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Series: Geometry Topology Seminar

To each three-component link in the 3-dimensional sphere we associate a characteristic map from the 3-torus to the 2-sphere, and establish a correspondence between the pairwise and Milnor triple linking numbers of the link and the Pontryagin invariants that classify its characteristic map up to homotopy. This can be viewed as a natural extension of the familiar fact that the linking number of a two-component link is the degree of its associated Gauss map from the 2-torus to the 2-sphere.In the case where the pairwise linking numbers are all zero, I will present an integral formula for the triple linking number analogous to the Gauss integral for the pairwise linking numbers. The integrand in this formula is geometrically natural in the sense that it is invariant under orientation-preserving rigid motions of the 3-sphere.

Monday, October 26, 2009 - 13:00 ,
Location: Skiles 255 ,
Chiu-Yen Kao ,
Ohio State University (Department of Mathematics) ,
kao@math.ohio-state.edu ,
Organizer: Sung Ha Kang

The Kadomtsev-Petviashvili (KP) equation is a two-dimensional dispersivewave equation which was proposed to study the stability of one solitonsolution of the KdV equation under the influence of weak transversalperturbations. It is well know that some closed-form solutions can beobtained by function which have a Wronskian determinant form. It is ofinterest to study KP with an arbitrary initial condition and see whetherthe solution converges to any closed-form solution asymptotically. Toreveal the answer to this question both numerically and theoretically, weconsider different types of initial conditions, including one-linesoliton, V-shape wave and cross-shape wave, and investigate the behaviorof solutions asymptotically. We provides a detail description ofclassification on the results. The challenge of numerical approach comes from the unbounded domain andunvanished solutions in the infinity. In order to do numerical computationon the finite domain, boundary conditions need to be imposed carefully.Due to the non-periodic boundary conditions, the standard spectral methodwith Fourier methods involving trigonometric polynomials cannot be used.We proposed a new spectral method with a window technique which will makethe boundary condition periodic and allow the usage of the classicalapproach. We demonstrate the robustness and efficiency of our methodsthrough numerous simulations.

Monday, October 26, 2009 - 10:00 ,
Location: Skiles 255 ,
Shea Vela-Vick ,
Columbia University ,
Organizer: John Etnyre

We will focus on the "toy model" of bordered Floer homology. Loosely speaking, this is bordered Floer homology for grid diagrams of knots. While the toy model unfortunately does not provide us with any knot invariants, it highlights many of the key ideas needed to understand the more general theory.
Note the different time and place!
This is a 1.5 hour talk.

Series: ACO Distinguished Lecture

Preceded with a reception at 4:10pm.

To come to grips with consciousness, I postulate that living entities in
general, and human beings in particular, are mechanisms... marvelous
mechanisms to be sure but not magical ones... just mechanisms. On this
basis, I look to explain some of the paradoxes of consciousness such as
Samuel Johnson's "All theory is against the freedom of the will; all
experience is for it."
I will explain concepts of self-awareness and free will from a mechanistic
view. My explanations make use of computer science and suggest why these
phenomena would exist even in a completely deterministic world. This is
particularly striking for free will.
The impressions of our senses, like the sense of the color blue, the sound
of a tone, etc. are to be expected of a mechanism with enormously many
inputs categorized into similarity classes of sight, sound, etc. Other
phenomena such as the "bite" of pain are works in progress. I show the
direction that my thinking takes and say something about what I've found and
what I'm still looking for. Fortunately, the sciences are discovering a
great deal about the brain, and their discoveries help enormously in guiding
and verifying the results of this work.

Series: Other Talks

The spectral properties of a graph are intimately related to its structure. This can be applied in the study of discrete isoperimetric problems and in the investigation of extremal and algorithmic questions for graphs. I will discuss several recent examples illustrating this theme.

Series: Other Talks

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; price equilibria in markets; optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysis of the evolution of various types of dynamic stochastic models. It is not known whether these problems can be solved in polynomial time. Despite their broad diversity, there are certain common computational principles that underlie different types of equilibria and connect many of these problems to each other. In this talk we will discuss some of these common principles and the corresponding complexity classes that capture them; the effect on the complexity of the underlying computational framework; and the relationship with other open questions in computation.

Series: Other Talks

From time to time a new algorithm comes along that causes a sensation in theoretical computer science or in an area of application because of its resolution of a long-standing open question, its surprising efficiency, its practical usefulness, the novelty of its setting or approach, the elegance of its structure, the subtlety of its analysis or its range of applications. We will give examples of algorithms that qualify for greatness for one or more of these reasons, and discuss how to equip students to appreciate them and understand their strengths and weaknesses.

Series: Probability Working Seminar

The talk is based on a 1992 paper by Yakov Sinai. He proves a localization property for random walks in the random potential known as Nechaev's model.

Series: ACO Seminar

The class of random regular graphs has been the focus of extensive study highlighting
the excellent expansion properties of its typical instance. For instance, it is well
known that almost every regular graph of fixed degree is essentially Ramanujan, and
understanding this class of graphs sheds light on the general behavior of expanders.
In this talk we will present several recent results on random regular graphs,
focusing on the interplay between their spectrum and geometry.
We will first discuss the relation between spectral properties and the abrupt
convergence of the simple random walk to equilibrium, derived from precise
asymptotics of the number of paths between vertices. Following the study of the graph
geometry we proceed to its random perturbation via exponential weights on the edges
(first-passage-percolation). We then show how this allows the derivation of various
properties of the classical Erd\H{o}s-R\'enyi random graph near criticality. Finally,
returning to the spectrum of random regular graph, we discuss the question of how
close they really are to being Ramanujan and conclude with related problems involving
random matrices.
Based on joint works with Jian Ding, Jeong Han Kim and Yuval Peres, with Allan Sly
and with Benny Sudakov and Van Vu.

Friday, October 23, 2009 - 15:00 ,
Location: Skiles 269 ,
Amey Kaloti ,
Georgia Tech ,
Organizer:

This is a 2 hour talk.

Abstract: Heegaard floer homology is an invariant of closed 3 manifolds defined by Peter
Ozsvath and Zoltan Szabo. It has proven to be a very strong invariant of 3 manifolds with
connections to contact topology. In these talks we will try to define the Heegaard Floer
homology without assuming much background in low dimensional topology. One more goal is
to present the combinatorial description for this theory.