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Series: CDSNS Colloquium

One dimensional discrete Schrödinger operators arise naturally in modeling
the motion of quantum particles in a disordered medium. The medium is
described by potentials which may naturally be generated by certain ergodic
dynamics. We will begin with two classic models where the potentials are
periodic sequences and i.i.d. random variables (Anderson Model). Then we
will move on to quasi-periodic potentials, of which the randomness is
between periodic and i.i.d models and the phenomena may become more subtle,
e.g. a metal-insulator type of transition may occur. We will show how the
dynamical object, the Lyapunov exponent, plays a key role in the spectral
analysis of these types of operators.

Series: Combinatorics Seminar

Various parameters of many models of random rooted trees are fairly
well understood if they relate to a near-root part of the tree or to global tree
structure. In recent years there has been a growing interest in the analysis
of the random tree fringe, that is, the part of the tree that is close to the
leaves. Distance from the closest leaf can be viewed as the protection level of
a vertex, or the seniority of a vertex within a network.
In this talk we will review a few recent results of this kind for a number of
tree varieties, as well as indicate the challenges one encounters when trying
to generalize the existing results. One tree variety, that of decreasing binary
trees, will be related to permutations, another one, phylogenetic trees, is
frequent in applications in molecular biology.

Friday, April 21, 2017 - 15:00 ,
Location: Skiles 254 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer:

A classical theorem of Arnold, Moser shows that in analytic families of
maps close to a rotation we can find maps which are smoothly conjugate
to rotations. This is one of the first examples of the KAM theory. We
aim to present an efficient numerical algorithm, and its implementation, which approximate the conjugations given by the Theorem

Series: ACO Student Seminar

Beginning with Szemerédi’s regularity lemma, the theory of graph
decomposition and graph limits has greatly increased our understanding
of large dense graphs and provided a framework for graph approximation.
Unfortunately, much of this work does not meaningfully extend to
non-dense graphs. We present preliminary work towards our goal of
creating tools for approximating graphs of intermediate degree (average
degree o(n) and not bounded). We give a new random graph model that
produces a graph of desired size and density that approximates the
number of small closed walks of a given sparse graph (i.e., small
moments of its eigenspectrum). We show how our model can be applied to
approximate the hypercube graph. This is joint work with Santosh
Vempala.

Series: Professional Development Seminar

A conversation with Adam Fox, former GT postdoc who secured his "dream job" as a tenure-track assistant professor at Western New England University, but who recently moved into industry as a Data Scientist.

Series: Stochastics Seminar

We consider rooted subgraph
extension counts, such as (a) the number of triangles containinga given vertex, or (b) the number of paths of length three connecting two
given vertices.
In 1989 Spencer gave sufficient conditions for the event that whp all
roots of the binomial random graph G(n,p) have the same asymptotic
number of extensions, i.e., (1 \pm \epsilon) times their expected
number.
Perhaps surprisingly, the question whether these conditions are
necessary has remained open. In this talk we briefly discuss our
qualitative solution of this problem for the `strictly balanced' case,
and mention several intriguing questions that remain open (which lie at the intersection of probability theory + discrete mathematics, and are of concentration inequality type).
Based on joint work in progress with Matas Sileikis

Series: Analysis Seminar

A well-known elementary linear algebra fact says that any linear
independent set of vectors in a finite-dimensional vector space cannot
have more elements than any spanning set. One way to obtain an analog of
this result in the infinite
dimensional setting is by replacing the comparison of cardinalities
with a more suitable concept - which is the concept of densities.
Basically one needs to compare the cardinalities locally everywhere and
then take the appropriate limits. We provide a rigorous
way to do this and obtain a universal density theorem that generalizes
many classical density results. I will also discuss the connection
between this result and the uncertainty principle in harmonic analysis.

Series: Research Horizons Seminar

In Fall 2017 I will teach `Random Discrete Structures', which is an advanced course in discrete probability and probabilistic combinatorics. The goal of this informal lecture is to give a brief outline of the topics we intend to cover in this course. Buzz-words include Algorithmic Local Locasz Lemma, Concentration Inequalities, Differential Equation Method, Interpolation method and Advanced Second Moment Method.

Series: Geometry Topology Seminar

Let S be a Riemann surface of type (p,1), p > 1. Let f be a point-pushing pseudo-Anosov map of S. Let t(f) denote the translation length of f on the curve complex for S. According to Masur-Minsky, t(f) has a uniform positive lower bound c_p that only depends on the genus p.Let F be the subgroup of the mapping class group of S consisting of point-pushing mapping classes. Denote by L(F) the infimum of t(f) for f in F pseudo-Anosov. We know that L(F) is it least c_p. In this talk we improve this result by establishing the inequalities .8 <= L(F) <= 1 for every genus p > 1.

Monday, April 17, 2017 - 14:00 ,
Location: Skiles 005 ,
Dr. Andre Souza ,
Georgia Tech ,
andre.souza@math.gatech.edu ,
Organizer: Molei Tao

In this talk we discuss how to find probabilities of extreme events in stochastic differential equations. One approach to calculation would be to perform a large number of simulations and gather statistics, but an efficient alternative is to minimize Freidlin-Wentzell action. As a consequence of the analysis one also determines the most likely trajectory that gave rise to the extreme event. We apply this approach to stochastic systems whose deterministic behavior exhibit chaos (Lorenz and Kuramoto-Sivashinsky equations), comment on the observed behavior, and discuss.