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Series: Math Physics Seminar

We generalize the Lp spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. We show that these bounds are optimal on any manifold in a very strong sense. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators and their optimality follows by semi-classical analysis.

Wednesday, October 3, 2018 - 14:00 ,
Location: Skiles 006 ,
Stephen Mckean ,
GaTech ,
Organizer: Anubhav Mukherjee

Many problems in algebraic geometry involve counting solutions to
geometric problems. The number of intersection points of two projective
planar curves and the number of lines on a cubic surface are two
classical problems in this enumerative
geometry. Using A1-homotopy theory, one can gain new insights to old
enumerative problems. We will outline some results in A1-enumerative
geometry, including the speaker’s current work on Bézout’s Theorem.

Series: Analysis Seminar

We prove a criterion for nondoubling parabolic measure to satisfy a weak reverse H¨older inequality
on a domain with time-backwards ADR boundary, following a result of Bennewitz-Lewis for nondoubling
harmonic measure.

Series: High Dimensional Seminar

The n-dimensional L^p Brunn-Minkowski inequality for p<1 , in particular the log-Brunn-Minkowski inequality, is proposed by Boroczky-Lutwak-Yang-Zhang in 2013, based on previous work of Firey and Lutwak . When it came out, it promptly became the major problem in convex geometry. Although some partial results on some specific convex sets are shown to be true, the general case stays wide open. In this talk I will present a breakthrough on this conjecture due to E. Milman and A Kolesnikov, where we can obeserve a beautiful interaction of PDE, operator theory, Riemannian geometry and all sorts of best constant estimates. They showed the validity of the local version of this inequality for orgin-symmtric convex sets with a C^{2} smooth boundary and strictly postive mean curvature, and for p between 1-c/(n^{3/2}) and 1. Their infinitesimal formulation of this inequality reveals the deep connection with the poincare-type inequalities. It turns out, after a sophisticated transformation, the desired inequality follows from an estimate of the universal constant in Poincare inequality on convex sets.

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.