Seminars and Colloquia by Series

Gluing data in chromatic homotopy theory

Series
Geometry Topology Seminar
Time
Monday, September 14, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Agnes Beaudry – University of Chicago
Understanding the stable homotopy groups of spheres is one of the great challenges of algebraic topology. They form a ring which, despite its simple definition, carries an amazing amount of structure. A famous theorem of Hopkins and Ravenel states that it is filtered by simpler rings called the chromatic layers. This point of view organizes the homotopy groups into periodic families and reveals patterns. There are many structural conjectures about the chromatic filtration. I will talk about one of these conjectures, the \emph{chromatic splitting conjecture}, which concerns the gluing data between the different layers of the chromatic filtration.

Projection on a Polyhedron

Series
Applied and Computational Mathematics Seminar
Time
Monday, September 14, 2015 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Associate Professor Hongchao Zhang – Department of Mathematics and Center for Computational & Technology (CCT) at Louisiana State University –
In this talk, we discuss a very efficient algorithm for projecting a point onto a polyhedron. This algorithm solves the projeciton problem through its dual and fully exploits the sparsity. The SpaRSA (Sparse Reconstruction by Separable Approximation) is used to approximately identify active constraints in the polyhedron, and the Dual Active Set Algorithm (DASA) is used to compute a high precision solution. Some interesting convergence properties and very promising numerical results compared with the state-of-the-art software IPOPT and CPLEX will be discussed in this talk.

Dynamics on valuation spaces and applications to complex dynamics

Series
CDSNS Colloquium
Time
Monday, September 14, 2015 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Willam T. Gignac – Georgia Tech (Math)
Let f be a rational self-map of the complex projective plane. A central problem when analyzing the dynamics of f is to understand the sequence of degrees deg(f^n) of the iterates of f. Knowing the growth rate and structure of this sequence in many cases enables one to construct invariant currents/measures for dynamical system as well as bound its topological entropy. Unfortunately, the structure of this sequence remains mysterious for general rational maps. Over the last ten years, however, an approach to the problem through studying dynamics on spaces of valuations has proved fruitful. In this talk, I aim to discuss the link between dynamics on valuation spaces and problems of degree/order growth in complex dynamics, and discuss some of the positive results that have come from its exploration.

Random Walks on the Symmetric Group, Likelihood Orders, and Involutions

Series
Combinatorics Seminar
Time
Friday, September 11, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Megan Bernstein – Georgia Tech
I will find upper and lower bounds for the mixing time as well as the likelihood order after sufficient time of the following involution walk on the symmetric group. Consider 2n cards on a table. Pair up all the cards. Ignore each pairing with probability $p \geq 1/2$. For any pair not being ignored, pick up the two cards and switch their spots. This walk is generated by involutions with binomially distributed two cycles. The upper bound of $\log_{2/(1+p)}(n)$ will result from spectral analysis using a combination of a series of monotonicity relations on the eigenvalues of the walk and the character polynomial for the representations of the symmetric group. A lower bound of $\log_{1/p}$ differs by a constant factor from the upper bound. This walk was introduced to study a conjecture about a random walk on the unitary group from the information theory of black holes.

Bounds on eigenvalues on riemannian manifolds

Series
CDSNS Colloquium
Time
Friday, September 11, 2015 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yannick Sire – John Hopkins University
I will describe several recent results with N. Nadirashvili where we construct extremal metrics for eigenvalues on riemannian surfaces. This involves the study of a Schrodinger operator. As an application, one gets isoperimetric inequalities on the 2-sphere for the third eigenvalue of the Laplace Beltrami operator.

3-coloring H-minor-free graphs with no large monochromatic components

Series
ACO Student Seminar
Time
Friday, September 11, 2015 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chun-Hung Liu – Princeton University
A graph is a minor of another graph if the former can be obtained from a subgraph of the latter by contracting edges. We prove that for every graph H, if H is not a minor of a graph G, then V(G) can be 3-colored such that the subgraph induced by each color class has no component with size greater than a function of H and the maximum degree of G. This answers a question raised by Esperet and Joret, generalizes their result for 3-coloring V(G) for graphs G embeddable in a fixed surface, and improves a result of Alon, Ding, Oporowski and Vertigan for 4-coloringing V(G) for H-minor free graphs G. As a corollary, we prove that for every positive integer t, if G is a graph with no K_{t+1} minor, then V(G) can be 3t-colored such that the subgraph induced by each color class has no component with size larger than a function of t. This improves a result of Wood for coloring V(G) by 3.5t+2 colors. This work is joint with Sang-il Oum.

Asymptotics for 2D critical first-passage percolation

Series
Stochastics Seminar
Time
Thursday, September 10, 2015 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Xuan Wang – School of Mathematics, Georgia Tech
We consider the first-passage percolation model defined on the square lattice Z^2 with nearest-neighbor edges. The model begins with i.i.d. nonnegative random variables indexed by the edges. Those random variables can be viewed as edge lengths or passage times. Denote by T_n the length (i.e. passage time) of the shortest path from the origin to the boundary of the box [-n,n] \times [-n,n]. We focus on the case when the distribution function of the edge weights satisfies F(0) = 1/2. This is sometimes known as the "critical case" because large clusters of zero-weight edges force T_n to grow at most logarithmically. We characterize the limit behavior of T_n under conditions on the distribution function F. The main tool involves a new relation between first-passage percolation and invasion percolation. This is joint work with Michael Damron and Wai-Kit Lam.

Packing and covering topological minors and immersions

Series
Graph Theory Seminar
Time
Thursday, September 10, 2015 - 13:35 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chun-Hung Liu – Princeton University
A set F of graphs has the Erdos-Posa property if there exists a function f such that every graph either contains k disjoint subgraphs each isomorphic to a member in F or contains at most f(k) vertices intersecting all such subgraphs. In this talk I will address the Erdos-Posa property with respect to three closely related graph containment relations: minor, topological minor, and immersion. We denote the set of graphs containing H as a minor, topological minor and immersion by M(H),T(H) and I(H), respectively. Robertson and Seymour in 1980's proved that M(H) has the Erdos-Posa property if and only if H is planar. And they left the question for characterizing H in which T(H) has the Erdos-Posa property in the same paper. This characterization is expected to be complicated as T(H) has no Erdos-Posa property even for some tree H. In this talk, I will present joint work with Postle and Wollan for providing such a characterization. For immersions, it is more reasonable to consider an edge-variant of the Erdos-Posa property: packing edge-disjoint subgraphs and covering them by edges. I(H) has no this edge-variant of the Erdos-Posa property even for some tree H. However, I will prove that I(H) has the edge-variant of the Erdos-Posa property for every graph H if the host graphs are restricted to be 4-edge-connected. The 4-edge-connectivity cannot be replaced by the 3-edge-connectivity.

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