Seminars and Colloquia by Series

Hyperbolic manifolds, algebraic K-theory and the extended Bloch group

Series
Geometry Topology Seminar
Time
Monday, September 14, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Christian ZickertUC Berkeley
A closed hyperbolic 3-manifold $M$ determines a fundamental classin the algebraic K-group $K_3^{ind}(C)$. There is a regulator map$K_3^{ind}(C)\to C/4\Pi^2Z$, which evaluated on the fundamental classrecovers the volume and Chern-Simons invariant of $M$. The definition of theK-groups are very abstract, and one is interested in more concrete models.The extended Bloch is such a model. It is isomorphic to $K_3^{ind}(C)$ andhas several interesting properties: Elements are easy to produce; thefundamental class of a hyperbolic manifold can be constructed explicitly;the regulator is given explicitly in terms of a polylogarithm.

Confoliations and contact structures on higher dimensions

Series
Geometry Topology Seminar
Time
Monday, September 14, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Dishant M. PancholiInternational Centre for Theoretical Physics, Trieste, Italy
After reviewing a few techniques from the theory of confoliation in dimension three we will discuss some generalizations and certain obstructions in extending these techniques to higher dimensions. We also will try to discuss a few questions regarding higher dimensional confoliations.

Cube knots and a homology theory from cube diagrams

Series
Geometry Topology Seminar
Time
Monday, April 20, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Scott BaldridgeLSU
In this talk we will introduce the notion of a cube diagram---a surprisingly subtle, extremely powerful new way to represent a knot or link. One of the motivations for creating cube diagrams was to develop a 3-dimensional "Reidemeister's theorem''. Recall that many knot invariants can be easily be proven by showing that they are invariant under the three Reidemeister moves. On the other hand, simple, easy-to-check 3-dimensional moves (like triangle moves) are generally ineffective for defining and proving knot invariants: such moves are simply too flexible and/or uncontrollable to check whether a quantity is a knot invariant or not. Cube diagrams are our attempt to "split the difference" between the flexibility of ambient isotopy of R^3 and specific, controllable moves in a knot projection. The main goal in defining cube diagrams was to develop a data structure that describes an embedding of a knot in R^3 such that (1) every link is represented by a cube diagram, (2) the data structure is rigid enough to easily define invariants, yet (3) a limited number of 5 moves are all that are necessary to transform one cube diagram of a link into any other cube diagram of the same link. As an example of the usefulness of cube diagrams we present a homology theory constructed from cube diagrams and show that it is equivalent to knot Floer homology, one of the most powerful known knot invariants.

The Sutured Embedded Contact Homology of S^1\times D^2

Series
Geometry Topology Seminar
Time
Monday, April 13, 2009 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Roman GolovkoUSC
We will define the sutured version of embedded contact homology for sutured contact 3-manifolds. After that, we will show that the sutured version of embedded contact homology of S^1\times D^2, equipped with 2n sutures of integral or infinite slope on the boundary, coincides with the sutured Floer homology.

Hypergeometric functions - the GKZ-perspective

Series
Geometry Topology Seminar
Time
Monday, April 13, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Uli WaltherPurdue University
Starting with some classical hypergeometric functions, we explain how to derive their classical univariate differential equations. A severe change of coordinates transforms this ODE into a system of PDE's that has nice geometric aspects. This type of system, called A-hypergeometric, was introduced by Gelfand, Graev, Kapranov and Zelevinsky in about 1985. We explain some basic questions regarding these systems. These are addressed through homology, combinatorics, and toric geometry.

Density of isoperimetric spectra

Series
Geometry Topology Seminar
Time
Monday, April 6, 2009 - 16:00 for 1 hour (actually 50 minutes)
Location
Emory, W306 MSC (Math and Science Center)
Speaker
Noel BradyUniversity of Oklahoma

Please Note: Joint meeting at Emory

A k--dimensional Dehn function of a group gives bounds on the volumes of (k+1)-balls which fill k--spheres in a geometric model for the group. For example, the 1-dimensional Dehn function of the group Z^2 is quadratic. This corresponds to the fact that loops in the euclidean plane R^2 (which is a geometric model for Z^2) have quadratic area disk fillings. In this talk we will consider the countable sets IP^{(k)} of numbers a for which x^a is a k-dimensional Dehn function of some group. The situation k \geq 2 is very different from the case k=1.

Contact geometry, open books and monodromy

Series
Geometry Topology Seminar
Time
Monday, April 6, 2009 - 15:00 for 1 hour (actually 50 minutes)
Location
Emory, W306 MSC (Math and Science Center)
Speaker
John EtnyreSchool of Mathematics, Georgia Tech

Please Note: Joint meeting at Emory

Recall that an open book decomposition of a 3-manifold M is a link L in M whose complement fibers over the circle with fiber a Seifert surface for L. Giroux's correspondence relates open book decompositions of a manifold M to contact structures on M. This correspondence has been fundamental to our understanding of contact geometry. An intriguing question raised by this correspondence is how geometric properties of a contact structure are reflected in the monodromy map describing the open book decomposition. In this talk I will show that there are several interesting monoids in the mapping class group that are related to various properties of a contact structure (like being Stein fillable, weakly fillable, . . .). I will also show that there are open book decompositions of Stein fillable contact structures whose monodromy cannot be factored as a product of positive Dehn twists. This is joint work with Jeremy Van Horn-Morris and Ken Baker.

Hyperbolic volume and the complexity of Heegaard splittings of 3-manifolds

Series
Geometry Topology Seminar
Time
Monday, March 30, 2009 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 269
Speaker
Yo'av Rieck University of Arkansas
Let M be a hyperbolic 3-manifold, that is, a 3-manifold admitting a complete, finite volume Riemannian metric of constant section curvature -1. Let S be a Heegaard surface in M, that is, M cut open along S consists of two handlebodies. Our goal is to prove that is the volume of M (denoted Vol(M)) if small than S is simple. To that end we define two complexities for Heegaard surfaces. The first is the genus of the surface (denoted g(S)) and the second is the distance of the surface, as defined by Hempel (denoted d(S)). We prove that there exists a constant K>0 so that for a generic manifold M, if g(S) \geq 76KVol(M) + 26, then d(S) \leq 2. Thus we see that for a generic manifold of small volume, either the genus of S is small or its distance is at most two. The term generic will be explained in the talk.

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