Seminars and Colloquia Schedule

Applications of the knot Floer complex to concordance

Series
Geometry Topology Seminar
Time
Monday, November 14, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jen Hom – Columbia University
We will use a new concordance invariant, epsilon, associated to the knot Floer complex, to define a smooth concordance homomorphism. Applications include a new infinite family of smoothly independent topologically slice knots, bounds on the concordance genus, and information about tau of satellites. We will also discuss various algebraic properties of this construction.

Domain decomposition methods for large problems of elasticity

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 14, 2011 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Olof Widlund – Courant Institute,New York University, Mathematics and Computer Science
The domain decomposition methods considered are preconditioned conjugate gradient methods designed for the very large algebraic systems of equations which often arise in finite element practice. They are designed for massively parallel computer systems and the preconditioners are built from solvers on the substructures into whichthe domain of the given problem is partitioned. In addition, to obtain scalability, there must be a coarse problem, with a small number of degrees of freedom for each substructure. The design of this coarse problem is crucial for obtaining rapidly convergent iterations and poses the most interesting challenge in the analysis.Our work will be illustrated by overlapping Schwarz methods for almost incompressible elasticity approximated by mixed finite element and mixed spectral element methods. These algorithms is now used extensively at the SANDIA, Albuquerque laboratories and were developed in close collaboration with Dr. Clark R. Dohrmann. These results illustrate two roles of the coarse component of the preconditioner.Currently, these algorithms are being actively developed for problems posed in H(curl) and H(div). This work requires the development of new coarse spaces. We will also comment on recent work on extending domain decomposition theory to subdomains with quite irregular boundaries.  This work is relevant because of the use of mesh partitioners in the decomposition of large finite element matrices. 

The control polyhedron of a rational Bezier surface

Series
Algebra Seminar
Time
Monday, November 14, 2011 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Luis Garcia – Sam Houston State University
Geometric modeling builds computer models for industrial design and manufacture from basic units, called patches, such as, Bézier curves and surfaces. The control polygon of a Bézier curve is well-defined and has geometric significance—there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. In this talk, I will present a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary. While our primary interest is to explain the meaning of control nets for the classical rational Bezier patches, we work in the generality of Krasauskas’ toric Bezier patches. Toric Bezier patches are multi-sided parametric patches based on the geometry of toric varieties and depend on a polytope and some weights. Our results rely upon a construction in computational algebraic geometry called a toric degeneration.

Randomized Approximation Schemes for Cuts and Flows in Capacitated Graphs

Series
High-Dimensional Phenomena in Statistics and Machine Learning Seminar
Time
Tuesday, November 15, 2011 - 16:00 for 1.5 hours (actually 80 minutes)
Location
skyles 006
Speaker
Yingyu Liang – School of Compter Science, Georgia tech
We will talk about how to approximate an arbitrary graph by a sparse graph with respect to cuts and flows, using random sampling techniques. More specifically, we will describe a near-linear-time randomized combinatorial construction that transforms any graph on n vertices into an O(n log n)-edge graph on the same vertices whose cuts have approximately the same value as the original graph's. The new graph can be used to accelerate cut and flow algorithms, leading to approximate solution on the original graph. The construction algorithms of the sparse graph are based on a general theorem analyzing the concentration of cut values near their expectation in random graphs.

CANCELLED (Multi-scale Model of CRISPR-induced Coevolutionary Dynamics -- Diversification at the Interface of Lamarck and Darwin)

Series
Mathematical Biology Seminar
Time
Wednesday, November 16, 2011 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Lauren Childs – Biology, Georgia Tech
The CRISPR (Clustered Regularly Interspaced Short Palindromic Repeats) system is a recently discovered immune defense in bacteria and archaea (hosts) that functions via directed incorporation of viral DNA intohost genomes. Here, we introduce a multi-scale model of dynamic coevolution between hosts and viruses in an ecological context that incorporates CRISPR immunity principles. We analyze the model to test whether and how CRISPR immunity induces host and viral diversification and maintenance of coexisting strains. We show that hosts and viruses coevolve to form highly diverse communities through punctuated replacement of extant strains. The populations have very low similarity over long time scales. However overshort time scales, we observe evolutionary dynamics consistent with incomplete selective sweeps of novel strains, recurrence of previously rare strains, and sweeps of coalitions of dominant host strains with identical phenotypes but different genotypes. Our explicit eco-evolutionary model of CRISPR immunity can help guide efforts to understand the drivers of diversity seen in microbial communities where CRISPR systems are active.

