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Series: CDSNS Colloquium

This concerns general gradient-like
dynamical systems
in Banach space with the property that there is a manifold along which
solutions move slowly compared to attraction in the transverse
direction. Conditions are given on the energy (or, more generally,
Lyapunov functional) that ensure solutions starting near
the manifold stay near for a long time or even forever. Applications are
given with the vector Allen-Cahn and Cahn-Morral equations. This is
joint work with Giorgio Fusco and Georgia Karali.

Monday, September 24, 2018 - 13:00 ,
Location: Skiles 006 ,
Miriam Kuzbary ,
Rice University ,
Organizer: Jennifer Hom

In this introductory talk I will outline the general landscape of Milnor’s invariants for links. First introduced in Milnor’s master’s thesis in 1954, these invariants capture fundamental information about links and have remained a fascinating object of study throughout the past half century. In the early 80s, Turaev and Porter independently proved their long-conjectured correspondence with Massey products of the link complement and in 1990, Tim Cochran introduced a beautiful construction to compute them using intersection theory. I will give an overview of these constructions and motivate the importance of these invariants, particularly for the study of links considered up to concordance.

Monday, September 24, 2018 - 13:55 ,
Location: Skiles 005 ,
Anthony Yezzi ,
Georgia Tech, ECE ,
Organizer: Sung Ha Kang

Following the seminal work of Nesterov, accelerated optimization methods (sometimes referred to as momentum methods) have been used to powerfully boost the performance of first-order, gradient-based parameter estimation in scenarios were second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with an attraction basin large enough to accommodate the initial overshoot. This behavior has made accelerated search methods particularly popular within the machine learning community where stochastic variants have been proposed as well. So far, however, accelerated optimization methods have been applied to searches over finite parameter spaces. We show how a variational setting for these finite dimensional methods (recently formulated by Wibisono, Wilson, and Jordan) can be extended to the infinite dimensional setting, both in linear functional spaces as well as to the more complicated manifold of 2D curves and 3D surfaces.

Series: Geometry Topology Seminar

Since its introduction in 1966 by Fox and Milnor the knot concordance group has been an invaluable algebraic tool for examining the relationships between 3- and 4- dimensional spaces. Though knots generalize naturally to links, this group does not generalize in a natural way to a link concordance group. In this talk, I will present joint work with Matthew Hedden where we define a link concordance group based on the “knotification” construction of Peter Ozsvath and Zoltan Szabo. This group is compatible with Heegaard Floer theory and, in fact, much of the work on Heegaard Floer theory for links has implied a study of these objects. Moreover, we have constructed a generalization of Milnor’s group-theoretic higher order linking numbers in a novel context with implications for our link concordance group.

Series: PDE Seminar

I will discuss the limit shapes for local minimizers of the Alt-Caffarelli energy. Fine properties of the associated pinning intervals, continuity/discontinuity in the normal direction, determine the formation of facets in an associated quasi-static motion. The talk is partially based on joint work with Charles Smart.

Series: Research Horizons Seminar

Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects. Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all. Can we characterize when? Can we even formalize what is a "formula"? I will try to answer these questions
by presenting many examples, results and open problems.
Note: This is an introductory general audience talk unrelated to the colloquium.

Series: High Dimensional Seminar

I shall tell about some background and known results in regards to the celebrated and fascinating Log-Brunn-Minkowski inequality, setting the stage for Xingyu to discuss connections with elliptiic operators a week later.

Series: Analysis Seminar

Abstract: Let $(M,g)$ be a compact Riemannian n-manifold without boundary. Consider
the corresponding $L^2$-normalized Laplace-Beltrami eigenfunctions. Eigenfunctions
of this type arise in physics as modes of periodic vibration
of drums and membranes. They also represent stationary states of a free
quantum particle on a Riemannian manifold. In the first part of the
lecture, I will give a survey of results which demonstrate how the
geometry of $M$ affects the behaviour of these special
functions, particularly their “size” which can be quantified by
estimating $L^p$ norms.
In joint work with Malabika Pramanik (U. British Columbia),
I will present in the second part of my lecture a result on the $L^p$ restriction of
these eigenfunctions to random Cantor-type
subsets of $M$. This, in some sense, is complementary to the smooth submanifold $L^p$ restriction results
of Burq-Gérard-Tzetkov ’06 (and later work of other authors). Our
method includes concentration inequalities from probability theory in
addition to the analysis of singular Fourier integral operators on
fractals.

Wednesday, September 26, 2018 - 14:00 ,
Location: Skiles 006 ,
Agniva Roy ,
Georgia Tech ,
Organizer: Sudipta Kolay

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.

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.

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: 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.

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.

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.