Seminar Calendar
for Graduate Student Topology events the next 12 months of Monday, August 1, 2005.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery. 
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Wednesday, August 31, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, August 31, 2005
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Submitted by bjwalker.
TBA (UIUC Math)
To Be Announced

Wednesday, September 7, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, September 7, 2005
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Submitted by bjwalker.
Shivi Bansal (UIUC)
Loop Spaces and Loop Groups
Abstract: We will begin with a review of last week's talk by Rekha. I will expand our discussion on Loop Spaces and we will try to explore how these definitions in Topology parallel the ones for Loop Groups in other areas of Math. The style will be very informal like last time, and the less you know about all this the more welcome you'll be.

Wednesday, September 14, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, September 14, 2005
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Submitted by bjwalker.
Christopher Lee (UIUC)
Toric Integrability and Classical Topological Invariants
Abstract: The study of integrable Hamiltonian systems has its roots in classical mechanics. We discuss a special type of integrable system based on the geodesic flow of a Riemannian metric on a compact manifold. The most fundamental topological obstructions to the existence of such systems can be formulated in terms of Betti numbers. While not necessary, some familiarity with the language of smooth manifold theory and differential forms may be beneficial. Additionally, we will ``borrow'' results from homotopy theory and algebraic geometry with wild abandon. I intend to make this talk as informal and introductory as possible.

Wednesday, September 21, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, September 21, 2005
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Submitted by bjwalker.
Rosona Eldred (UIUC Math)
k-Chainlets
Abstract: A k-chainlet is an element of a normed vector space of equivalence classes of formal sums of k-cells. Chainlets include manifolds, rough domains (e.g. fractals), soap films, foliations, and Euclidean space. Taking limits allow for a discrete exterior calculus to be developed at a point. We will present some of the chainlet geometry and calculus.

Wednesday, September 28, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, September 28, 2005
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Submitted by bjwalker.
David Lipsky (UIUC Math)
Cobordism and Pontryagin-Thom Construction
Abstract: Two closed manifolds M and N are said to be cobordant if there exists a compact manifold W whose boundary is diffeomorphic to the disjoint union of M and N. This is a surprisingly rich invariant, with close ties to homotopy theory. I will present a version of the Pontryagin-Thom construction, which identifies cobordism with the homotopy groups of a certain spectrum. Time permitting, I will also outline a proof that every mod-2 singular homology class in a space can be realized as the continuous image of a closed manifold.

Wednesday, October 5, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, October 5, 2005
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Submitted by bjwalker.
Hsian-Yang Chen (UIUC Math)
Hopf Invariant Problem
Abstract: Hopf invariant problem is seeking the existence of certain elements in homotopy group of spheres. We will first discuss its algebraic and geometric equivalence. Then we will present a K-therotic proof.

Wednesday, October 12, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, October 12, 2005
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Submitted by bjwalker.
Marc Harper (UIUC Math)
Introduction to Model Categories
Abstract: This is an expository talk introducing the notions of model category and homotopy category. We will discuss applications and examples, such as the category of chain complexes of modules over a ring, and homotopy theory in this context.

Wednesday, October 19, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, October 19, 2005
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Submitted by bjwalker.
Barry Walker (UIUC Math)
Complex Oriented Cohomology Theories
Abstract: An interesting class of cohomology theories to study are complex oriented theories. These theories have Thom classes, Chern classes, and a formal group law (of dimension one) attached to them. We will give the basic definitions of the theory and discuss how these invariants arise. We'll try to give some helpful examples along the way in an attempt to keep the abstraction from getting out of control. For those interested, geometry will show up via finite dimensional complex vector bundles over connected spaces and how such things are classified.

Wednesday, October 26, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, October 26, 2005
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Submitted by bjwalker.
Valerie Peterson (UIUC Math)
The Topology of Reconfiguration
Abstract: There are a number of settings in which a dynamic system coordinates local rules in order to effect global changes in the states of the system. Plainly said, the system reconfigures itself. I will give a general definition of a reconfigurable system and several examples of how and where they arise (in robotics, biology and computer science, to name a few places). I will also define an abstract cubical complex - the state complex - that keeps track of local moves and global states, as well as which moves "commute". Time permitting, I also plan to prove that there are only finitely many homotopy classes of time-optimal paths between two points on the state complex of a reconfigurable system, due to the presence of non-positive curvature. CAT(0) geometry and covering space theory are the main tools used in the result, but prior knowledge (of the geometry at least!) will not be assumed. This is joint work with R. Ghrist.

Wednesday, November 2, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, November 2, 2005
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Submitted by bjwalker.
Christopher Lee (UIUC Math)
Contact Toric Manifolds and Cosphere Bundles
Abstract: As was (hastily) outlined in the previous talk in this series, the connection between contact toric manifolds and cosphere bundles of Riemannian manifolds is integral to the study of toric integrable geodesic flows. I will begin by giving a brief overview of contact toric manifolds, including the notion of contact moment maps and the statement of a classification theorem of such manifolds due to E. Lerman. In particular, we will concentrate on a certain class of contact toric manifolds; those of so-called Reeb type. We will try to answer the question: Can a contact toric manifold of Reeb type with dimension greater than 5 arise as a cosphere bundle? While this can be viewed as a continuation of a previous talk, I will strive to make it as accessible as possible.

