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Series: Geometry Topology Seminar

Series: PDE Seminar

In this talk we will introduce two models for the movement of a small droplet over a substrate: the thin film equation and the quasi static approximation. By tracking the motion of the apparent support of solutions to the thin film equation, we connect these two models. This connection was expected from Tanner's law: the edge velocity of a spreading thin film on a pre-wetted solid is approximately proportional to the cube of the slope at the inflection. This is joint work with Prof. Antoine Mellet.

Series: Analysis Seminar

Valuations are finitely additive measures on convex compact subsets of a finite dimensional vector space. The theory of valuations originates in convex geometry. Valuations continuous in the Hausdorff metric play a special role, and we will concentrate in the talk on this class of valuations. In recent years there was a considerable progress in the theory and its applications. We will describe some of the progress with particular focus on the multiplicative structure on valuations and its applications to kinematic formulas of integral geometry.

Series: High Dimensional Seminar

The celebrated Hadwiger's theorem says that linear combinations of intrinsic volumes on convex sets are the only isometry invariant continuous valuations(i.e. finitely additive measures). On the other hand H. Weyl has extended intrinsic volumes beyond convexity, to Riemannian manifolds. We try to understand the continuity properties of this extension under theGromov-Hausdorff convergence (literally, there is no such continuityin general). First, we describe a new conjectural compactification of the set of all closed Riemannian manifolds with given upper bounds on dimensionand diameter and lower bound on sectional curvature. Points of thiscompactification are pairs: an Alexandrov space and a constructible(in the Perelman-Petrunin sense) function on it. Second, conjecturally all intrinsic volumes extend by continuity to this compactification. No preliminary knowledge of Alexandrov spaces will be assumed, though it will be useful.

Series: Intersection Theory Seminar

We will cover the first half of chapter 7 of Eisenbud and Harris, 3264 and All That.Topics: singular hypersurfaces and the universal singularity, bundles of principal parts, singular elements of a pencil, singular elements of linear series in general.

Series: Stochastics Seminar

In first-passage percolation (FPP), one assigns i.i.d. weights
to the edges of the cubic lattice Z^d and analyzes the induced weighted
graph metric. If T(x,y) is the distance between vertices x and y, then a
primary question in the model is: what is the order of the fluctuations
of T(0,x)? It is expected that the variance of T(0,x) grows like the
norm of x to a power strictly less than 1, but the best lower bounds
available are (only in two dimensions) of order \log |x|.
This result was found in the '90s and there has not been any improvement
since. In this talk, we discuss the problem of getting stronger
fluctuation bounds: to show that T(0,x) is with high probability not
contained in an interval of size o(\log |x|)^{1/2}, and similar
statements for FPP in thin cylinders. Such a statement has been proved
for special edge-weight distributions by Pemantle-Peres ('95) and
Chatterjee ('17). In work with J. Hanson, C. Houdré, and C. Xu, we
extend these bounds to general edge-weight distributions. I will explain
some of the methods we use, including an old and elementary ``small
ball'' probability result for functions on the hypercube.

Friday, January 25, 2019 - 12:00 ,
Location: Skiles 005 ,
Tim Duff ,
Georgia Tech ,
Organizer: Trevor Gunn

Bring a laptop with Macaulay 2 installed.

Series: ACO Student Seminar

We present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired byrecent results in algorithmic discrepancy theory. We develop and present toolsfor analyzing our new rounding algorithms, utilizing mathematical machineryfrom the theory of Brownian motion, complex analysis, and partial differentialequations. We will present our method to several classical problems, including Max-Cut, Max-di-cut and Max-2-SAT, and derive new algorithms that are competitive with the best known results. In particular, we show that the basic algorithm achieves 0.861-approximation for Max-cut and a natural variant of the algorithm achieve 0.878-approximation, matching the famous Goemans-Williamson algorithm upto first three decimal digits. This is joint work with Abbas-Zadeh, Nikhil Bansal, Guru Guruganesh, Sasho Nikolov and Roy Schwartz.

Series: Algebra Seminar

An amoeba is the image of a subvariety X of an algebraic torus under the logarithmic moment map. Nisse and Sottile conjectured that the (real) dimension of an amoeba is smaller than the expected one, namely, two times the complex dimension of X, precisely when X has certain symmetry with respect to toric actions. We prove their conjecture and derive a formula for the dimension of an amoeba. We also provide a connection with tropical geometry. This is joint work with Jan Draisma and Johannes Rau.

Series: Combinatorics Seminar

The importance of analyzing big data and in particular very large networks has shown that the traditional notion of a fast algorithm, one that runs in polynomial time, is often insufficient. This is where property testing comes in, whose goal is to very quickly distinguish between objects that satisfy a certain property from those that are ε-far from satisfying that property. It turns out to be closely related to major developments in combinatorics, number theory, discrete geometry, and theoretical computer science. Some of the most general results in this area give "constant query complexity" algorithms, which means the amount of information it looks at is independent of the input size. These results are proved using regularity lemmas or graph limits. Unfortunately, typically the proofs come with no explicit bound for the query complexity, or enormous bounds, of tower-type or worse, as a function of 1/ε, making them impractical. We show by entirely new methods that for permutations, such general results still hold with query complexity only polynomial in 1/ε. We also prove stronger results for graphs through the study of new metrics. These are joint works with Jacob Fox.