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Series: CDSNS Colloquium

Given a dynamical system (in finite or infinite dimension) it is very natural to look for finite dimensional invariant subspaces on which the dynamics is very simple. Of particular interest are the invariant tori on which the dynamics is conjugated to a linear one. The problem of persistence under perturbations of such objects has been widely studied starting form the 50's, and this gives rise to the celebrated KAM theory. The aim of this talk is to give an overview of the main difficulties and strategies, having in mind the application to PDEs.

Series: Combinatorics Seminar

The flow polytope associated to an acyclic graph is the set of all
nonnegative flows on the edges of the graph with a fixed netflow at each
vertex. We will examine flow polytopes arising from permutation matrices,
alternating sign matrices and Tesler matrices. Our inspiration is the
Chan-Robins-Yuen polytope (a face of the polytope of doubly-stochastic
matrices), whose volume is equal to the product of the first n Catalan
numbers (although there is no known combinatorial proof of this fact!). The
volumes of the polytopes we study all have nice product formulas.

Series: Combinatorics Seminar

The theme of this talk is walks in a random environment of "signposts"
altered by the walker. I'll focus on three related examples:
1. Rotor walk on Z^2. Your initial signposts are independent with the
uniform distribution on {North,East,South,West}. At each step you rotate
the signpost at your current location clockwise 90 degrees and then follow
it to a nearest neighbor. Priezzhev et al. conjectured that in n such steps
you will visit order n^{2/3} distinct sites. I'll outline an elementary
proof of a lower bound of this order. The upper bound, which is still open,
is related to a famous question about the path of a light ray in a grid of
randomly oriented mirrors. This part is joint work with Laura Florescu and
Yuval Peres.
2. p-rotor walk on Z. In this walk you flip the signpost at your current
location with probability 1-p and then follow it. I'll explain why your
scaling limit will be a Brownian motion perturbed at its extrema. This part
is joint work with Wilfried Huss and Ecaterina Sava-Huss.
3. p-rotor walk on Z^2. Rotate the signpost at your current location
clockwise with probability p and counterclockwise with probability 1-p, and
then follow it. This walk “organizes” its environment of signposts. The
stationary environment is an orientation of the uniform spanning forest,
plus one additional edge. This part is joint work with Swee Hong Chan, Lila
Greco and Boyao Li.

Friday, April 7, 2017 - 15:05 ,
Location: Skiles 254 ,
Prof. Rafael de la Llave ,
School of Math, Georgia Tech ,
Organizer: Jiaqi Yang

It is well known that periodic orbits give all the information about dynamical systems, at least for expanding maps, for which the periodic orbits are dense. This turns out to be true in dimensions 1 and 2, and false in dimension 4 or higher.We will present a proof that two $C^\infty$ expanding maps of the circle, which are topologically equivalent are $C^\infty$ conjugate if and only if the derivatives or the return map at periodic orbits are the same.

Series: Stochastics Seminar

We discuss scaling methods
which can be used to solve low mode control problems for nonlinear
partial differential equations. These methods lead naturally to a
infinite-dimensional generalization of the notion of saturation,
originally due to Jurdjevic and Kupka in the finite-dimensional setting
of ODEs. The methods will be highlighted by applying them to specific
equations, including reaction-diffusion equations, the 2d/3d
Euler/Navier-Stokes equations and the 2d Boussinesq equations.
Applications to support properties of the laws solving randomly-forced
versions of each of these equations will be noted.

Series: ACO Student Seminar

We study stable marriage where individuals strategically submit private preference information to a publicly known stable marriage algorithm. We prove that no stable marriage algorithm ensures actual stability at every Nash equilibrium when individuals are strategic. More specifically, we show that any rational marriage, stable or otherwise, can be obtained at a Nash equilibrium. Thus the set of Nash equilibria provides no predictive value nor guidance for mechanism design. We propose the following new minimal dishonesty equilibrium refinement, supported by experimental economics results: an individual will not strategically submit preference list L if there exists a more honest L' that yields as preferred an outcome. Then for all marriage algorithms satisfying monotonicity and IIA, every minimally dishonest equilibrium yields a sincerely stable marriage. This result supports the use of algorithms less biased than the (Gale-Shapley) man-optimal, which we prove yields the woman-optimal marriage in every minimally dishonest equilibrium. However, bias cannot be totally eliminated, in the sense that no monotonic IIA stable marriage algorithm is certain to yield the egalitarian-optimal marriage in a minimally dishonest equilibrium – thus answering a 28-year old open question of Gusfield and Irving's in the negative. Finally, we show that these results extend to student placement problems, where women are polygamous and honest, but not to admissions problems, where women are both polygamous and strategic.
Based on joint work with Craig Tovey at Georgia Tech.

Series: Stochastics Seminar

Spectral algorithms are a powerful method for detecting low-rank structure
in dense random matrices and random graphs. However, in certain problems
involving sparse random graphs with bounded average vertex degree, a naive
spectral analysis of the graph adjacency matrix fails to detect this
structure. In this talk, I will discuss a semidefinite programming (SDP)
approach to address this problem, which may be viewed both as imposing a
delocalization constraint on the maximum eigenvalue problem and as a
natural convex relaxation of minimum graph bisection. I will discuss
probabilistic results that bound the value of this SDP for sparse
Erdos-Renyi random graphs with fixed average vertex degree, as well as an
extension of the lower bound to the two-group stochastic block model. Our
upper bound uses a dual witness construction that is related to the
non-backtracking matrix of the graph. Our lower bounds analyze the behavior
of local algorithms, and in particular imply that such algorithms can
approximately solve the SDP in the Erdos-Renyi setting.
This is joint work with Andrea Montanari.

Wednesday, April 5, 2017 - 14:05 ,
Location: Skiles 006 ,
Sudipta Kolay ,
Georgia Tech ,
Organizer: Justin Lanier

Continuing from last time, we will discuss Hilden and Montesinos' result
that every smooth closed oriented three manifold is a three fold
branched cover over the three sphere, and also there is a representation
by bands.

Series: Analysis Seminar

It was shown by Keith Ball that the maximal section of an n-dimensional
cube is \sqrt{2}. We show the analogous sharp bound for a maximal
marginal of a product measure with bounded density. We also show an
optimal bound for all k-codimensional marginals in this setting,
conjectured by Rudelson and Vershynin. This bound yields a sharp small
ball inequality for the length of a projection of a random vector. This
talk is based on the joint work with G. Paouris and P. Pivovarov.

Series: Research Horizons Seminar

I
will continue the discussion on the group actions of the graph Jacobian
on the set of spanning trees. After reviewing the basic definitions, I
will explain how polyhedral geometry leads to a new family of such
actions.
These actions can be described combinatorially, but proving that they
are simply transitive uses geometry in an essential way. If time
permits, I will also explain the following surprising connection: the
canonical group action for a plane graph (via rotor-routing
or Bernardi process) is related to the canonical tropical geometric
structure of its dual graph. This is joint work with Spencer Backman and
Matt Baker.