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Series: Stochastics Seminar

This talk concerns to spectral gap of random regular graphs. First, we prove that almost all bipartite biregular graphs are almost Ramanujan by
providing a tight upper bound for the non trivial eigenvalues of its adjacency operator, proving Alon's Conjecture for this family of graphs. Also, we use a spectral algorithm to recover hidden communities in a random network model we call regular stochastic block model. Our proofs rely on a technique introduced recently by Massoullie, which we developed for random
regular graphs.

Series: Graph Theory Seminar

Let G be a graph containing 5 different vertices a0, a1, a2, b1 and b2.
We say that (G, a0, a1, a2, b1, b2) is feasible if G contains disjoint
connected subgraphs G1, G2, such that {a0, a1, a2}⊆V(G1)
and {b1, b2}⊆V(G2).
In this talk, we will continue our discussion on
the operations we use for characterizing feasible (G, a0, a1, a2, b1,
b2). If time permits, we will also discuss useful structures for
obtaining that characterization, such as frame, ideal frame, and
framework. Joint work with Changong Li, Robin Thomas, and
Xingxing Yu.

Wednesday, September 13, 2017 - 13:55 ,
Location: Skiles 006 ,
Hyun Ki Min ,
Georgia Tech ,
Organizer: Jennifer Hom

The Weeks manifold W is a closed orientable hyperbolic 3-manifold with the smallest volume. Understanding contact structures on hyperbolic 3-manifolds is one of problems in contact topology. Stipsicz previously showed that there are 4 non-isotopic tight contact structures on the Weeks manifold. In this talk, we will exhibit 7 non-isotopic tight contact structures on W with non-vanishing Ozsvath-Szabo invariants.

Series: Analysis Seminar

A sparse bound is a novel method to bound a bilinear
form. Such a bound gives effortless weighted inequalities, which are
also easy to quantify. The range of forms which admit a sparse bound is
broad. This short survey of the subject will include the case of
spherical averages, which has a remarkably easy proof.

Series: Research Horizons Seminar

Antibiotics have greatly reduced morbidity and mortality from
infectious diseases. Although antibiotic resistance is not a new
problem, it breadth now constitutes asignificant threat to human health.
One strategy to help combat resistance is to find novel
ways of using obsolete antibiotics. For strains of E. coli and P.
aeruginosa, pairs of antibiotics have been found where evolution of
resistance to one increases, sometimes significantly, sensitivity to the
other. These researchers
have proposed cycling such
pairs to treat infections. Similar strategies are being investigated to
treat cancer. Using systems of ODEs, we model several possible treatment
protocols using pairs and triples of such antibiotics, and investigate
the speed of ascent of multiply resistant
mutants. Rapid ascent would doom this strategy. This is joint work with
Klas Udekwu (Stockholm University).

Series: Geometry Topology Seminar

Series: Combinatorics Seminar

Among the most studied tree growth processes there are recursive trees and
linear preferential attachment trees. The study of these two models is
motivated by the need of understanding the evolution of social networks. A
key feature of social networks is the presence of vertices that serve as
hubs, connecting large parts of the network. While such type of vertices
had been widely studied for linear preferential attachment trees, analogous
results for recursive trees were missing.
In this talk, we will present joint laws for both the number and depth of
vertices with near-maximal degrees and comment on the possibilities that
our methods open for future research.
This is joint work with Louigi Addario-Berry.

Series: Stochastics Seminar

On the two-dimensional square lattice, assign i.i.d. nonnegative weights to the
edges with common distribution F. For which F is there an infinite self-avoiding path with
finite total weight? This question arises in first-passage percolation, the study of the
random metric space Z^2 with the induced random graph metric coming from the above edge-weights. It has long been known that there is no such infinite path when F(0)<1/2
(there are only finite paths of zero-weight edges), and there is one when F(0)>1/2 (there
is an infinite path of zero-weight edges). The critical case, F(0)=1/2, is considerably
more difficult due to the presence of finite paths of zero-weight edges on all scales. I will
discuss work with W.-K. Lam and X. Wang in which we give necessary and sufficient
conditions on F for the existence of an infinite finite-weight path. The methods involve
comparing the model to another one, invasion percolation, and showing that geodesics
in first-passage percolation have the same first order travel time as optimal paths in an
embedded invasion cluster.

Series: Graph Theory Seminar

Let $G$ be a graph containing 5 different vertices $a_0, a_1, a_2, b_1$ and $b_2$. We say that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible if $G$ contains disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We give a characterization for $(G,a_0,a_1,a_2,b_1,b_2)$ to be feasible, answering a question of Robertson and Seymour. This is joint work with Changong Li, Robin Thomas, and Xingxing Yu.In this talk, we will discuss the operations we will use to reduce $(G,a_0,a_1,a_2,b_1,b_2)$ to $(G',a_0',a_1',a_2',b_1',b_2')$ with $|V(G)|+|E(G)|>|V(G')|+E(G')$, such that $(G,a_0,a_1,a_2,b_1,b_2)$ is feasible iff $(G',a_0',a_1',a_2'b_1',b_2')$ is feasible.

Series: School of Mathematics Colloquium

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random particle simulations, flocks and mills, and their qualitative behavior. Qualitative properties of local minimizers of the interaction energies are crucial in order to understand these complex behaviors. Compactly supported global minimizers determine the flock patterns whose existence is related to the classical H-stability in statistical mechanics and the classical obstacle problem for differential operators.