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Series: Geometry Topology Seminar

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.

Monday, April 20, 2009 - 13:00 ,
Location: Skiles 255 ,
Tiancheng Ouyang ,
Brigham Young ,
Organizer: Chongchun Zeng

In this talk, I will show many interesting orbits in 2D and 3D of the N-body problem. Some of them do not have symmetrical property nor with equal masses. Some of them with collision singularity. The methods of our numerical optimization lead to search the initial conditions and properties of preassigned orbits. The variational methods will be used for the prove of the existence.

Series: Combinatorics Seminar

Let G be a graph and K be a field. We associate to G a projective toric variety X_G over K, the cut variety of the graph G. The cut ideal I_G of the graph G is the ideal defining the cut variety. In this talk, we show that, if G is a subgraph of a subdivision of a book or an outerplanar graph, then the minimal generators are quadrics. Furthermore we describe the generators of the cut ideal of a subdivision of a book.

Friday, April 17, 2009 - 15:00 ,
Location: Skiles 269 ,
Thang Le ,
School of Mathematics, Georgia Tech ,
Organizer: John Etnyre

These are two hour lectures.

We will develop general theory of quantum invariants based on sl_2 (the simplest Lie algebra): The Jones polynomials, the colored Jones polynomials, quantum sl_2 groups, operator invariants of tangles, and relations with the Alexander polynomial and the A-polynomials. Optional: Finite type invariants and the Kontsevich integral.

Friday, April 17, 2009 - 13:00 ,
Location: Skiles 255 ,
Gilad Lerman ,
University of Minnesota ,
Organizer: Sung Ha Kang

Note special day.

We propose a fast multi-way spectral clustering algorithm for multi-manifold data modeling, i.e., modeling data by mixtures of manifolds (possibly intersecting). We describe the supporting theory as well as the practical choices guided by it. We first develop the case of hybrid linear modeling, i.e., when the underlying manifolds are affine subspaces in a Euclidean space, and then we extend this setting to more general manifolds. We exemplify the practical use of the algorithm by demonstrating its successful application to problems of motion segmentation.

Series: School of Mathematics Colloquium

Archimedes principle may be used to predict if and how certain solid objects float in a liquid bath. The principle, however, neglects to consider capillary forces which can sometimes play an important role. We describe a recent generalization of the principle and how the standard textbook presentation of Archimedes' work may have played a role in delaying the discovery of such generalizations to this late date.

Series: Stochastics Seminar

In binary classification problems, the goal is to estimate a function g*:S -> {-1,1} minimizing the generalization error (or the risk)
L(g):=P{(x,y):y \neq g(x)},
where P is a probability distribution in S x {-1,1}. The distribution P is unknown and estimators \hat g of g* are based on a finite number of independent random couples (X_j,Y_j) sampled from P. It is of interest to have upper bounds on the excess risk
{\cal E}(\hat g):=L(\hat g) - L(g_{\ast})
of such estimators that hold with a high probability and that take into account reasonable measures of complexity of classification problems (such as, for instance, VC-dimension). We will discuss several approaches (both old and new) to excess risk bounds in classification, including some recent results on excess risk in so called active learning.

Series: ACO Student Seminar

Grotzsch's Theorem states that every triangle-free planar graph is
3-colorable.
Thomassen conjectured that every triangle-free planar graph has
exponentially many distinct 3-colorings. He proved that it has at least
2^{n^{1/12}/20000} distinct 3-colorings where n is the number of vertices.
We show that it has at least 2^{\sqrt{n/600}} distinct 3-colorings.
Joint work with Arash Asadi and Robin Thomas.

Series: Research Horizons Seminar

This talk will focus on mathematical approaches using PDE and variational models for image processing. I will discuss general problems arising from image reconstructions and segmentation, starting from Total Variation minimization (TV) model and Mumford-Shah segmentation model, and present new models from various developments. Two main topics will be on variational approaches to image reconstruction and multi-phase segmentation. Many challenges and various problems will be presented with some numerical results.

Wednesday, April 15, 2009 - 11:00 ,
Location: Skiles 255 ,
Igor Belykh ,
University of Georgia ,
Organizer: