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Series: Research Horizons Seminar

After briefly describing my research interests, I’ll speak on two results that involve points moving around on surfaces. The first result shows how to “hear the shape of a billiard table.” A point bouncing around a polygon encodes a sequence of edges. We show how to recover geometric information about the table from the collection of all such bounce sequences. This is joint work with Calderon, Coles, Davis, and Oliveira. The second result answers the question, “Given n distinct points in a closed ball, when can a new point be added in a continuous fashion?” We answer this question for all values of n and for all dimensions. Our results generalize the Brouwer fixed point theorem, which gives a negative answer when n=1. This is joint work with Chen and Gadish.

Series: Geometry Topology Seminar

The knot group has played a central role in classical knot theory
and has many nice properties, some of which fail in interesting ways for
knotted surfaces. In this talk we'll introduce an invariant of
knotted surfaces called ribbon genus, which measures the failure of a
knot group to 'look like' a classical knot group. We will show that
ribbon genus is equivalent to a property of the group called Wirtinger
deficiency. Then we will investigate some examples
and conclude by proving a connection with the second homology of the
knot group.

Series: Geometry Topology Seminar

I'll describe a way to construct an A-infinity category associated to a
contact manifold, analogous to a Fukaya category for a symplectic
manifold. The objects of this category are Legendrian submanifolds
equipped with augmentations. Currently we're focusing on standard
contact R^3 but we're hopeful that we can extend this to other contact
manifolds. I'll discuss some properties of this contact Fukaya category,
including generation by unknots and a potential application to proving
that ``augmentations = sheaves''. This is joint work in progress with
Tobias Ekholm and Vivek Shende.

Monday, October 1, 2018 - 13:55 ,
Location: Skiles 005 ,
Dr. Andre Wibisono ,
Georgia Tech CS ,
Organizer: Molei Tao

Accelerated gradient methods play a central role in optimization, achieving the optimal convergence rates in many settings. While many extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this work, we study accelerated methods from a continuous-time perspective. We show there is a Bregman Lagrangian functional that generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that in continuous time, these accelerated methods correspond to traveling the same curve in spacetime at different speeds. This is in contrast to the family of rescaled gradient flows, which correspond to changing the distance in space. We show how to implement both the rescaled and accelerated gradient methods as algorithms in discrete time with matching convergence rates. These algorithms achieve faster convergence rates for convex optimization under higher-order smoothness assumptions. We will also discuss lower bounds and some open questions. Joint work with Ashia Wilson and Michael Jordan.

Friday, September 28, 2018 - 15:05 ,
Location: Skiles 156 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk I will present a proof of a generalization of a theorem by
Siegel, about the existence of an analytic conjugation between an
analytic map, $f(z)=\Lambda z +\hat{f}(z)$, and a linear map, $\Lambda
z$, in $\mathbb{C}^n$. This proof illustrates a standar technique used
to deal with small divisors problems. I will be following the work of E.
Zehnder. This is a continuation of last week talk.

Series: Combinatorics Seminar

In 1973 Erdos asked whether there are n-vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g \ge 4 for which some g-element vertex-set contains at least g-2 triples.)
We answer this question, by showing existence of approximate Steiner triple systems with arbitrary high girth. More concretely, for any fixed \ell \ge 4 we show that a natural constrained random process typically produces a partial Steiner triple system with (1/6-o(1))n^2 triples and girth larger than \ell. The process iteratively adds random triples subject to the constraint that the girth remains larger than \ell. Our result is best possible up to the o(1)-term, which is a negative power of n.
Joint work with Tom Bohman.

Series: ACO Student Seminar

Given a finite partially ordered set (poset) of width $w$, Dilworth's theorem gives an existence and minimality of a chain partition of size $w$. First-Fit is an online algorithm for creating a chain partition of a poset. Given a linear ordering of the points of the poset, $v_1, \cdots, v_n$, First-Fit assigns the point $v_i$ to the first available chain in the chain partition of the points $v_1, \cdots, v_{i-1}$. It is a known fact that First-Fit has unbounded performance over the family of finite posets of width 2. We give a complete characterization of the family of finite posets in which First-Fit performs with subexponential efficiency in terms of width. We will also review what is currently known on the family of posets in which First-Fit performs with polynomial efficiency in terms of width. Joint work with Kevin Milans.

Thursday, September 27, 2018 - 13:30 ,
Location: Skiles 006 ,
Stephen McKean ,
Georgia Tech ,
Organizer: Trevor Gunn

Bézout’s Theorem is the classical statement that generic curves of degree c and d intersect in cd points. However, this theorem requires that we work over an algebraically closed field. Using some tools from A^1-algebraic topology, we will give an arithmetic generalization of Bézout’s Theorem. We will also describe the geometric implications of this generalization over the reals.

Series: School of Mathematics Colloquium

Given a convex polytope P, what is the number of integer points in P? This problem is of great interest in combinatorics and discrete geometry, with many important applications ranging from integer programming to statistics. From computational point of view it is hopeless in any dimensions, as the knapsack problem is a special case. Perhaps surprisingly, in bounded dimension the problem becomes tractable. How far can one go? Can one count points in projections of P, finite intersections of such projections, etc? We will survey both classical and recent results on the problem, emphasizing both algorithmic and complexity aspects. Some elegant hardness results will make an appearance in dimension as small as three. If time permits, we will discuss connections to Presburger Arithmetic and decidability problems for irrational polyhedra. Joint work with Danny Nguyen.

Series: Graph Theory Working Seminar

A classical result of Edwards says that every m-edge graph has a 2-cut of size m/2+Ω(√m), and this is best possible. We will continue our discussion about recent results on analogues of Edwards’ result and related problems in hypergraphs.