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Series: CDSNS Colloquium

We introduce a change of coordinates allowing to capture in a fixed reference frame the profile of travelling wave solutions for nonlinear parabolic equations. For nonlinearities of bistable type the asymptotic travelling wave profile becomes an equilibrium state for the augmented reaction-diffusion equation. In the new equation, the profile of the asymptotic travelling front and its propagation speed emerge simultaneously as time evolves. Several numerical experiments illustrate the effciency of the method.

Series: Other Talks

We will state Serre's fundamental finiteness and vanishing results for the cohomology
of coherent sheaves on a projective algebraic variety. As an application, we'll prove that the
constant term of the Hilbert Polynomial does not depend on the projective embedding, a fact which
is hard to understand using classical (non-cohomological) methods.

Series: Algebra Seminar

This talk will start with an introduction to the area of numerical algebraic geometry. The homotopy continuation algorithms that it currently utilizes are based on heuristics: in general their results are not certified. Jointly with Carlos Beltran, using recent developments in theoretical complexity analysis of numerical computation, we have implemented a practical homotopy tracking algorithm that provides the status of a mathematical proof to its approximate numerical output.

Series: Geometry Topology Seminar

In 1979 Valiant gave algebraic analogs to algorithmic complexity problem such as $P \not = NP$. His central conjecture concerns the determinantal complexity of the permanents. In my lecture I shall propose geometric and algebraic methods to attack this problem and other lower bound problems based on the elusive functions approach by Raz. In particular I shall give new algorithms to get lower bounds for determinantal complexity of polynomials over $Q$, $R$ and $C$.

Monday, November 23, 2009 - 13:00 ,
Location: Skiles 255 ,
Xiaoming Huo ,
Georgia Tech (School of ISyE) ,
xiaoming@isye.gatech.edu ,
Organizer: Sung Ha Kang

Many algorithms were proposed in the past ten years on utilizing manifold structure for dimension reduction. Interestingly, many algorithms ended up with computing for eigen-subspaces. Applying theorems from matrix perturbation, we study the consistency and rate of convergence of some manifold-based learning algorithm. In particular, we studied local tangent space alignment (Zhang & Zha 2004) and give a worst-case upper bound on its performance. Some conjectures on the rate of convergence are made. It's a joint work with a former student, Andrew Smith.

Series: Combinatorics Seminar

Linkage involves finding a set of internally disjoint paths in a graph with specified endpoints. Given graphs G and H, we say G is H-linked if for every injective mapping f:V(H) -> V(G) we can find a subgraph H' of G which is a subdivision of H, with f(v) being the vertex of H' corresponding to each vertex v of H. We describe two results on H-linkage for small graphs H.

(1) Goddard showed that 4-connected planar triangulations are 4-ordered, or in other words C_4-linked. We strengthen this by showing that 4-connected planar triangulations are (K_4-e)-linked.

(2) Xingxing Yu characterized certain graphs related to P_4-linkage. We use his characterization to show that every 7-connected graph is P_4-linked, and to construct 6-connected graphs that are not P_4-linked.

This is joint work with Michael D. Plummer and Gexin Yu.

Series: SIAM Student Seminar

Let X_1, X_2,...,X_n be a sequence of i.i.d random variables with
values in a finite alphabet {1,...,m}. Let LI_n be the length of the
longest increasing subsequence of X_1,...,X_n. We shall express the
limiting distribution of LI_n as functionals of m and (m-1)-
dimensional Brownian motions as well as the largest eigenvalue of
Gaussian Unitary Ensemble (GUE) matrix. Then I shall illustrate
simulation study of these results

Series: PDE Seminar

One of the challenges in the study of transonic flows is the understanding of
the flow behavior near the sonic state due to the severe degeneracy of the
governing equations. In this talk, I will discuss the well-posedness theory of a
degenerate free boundary problem for a quasilinear second elliptic equation
arising from studying steady subsonic-sonic irrotational compressible flows in a convergent nozzle. The flow speed is sonic at the free boundary where the potential flow equation becomes degenerate. Both existence and uniqueness will be shown and optimal regularity will be obtained. Smooth transonic flows in deLaval nozzles
will also be discussed. This is a joint work with Chunpeng Wang.

Series: Stochastics Seminar

Recently functional data analysis has received considerable attention in
statistics research and a number of successful applications have been reported, but
there has been no results on the inference of the global shape of the mean regression
curve. In this paper, asymptotically simultaneous confidence band is obtained for the
mean trajectory curve based on sparse longitudinal data, using piecewise constant
spline estimation. Simulation experiments corroborate the asymptotic theory.

Series: Graph Theory Seminar

Several interesting models of random partial orders can be described via a
process that builds the partial order one step at a time, at each point
adding a new maximal element. This process therefore generates a linear
extension of the partial order in tandem with the partial order itself. A
natural condition to demand of such processes is that, if we condition on
the occurrence of some finite partial order after a given number of steps,
then each linear extension of that partial order is equally likely. This
condition is called "order-invariance".
The class of order-invariant processes includes processes generating a
random infinite partial order, as well as those that amount to taking a
random linear extension of a fixed infinite poset.
Our goal is to study order-invariant processes in general. In this talk, I
shall focus on some of the combinatorial problems that arise.
(joint work with Malwina Luczak)