Wednesday, November 9, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, November 9, 2005
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Submitted by bjwalker.
Shivi Bansal (UIUC Math)
Interesection Homology
Abstract: We will begin with a review of some (co)homology theories like singular, simplicial, sheaf etc and some theorems like Poincare Duality valid for nonsingular spaces. By considering some examples we will see how these theorems fail to hold if the space has singularities. Intersection (co)homology (IC) provides a setting where these theorems are true even for spaces with singularities. We will conclude with a discussion of sheaf theoretic versions of IC and perverse sheaves. The talk should be of interest to survivors of courses like Algebraic Topology or any version of Algebraic Geometry.

Wednesday, November 16, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, November 16, 2005
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Submitted by bjwalker.
Christian Haesemeyer (UIUC Math)
Topology of algebraic varieties
Abstract: Projective algebraic varieties have many special topological properties. In this talk, we will explore some of these properties of complex projective manifolds and present some of their applications (like the classification of the homotopy types of complete interesections of hypersurfaces). Time allowing, we will also look at the topology of varieties defined over subfields of the complex numbers (for example, the homotopy type of their complex points depends on the choice of embedding of the subfield, even though the homology type does not).

Wednesday, November 30, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, November 30, 2005
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Submitted by bjwalker.
(UIUC Math)
No Seminar this week

Wednesday, December 7, 2005

Graduate Student Topology and Geometry Seminar
3:00 pm   in 441 Altgeld Hall,  Wednesday, December 7, 2005
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Submitted by bjwalker.
Nora Ganter (UIUC Math)
An Introduction to Orbifolds and Groupoids
Abstract: Following an introductory paper by Moerdijk, I will explain what an orbifold is.

Thursday, January 26, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, January 26, 2006
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Submitted by crlee.
Shivi Bansal (UIUC Math)
Some Category Theory
Abstract: We will start the semester gently by talking about some general ideas like triangulated categories and their truncations. The goal is to lead up to perverse sheaves and hopefully discuss some applications in topology/geometry.

Thursday, February 2, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, February 2, 2006
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Submitted by crlee.
Shivi Bansal   [email] (UIUC Math)
Some category theory (cont'd)
Abstract: We will finish describing triangulated categories, their truncations and cores. To wrap up, we will see an important example of a core of a triangulated category.

Thursday, February 9, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, February 9, 2006
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Submitted by crlee.
David Lispky (UIUC Math)
Exotic 7-spheres
Abstract: I will present Milnor's paper "On Manifolds Homeomorphic to the 7-Sphere." In this paper, Milnor defines a diffeomorphism invariant "lambda" for certain smooth 7-dimensional manifolds. He then constructs, for each integer k, a smooth manifold M_k homeomorphic to the 7-sphere. Amazingly enough, lambda(M_k) varies with k, which means that homeomorphic smooth manifolds need not be diffeomorphic. In particular, there exist several distinct smooth structures on the 7-sphere. I will cover some background material and the beginning of Milnor's construction this week, and will continue next week if needed.

Thursday, February 16, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, February 16, 2006
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Submitted by crlee.
David Lipsky (UIUC Math)
Exotic 7-spheres (cont'd)

Thursday, February 23, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, February 23, 2006
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Submitted by crlee.
Christopher Lee (UIUC Math)
Distributions, integrability and contact geometry
Abstract: I will begin by discussing some basics in the theory of distributions, starting with the definition and making some remarks on integral/involutive distributions, the Frobenius theorem, and curvature of distributions. Following this, we will examine special distributions: contact structures on manifolds.

Thursday, March 2, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, March 2, 2006
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Submitted by crlee.
Daniel Zaharopol (UIUC Math)
An Introduction to Equivariant Cohomology
Abstract: I will provide a brief introduction to equivariant cohomology, based on the development in Guillemin and Sternberg's Supersymmetry and Equivariant de Rham Theory. Roughly speaking, equivariant cohomology allows one to consider the cohomology of a space relative to a group action (for us, a compact Lie group), even when that action is not free. Some comfort with differential forms and de Rham cohomology will be the only prerequisite.

Thursday, March 9, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, March 9, 2006
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Submitted by crlee.
Joe Montgomery (UIUC Math)
An Introduction to the method of moving frames via curves and surfaces
Abstract: I will give a very brief introduction to E. Cartan's method of moving frames in differential geometry. Topics will include the structure equations for Euclidian space, Gauss' Theorema Egregium, the Gauss map, and the Gauss-Bonnet theorem. If time permits, I will examine some nice relations between complex geometry and mininmal surfaces.

Thursday, March 16, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, March 16, 2006
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Submitted by crlee.
Rosona Eldred (UIUC Math)
A little about the Teichmuller Space of a surface
Abstract: I will present a gentle introduction to foliations, hyperbolic structures and the Teichmuller space of a surface, proving a few results. Very little background is assumed.

Thursday, April 6, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, April 6, 2006
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Submitted by crlee.
Hsian-Yang Chen (UIUC Math)
What is Elliptic Cohomology?
Abstract: Elliptic chomology arises from the complex cobordism theory, whose coefficient ring classifies the formal group laws. Also, every elliptic curve inherits naturally a formal group law. In this talk, I will explain these terminologies, construct elliptic cohomology by connecting these objects, and discuss some variants.

Thursday, April 13, 2006

Graduate Student Topology & Geometry
2:00 pm   in 443 Altgeld Hall,  Thursday, April 13, 2006
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Submitted by crlee.
Christopher Lee   [email] (UIUC Math)
Folded symplectic structures
Abstract: I will introduce the notion of a folded symplectic form on an even dimensional smooth manifold. Essentially, a folded symplectic form is a closed two form whose top power is satisfies a certain transversality condition, making its zero locus a smooth hypersurface of the manifold. We then consider the two-form to be symplectic "away from the hypersurface". Following a formalization of these notions, we will consider examples.