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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)

Series: School of Mathematics Colloquium

Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. In this talk, we focus on understanding how an RNA viral genome can fold into the dodecahedral cage known from experimental data. Using strings and trees as a combinatorial model of RNA folding, we give mathematical results which yield insight into RNA structure formation and suggest new directions in viral capsid assembly. We also illustrate how the interaction between discrete mathematics and molecular biology motivates new combinatorial theorems as well as advancing biomedical applications.

Series: Analysis Seminar

It is well known that a needle thrown at random has zero
probability of intersecting any given irregular planar set of finite
1-dimensional Hausdorff measure. Sharp quantitative estimates for fine open
coverings of such sets are still not known, even for such sets as the
Sierpinski gasket and the 4-corner Cantor set (with self-similarities 1/4
and 1/3). In 2008, Nazarov, Peres, and Volberg provided the sharpest known
upper bound for the 4-corner Cantor set. Volberg and I have recently used
the same ideas to get a similar estimate for the Sierpinski gasket. Namely,
the probability that Buffon's needle will land in a 3^{-n}-neighborhood of
the Sierpinski gasket is no more than C_p/n^p, where p is any small enough
positive number.

Series: Other Talks

In the 60s, Grothendieck had the remarkable idea of introducing a new kind of topology where open coverings of X are no longer collections of subsets of X, but rather certain maps from other spaces to X. I will give some examples to show why this is reasonable and what one can do with it.

Series: Research Horizons Seminar

The Research Horizons seminar this week will be a panel discussion on
the academic job market for mathematicians. The discussion will begin
with an overview by Doug Ulmer of the hiring process, with a focus on
the case of research-oriented universities. The panel will then take
questions from the audience. Professor Wick was hired last year at
Tech, so has recently been on the students' side of the process.
Professor Harrell has been involved with hiring at Tech for many years
and can provide a perspective on the university side of the process.

Series: Mathematical Biology Seminar

Host: Meghan Duffy (School of Biology, Georgia Tech)

Why do parasites cause disease? Theory has shown that natural selection could select for virulent parasites if virulence is correlated with between-host parasite transmission. Because ecological conditions may affect virulence and transmission, theory further predicts that adaptive levels of virulence depend on the specific environment in which hosts and parasites interact. To test these predictions in a natural system, we study monarch butterflies (Danaus plexippus) and their protozoan parasite (Ophryocystis elektroscirrha). Our studies have shown that more virulent parasites obtain greater between-host transmission, and that parasites with intermediate levels of virulence obtain highest fitness. The average virulence of wild parasite isolates falls closely to this optimum level, providing additional support that virulence can evolve as a consequence of natural selection operating on parasite transmission. Our studies have also shown that parasites from geographically separated populations differ in their virulence, suggesting that population-specific ecological factors shape adaptive levels of virulence. One important ecological factor is the monarch larval host plants in the milkweed family. Monarch populations differ in the milkweed species they harbor, and experiments have shown that milkweeds can alter parasite virulence. Our running hypothesis is that plant availability shapes adaptive levels of parasite virulence in natural monarch populations. Testing this hypothesis will improve our understanding of why some parasites are more harmful than others, and will help with predicting the consequences of human actions on the evolution of disease.