy theory in joint work with A. Bahri, M. Bendersky, and S. Gitler. These elementary properties provide easily accessible explantions for certain properties of moment-angle complexes as well as further, delicate features

Coffee

Speaker:

"Coffee will be served"

Time: 15:00

Room: MC 109A

Distinguished Lecture

Speaker: Fred Cohen (Rochester)

"Spaces of homomorphisms and representations"

Time: 15:30

Room: MC 108

The main direction of this topic is the structure for the space of homomorphisms from a free abelian group to a Lie group as well as associated quotients spaces obtained from the adjoint representation, the associated space of representations. These spaces admit the additional structure of a simplicial space at the heart of work of Jardine. What is the fundamental group or the first homology group of the associated space ? This deceptively elementary question as well as more global information is the subject of this talk based on joint work with A. Adem, E. Torres, and J. Gomez.

Algebra Seminar

Speaker: Guillermo Mantilla (University of Wisconsin)

"Integral trace forms associated to cubic extensions"

Time: 15:30

Room: MC 106

Given a nonzero integer d, we know, by Hermite's Theorem, that there exist only finitely many cubic number fields of discriminant d. A natural question is, how to refine the discriminant in such way that we can tell, when two of these fields are isomorphic. Here we consider the binary quadratic form q_K: Tr_{K/ mathbb{Q}}(x2)|_{O^0_K}, and we show that if d is a positive fundamental discriminant, then the isomorphism class of q_K, as a quadratic form over Z², gives such a refinement.

Geometry and Topology

Speaker: Alexandra Pettet (Michigan)

"Dynamics of Out(F): twisting out fully irreducible automorphisms"

Time: 15:30

Room: MC 107

The outer automorphism group Out(F) of a free group F of finite rank shares many properties with the mapping class group of a surface, however the techniques for studying these groups are generally quite different. Analogues of the pseudo-Anosov elements of the mapping class group are the so-called fully irreducible automorphisms, which exhibit north-south dynamics on Culler-Vogtmann's Outer Space. We will explain a method for constructing these automorphisms and suggest why this construction should be useful. This is joint work with Matt Clay (University of Oklahoma).

Geometry and Topology

Speaker: Jean-Francois Lafont (Ohio State University)

"Hyperbolic groups act on high-dimensional spheres"

Time: 12:30

Room: MC 107

I'll show that every torsion-free delta-hyperbolic group supports a non-trivial topological action on a high-dimensional ball. Aside from a bad limit set in the boundary of the ball, this action is well-behaved. This was joint work with Tom Farrell.

Analysis Seminar

Speaker: Eduardo Gonzalez (University of Massachusetts Boston)

"Compactness of the moduli space of symplectic vortices and gauged- Gromov Witten invariants"

Time: 15:30

Room: MC 108

Let X be a symplectic manifold and G a Lie group acting in a Hamiltonian fashion with a moment map f. Let P denote a principal G-bundle over a surface with area form V. A pair (A,u) of a connection A on P and a section u of the associated bundle P(X):=P imes_G X is a gauged pseudo-holomorphic map if it satisfies the A-twisted Cauchy-Rieman equation. The space of vortices is the quotient of gauged pseudo-holomorphic map by Aut(P). We will give a brief introduction to moduli spaces of curves arising from Gromov-Witten theory, including some Fredholm theory. We will show that under some good choices this moduli space can be compactified and get an orbifold structure. This is work in progress with A. Ott, C. Woodward and F. Ziltiner.

Coffee

Speaker: (Western)

"Coffee will be served"

Time: 15:00

Room: MC 109A

Colloquium

Speaker: Eduardo Gonzales (University of Massachusetts Boston)

"Symplectic Vortices and Equivariant Gromov-Witten Theory"

Time: 15:30

Room: MC 108

Gromov-Witten theory for projective varieties has been the subject of intense research since its intruduction. Many important results in the area (eg. Givental's mirror theorem) were proven using equivariant localization techniques from topology, using natural group actions. For a symplectic manifold X, GW invariants are defined using moduli spaces pseudoholomophic maps u:Σ o X from a Riemann surface Σ to X. Suppose that a compact Lie group is acting on X in a Hamiltonian way. After an introduction to the general theory, I will introduce the "space of vortices" which are pairs (A,u) of a connection over a principal bundle P, and a section u:Σ satisfying certain "gauged" equations. Using these spaces one can define gauged Gromov-Witten invariants, which are an equivariant version of Gromov-Witten invariants. This theory depends on a choice of area form on Σ as well as Σ. I will describe joint work with C. Woodward regarding the dependency on the area form, as well as joint work with Ott, Ziltener and Woodward on punctured surfaces. I will also give relations with other equivariant theories and I will discuss a conjecture related to the Gromov-Witten theory of symplectic quotients.

Colloquium

Speaker: Jean-Francois Lafont (Ohio State University)

"Introduction to simplicial volume"

Time: 15:30

Room: MC 108

Simplicial volume is an invariant of closed manifolds that measures how efficiently the manifold can be "triangulated over the reals". I will discuss various geometric and topological consequences that result from positivity of the simplicial volume. Finally, I will end by giving an indication of the proof (joint with Ben Schmidt) that locally symmetric spaces of non-compact type have positive simplicial volume.

Algebra Seminar

Speaker: Richard Kane (Western)

"Cobordism and BP theory plus K-theory continued"

Time: 15:30

Room: MC 106

Generalized cohomology theories have been studied continuously since the late 1950's. My lecture of several weeks ago was devoted to a brief survey of topological K-theory, the first generalized cohomology theory to be developed and utilized. The present lecture will begin by discussing another family of generalized cohomology theories which have also been extensively studied - cobordism theory and its associated theories, notably Brown-Peterson theory and Morava K-theory. This work includes major contributions by Thom, Novikov and Quillen. A major theme of this discussion will be the algebra of cohomology operations associated with each of these theories.

As we will see, topological K-theory can also be fitted into the framework of this family of cohomology theories. In addition we will return to K-theory and discuss both operations and Adams operations. However these operations will turn out out be rather distinct from the ones considered for cobordism and BP theory.