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Series: Geometry Topology Seminar

The uniform thickness property (UTP) is a property of knots embeddedin the 3-sphere with the standard contact structure. The UTP was introduced byEtnyre and Honda, and has been useful in studying the Legendrian and transversalclassification of cabled knot types. We show that every iterated torus knotwhich contains at least one negative iteration in its cabling sequence satisfiesthe UTP. We also conjecture a complete UTP classification for iterated torusknots, and fibered knots in general.

Monday, September 21, 2009 - 13:00 ,
Location: Skiles 255 ,
Yuliya Babenko ,
Department of Mathematics and Statistics, Sam Houston State University ,
Organizer: Doron Lubinsky

In this talk we first present the exact asymptotics of the optimal
error in the weighted L_p-norm, 1\leq p \leq \infty, of linear spline
interpolation of an arbitrary bivariate function f \in C^2([0,1]^2). We
further discuss the applications to numerical integration and adaptive
mesh generation for finite element methods, and explore connections
with the problem of approximating the convex bodies by polytopes. In
addition, we provide the generalization to asymmetric norms.
We give a brief review of known results and introduce a series of new
ones. The proofs of these results lead to algorithms for the
construction of asymptotically optimal sequences of triangulations for
linear interpolation.
Moreover, we derive similar results for other classes of splines and
interpolation schemes, in particular for splines over rectangular
partitions.
Last but not least, we also discuss several multivariate
generalizations.

Series: CDSNS Colloquium

Fourier's Law assert that the heat flow through a point in a solid is proportional to the temperature gradient at that point. Fourier himself thought that this law could not be derived from the mechanical properties of the elementary constituents (atoms and electrons, in modern language) of the solid. On the contrary, we now believe that such a derivation is possible and necessary. At the core of this change of opinion is the introduction of probability in the description. We now see the microscopic state of a system as a probability measure on phase space so that evolution becomes a stochastic process. Macroscopic properties are then obtained through averages. I will introduce some of the models used in this research and discuss their relevance for the physical problem and the mathematical results one can obtain.

Friday, September 18, 2009 - 14:00 ,
Location: Skiles 269 ,
John Etnyre ,
Georgia Tech ,
Organizer:

We will discuss how to put a hyperbolic structure on various surface and 3-manifolds. We will being by discussing isometries of hyperbolic space in dimension 2 and 3. Using our understanding of these isometries we will explicitly construct hyperbolic structures on all close surfaces of genus greater than one and a complete finite volume hyperbolic structure on the punctured torus. We will then consider the three dimensional case where we will concentrate on putting hyperbolic structures on knot complements. (Note: this is a 1.5 hr lecture)

Series: Graph Theory Seminar

This is the third session in this series and a special effort will be made to make it self contained ... to the fullest extent possible.With Felsner and Li, we proved that the dimension of the adjacency poset of a graph is bounded as a function of the genus. For planar graphs, we have an upper bound of 8 and for outerplanar graphs, an upper bound of 5. For lower bounds, we have respectively 5 and 4. However, for bipartite planar graphs, we have an upper bound of 4, which is best possible. The proof of this last result uses the Brightwell/Trotter work on the dimension of thevertex/edge/face poset of a planar graph, and led to the following conjecture:For each h, there exists a constant c_h so that if P is a poset of height h and the cover graph of P is planar, then the dimension of P is at most c_h.With Stefan Felsner, we have recently resolved this conjecture in the affirmative. From below, we know from a construction of Kelly that c_h must grow linearly with h.

Series: Other Talks

In these talks we will introduced the basic definitions and examples of presheaves, sheaves and sheaf spaces. We will also explore various constructions and properties of these objects.

Series: Research Horizons Seminar

(joint work with Csaba Biro, Dave Howard, Mitch Keller and Stephen Young. Biro and Young finished their Ph.D.'s at Georgia Tech in 2008. Howard and Keller will graduate in spring 2010)

Motivated by questions in algebra involving what is called "Stanley" depth, the following combinatorial question was posed to us by Herzog: Given a positive integer n, can you partition the family of all non-empty subsets of {1, 2, ..., n} into intervals, all of the form [A, B] where |B| is at least n/2. We answered this question in the affirmative by first embedding it in a stronger result and then finding two elegant proofs. In this talk, which will be entirely self-contained, I will give both proofs. The paper resulting from this research will appear in the Journal of Combinatorial Theory, Series A.

Series: ACO Student Seminar

I will describe a simple scheme for generating a valid inequality for a
stochastic integer programs from a given valid inequality for its
deterministic counterpart. Applications to stochastic lot-sizing problems
will be discussed. This is joint work with Yongpei Guan and George Nemhauser
and is based on the following two papers
(1) Y. Guan, S. Ahmed and G.L. Nemhauser. "Cutting planes for multi-stage
stochastic integer programs," Operations Research, vol.57, pp.287-298, 2009
(2)
Y. Guan, S. Ahmed and G. L. Nemhauser. "Sequential pairing of mixed integer
inequalities," Discrete Optimization, vol.4, pp.21-39, 2007
This is a joint DOS/ACO seminar.

Series: PDE Seminar

Many problems in Geometry, Physics and Biology are described by nonlinear partial differential equations of second order or four order. In this talk I shall mainly address the blow-up phenomenon in a class of fourth order equations from conformal geometry and some Liouville systems from Physics and Ecology. There are some challenging open problems related to these equations and I will report the recent progress on these problems in my joint works with Gilbert Weinstein and Chang-shou Lin.

Series: Geometry Topology Seminar

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.