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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.

Series: GT-MAP Seminars

Robotic snakes have the potential to navigate areas or environments that would be more challenging for traditionally engineered robots. To realize their potential requires deriving feedback control and path planning algorithms applicable to the diverse gait modalities possible. In turn, this requires equations of motion for snake movement that generalize across the gait types and their interaction dynamics. This talk will discuss efforts towards both obtaining general control equations for snake robots, and controlling them along planned trajectories. We model three-dimensional time- and spatially-varying locomotion gaits, utilized by snake-like robots, as planar continuous body curves. In so doing, quantities relevant to computing system dynamics are expressed conveniently and geometrically with respect to the planar body, thereby facilitating derivation of governing equations of motion. Simulations using the derived dynamics characterize the averaged, steady-behavior as a function of the gait parameters. These then inform an optimal trajectory planner tasked to generate viable paths through obstacle-strewn terrain. Discrete-time feedback control successfully guides the snake-like robot along the planned paths.

Series: Combinatorics Seminar

The fundamental EKR theorem states that, when n≥2r, no pairwise intersecting family of r-subsets of {1,2,...,n} is larger than the family of all r-subsets that each contain some fixed x (star at x), and that a star is strictly largest when n>2r. We will discuss conjectures and theorems relating to a generalization to graphs, in which only independent sets of a graph are allowed. In joint work with Kamat, we give a new proof of EKR that is injective, and also provide results on a special class of trees called spiders.

Series: GT-MAP Seminars

Robotic locomotive mechanisms designed to mimic those of their biological counterparts differ from traditionally engineered systems. Though both require overcoming non-holonomic properties of the interaction dynamics, the nature of their non-holonomy differs. Traditionally engineered systems have more direct actuation, in the sense that control signals directly lead to generated forces or torques, as in the case of rotors, wheels, motors, jets/ducted fans, etc. In contrast, the body/environment interactions that animals exploit induce forces or torque that may not always align with their intended direction vector.Through periodic shape change animals are able to effect an overall force or torque in the desired direction. Deriving control equations for this class of robotic systems requires modelling the periodic interaction forces, then applying averaging theory to arrive at autonomous nonlinear control models whose form and structure resembles that of traditionally engineered systems. Once obtained, classical nonlinear control methods may be applied, though some attention is required since the control can no longer apply at arbitrary time scales.The talk will cover the fundamentals of averaging theory and efforts to identify a generalized averaging strategy capable of recovering the desired control equations. Importantly, the strategy reverses the typical approach to averaged expansions, which significantly simplifies the procedure. Doing so provides insights into feedback control strategies available for systems controlled through time-periodic signals.

Series: Geometry Topology Seminar

This is joint work with Hyam Rubinstein. Matveev and Piergallini independently show that the set of triangulations of a three-manifold is connected under 2-3 and 3-2 Pachner moves, excepting triangulations with only one tetrahedron. We give a more direct proof of their result which (in work in progress) allows us to extend the result to triangulations of four-manifolds.