Discrete Schroedinger Problem.

Series
Research Horizons Seminar
Time
Wednesday, November 16, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005.
Speaker
Manwah Lilian Wong – Georgia Tech
We will discuss the discrete Schroedinger problem on the integer line and on graphs. Starting from the definition of the discrete Laplacian on the integer line, I will explain why the problem is interesting, how the discrete case relates to the continuous case, and what the open problems are. Recent results by the speaker (joint with Evans Harrell) will be presented.The talk will be accessible to anyone who knows arithmetic and matrix multiplications.

Empirical likelihood and Extremes

Series
Dissertation Defense
Time
Wednesday, November 16, 2011 - 15:00 for 1.5 hours (actually 80 minutes)
Location
Skiles 171
Speaker
Yun Gong – School of Mathematics, Georgia Tech

Advisor: Liang Peng

In 1988, Owen introduced empirical likelihood as a nonparametric method for constructing confidence intervals and regions. It is well known that empirical likelihood has several attractive advantages comparing to its competitors such as bootstrap: determining the shape of confidence regions automatically; straightforwardly incorporating side information expressed through constraints; being Bartlett correctable. In this talk, I will discuss some extensions of the empirical likelihood method to several interesting and important statistical inference situations including: the smoothed jackknife empirical likelihood method for the receiver operating characteristic (ROC) curve, the smoothed empirical likelihood method for the conditional Value-at-Risk with the volatility model being an ARCH/GARCH model and a nonparametric regression respectively. Then, I will propose a method for testing nested stochastic models with discrete and dependent observations.

Circuits in medial graphs and bipartite partial duals

Series
Graph Theory Seminar
Time
Thursday, November 17, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Iain Moffatt – University of South Alabama
A classical result in graph theory states that, if G is a plane graph, then G is Eulerian if and only if its dual, G*, is bipartite. I will talk about an extension of this well-known result to partial duality. (Where, loosely speaking, a partial dual of an embedded graph G is a graph obtained by forming the dual with respect to only a subset of edges of G.) I will extend the above classical connection between bipartite and Eulerian plane graphs, by providing a necessary and sufficient condition for the partial dual of a plane graph to be Eulerian or bipartite. I will then go on to describe how the bipartite partial duals of a plane graph G are completely characterized by circuits in its medial graph G_m. This is joint work with Stephen Huggett.

Planar and Hamiltonian Cover Graphs

Series
Dissertation Defense
Time
Friday, November 18, 2011 - 13:00 for 2 hours
Location
Skiles 005
Speaker
Noah Streib – School of Mathematics, Georgia Tech
This dissertation has two principal components: the dimension of posets with planar cover graphs, and the cartesian product of posets whose cover graphs have hamiltonian cycles that parse into symmetric chains. Posets of height two can have arbitrarily large dimension. In 1981, Kelly provided an infinite sequence of planar posets that shows that the dimension of planar posets can also be arbitrarily large. However, the height of the posets in this sequence increases with the dimension. In 2009, Felsner, Li, and Trotter conjectured that for each integer h \geq 2, there exists a least positive integer c_h so that if P is a poset having a planar cover graph (hence P is a planar poset as well) and the height of P is h, then the dimension of P is at most c_h. In the first principal component of this dissertation we prove this conjecture. We also give the best known lower bound for c_h, noting that this lower bound is far from the upper bound. In the second principal component, we consider posets with the Hamiltonian Cycle--Symmetric Chain Partition (HC-SCP) property. A poset of width w has this property if its cover graph has a Hamiltonian cycle which parses into w symmetric chains. This definition is motivated by a proof of Sperner's Theorem that uses symmetric chains, and was intended as a possible method of attack on the Middle Two Levels Conjecture. We show that the subset lattices have the HC-SCP property by showing that the class of posets with the strong HC-SCP property, a slight strengthening of the HC-SCP property, is closed under cartesian product with a two-element chain. Furthermore, we show that the cartesian product of any two posets from this class has the HC-SCP property.

Symmetric chain decomposition for cyclic quotients of Boolean algebras and relation to cyclic crystals

Series
Combinatorics Seminar
Time
Friday, November 18, 2011 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Patricia Hersh – North Carolina State University, Raleigh, NC
The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of Kashiwara's sl_2 lowering operator in the theory of crystal bases. The talk will include a survey of related past work on symmetric chain decomposition and unimodality by Greene-Kleitman, Griggs-Killian-Savage, Proctor, Stanley and others as well as a discussion of open questions that still remain. This is joint work with Anne Schilling.