Seminars & Colloquia

Thomas Hull (Merrimack College) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"Configuration spaces for flat vertex folds"

Abstract: Given a flat origami crease pattern, it is very much an open question to determine the number of ways it can fold flat, where each is distinguished by gaving a different mountain-valley assignment. Recursive formulas exist for counting the number of ways a single, flat-foldable vertex in a crease pattern can fold flat, but that's pretty much where our knowledge on the matter ends. In this talk we will delve more deeply into the single vertex case by trying to characterize flat vertex folds of a given degree into classes whose membership is determined by the number of ways they can fold flat.

    This leads to descriptions of the configuration spaces for flat vertex folds of a given degree. In general, the configuration space of a flat vertex fold of degree 2n will be a 2n-1-dimensional polytope with collections of lower-dimensional subspaces representing the various classes. We will describe the n=2 and 3 cases and eventually see that the polytope for the degree 2n case is the sum of two (n-1)-dimensional simplices.

Christian Zickert (Columbia University) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"The volume and Chern-Simons invariant of a hyperbolic manifold"

Abstract: Let M be a hyperbolic manifold. If M is complete and of finite volume, it follows from Mostow rigidity that the volume is a topological invariant of M.

The Chern-Simons invariant is defined by integrating a certain 3-form over a section of the orthonormal frame bundle. It can be regarded as the imaginary part of a complex volume with the real part being the usual volume. In this talk we shall discuss methods of computing the complex volume from purely topological descriptions of M. As a result, we obtain a very efficient algorithm for computing the Chern-Simons invariant.

Ken Bromberg (University of Utah) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"The topologyof deformation spaces of hyperbolic 3-manifolds"

Abstract: An infinite volume hyperbolic 3-manifold has many distinct hyperbolic structures. The deformation space of all such hyperbolic structures is a fractal object that has many similarities with the more well-known Mandelbrot set. We will survey what is known about the topology of these deformation spaces and give some indication of the techniques that are used in their study.

William Jaco (Oklahoma State University) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"Decision problems, algorithms and computational complexity"

Abstract: We will present an exposition on three decision problems coming from 3-manifold topology. The selected problems are: "Recognition of the 3-sphere", "the knot triviality problem", and the "word problem". These three problems provide a nice overview of the theory and methods used in approaching decision problems and algorithms in 3-manifold topology.

"Lattice Polygons and the Number 12"

Abstract:
Let P be a convex lattice polygon with one interior lattice point. The number of lattice points on the boundary of P plus the corresponding number for the dual polygon (to be defined) always equals 12. We will sketch four different proofs of this result, and explain why this 12 is the same as the 12 that arises in various other branches of mathematics.

Mike King (Boston College) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"Cluster Algebras and Triangulations"

 Abstract:
Cluster algebras, introduced by Fomin and Zelevinsky in 2000, carry a rich combinatorial structure. In many settings, this structure is closely related to the combinatorics of triangulations of a polygon or of a surface. I will review the definition of a cluster algebra and survey some of these connections to triangulations. If time permits, I will show how the configuration space of n vectors in the plane carries a cluster algebra structure arising from triangulations of the n-gon.

Slava Krushkal (University of Virginia) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"An Approach to Chromatic Polynomial via Quantum Topology"

Abstract:
In the 1960s Tutte discovered several remarkable properties of the chromatic polynomial related to the golden ratio. I will explain how these results may be proved and generalized using quantum topology and algebras underlying them. The talk will not assume any special background in topology or combinatorics.

Richard Schwartz (Brown University) will speak in 309 Carney Hall.  Refreshments at 4:00 p.m, followed by a talk at 4:15.

"Outer Billiards on Kites"

Abstract: "Outer billiards is a simple dynamical system defined relative to a convex set in the plane. B.H. Neumann introduced this system in the 1950s and J. Moser popularized it in the 1970s as a toy model for celestial mechanics. All along, one of the central questions has been the following: Does there exist a convex shape for which outer billiards has some unbounded orbits? In my talk, I will give a colorful computer demonstration of outer billiards on polygons and, along the way, explain my recent solution to the problem of unbounded orbits: Outer billiards has unbounded orbits when defined relative to every irrational kite. (A kite is a kite-shaped quadrilateral. It is irrational iff it is not affine equivalent to a lattice polygon.)"