Seminars and Colloquia by Series

Non-Archimedean Hyperbolicity and Applications

Series
Algebra Seminar
Time
Monday, January 28, 2019 - 12:50 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Jackson Morrow – Emory university –
The conjectures of Green—Griffths—Lang predict the precise interplay between different notions of hyperbolicity: Brody hyperbolic, arithmetically hyperbolic, Kobayashi hyperbolic, algebraically hyperbolic, groupless, and more. In his thesis (1993), W.~Cherry defined a notion of non-Archimedean hyperbolicity; however, his definition does not seem to be the "correct" version, as it does not mirror complex hyperbolicity. In recent work, A.~Javanpeykar and A.~Vezzani introduced a new non-Archimedean notion of hyperbolicity, which ameliorates this issue, and also stated a non-Archimedean variant of the Green—Griffths—Lang conjecture. In this talk, I will discuss complex and non-Archimedean notions of hyperbolicity as well as some recent progress on the non-Archimedean Green—Griffths—Lang conjecture. This is joint work with Ariyan Javanpeykar (Mainz) and Alberto Vezzani (Paris 13).

Property testing and removal lemma

Series
Combinatorics Seminar
Time
Friday, January 25, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Fan Wei – Stanford University
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.

The dimension of an amoeba

Series
Algebra Seminar
Time
Friday, January 25, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Chi Ho Yuen – University of Bern –
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.

Bridge trisections and minimal genus

Series
Geometry Topology Working Seminar
Time
Friday, January 25, 2019 - 14:00 for 2 hours
Location
Skiles 006
Speaker
Peter Lambert-Cole – Georgia Insitute of Technology
The classical degree-genus formula computes the genus of a nonsingular algebraic curve in the complex projective plane. The well-known Thom conjecture posits that this is a lower bound on the genus of smoothly embedded, oriented and connected surface in CP^2. The conjecture was first proved twenty-five years ago by Kronheimer and Mrowka, using Seiberg-Witten invariants. In this talk, we will describe a new proof of the conjecture that combines contact geometry with the novel theory of bridge trisections of knotted surfaces. Notably, the proof completely avoids any gauge theory or pseudoholomorphic curve techniques.

Sticky Brownian Rounding and its Applications to Optimization Problems

Series
ACO Student Seminar
Time
Friday, January 25, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Mohit Singh – ISyE, Georgia Tech –
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.

Lower bounds for fluctuations in first-passage percolation

Series
Stochastics Seminar
Time
Thursday, January 24, 2019 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
M. Damron – SOM, GaTech
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.

Singular Elements of Linear Series

Series
Intersection Theory Seminar
Time
Thursday, January 24, 2019 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Stephen McKean – Georgia Tech
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.

Few conjectures on intrinsic volumes on Riemannian manifolds and Alexandrov spaces

Series
High Dimensional Seminar
Time
Wednesday, January 23, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Semyon Alesker – Tel Aviv University –

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.

Dynamics and Topology of Contact 3-Manifolds with negative $\alpha$-Sectional Curvature: Lecture 2

Series
Geometry Topology Student Seminar
Time
Wednesday, January 23, 2019 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Surena Hozoori – Georgia Institute of Technology –
In this series of (3-5) lectures, I will talk about different aspects of a class of contact 3-manifolds for which geometry, dynamics and topology interact subtly and beautifully. The talks are intended to include short surveys on "compatibility", "Anosovity" and "Conley-Zehnder indices". The goal is to use the theory of Contact Dynamics to show that conformally Anosov contact 3-manifolds (in particular, contact 3-manifolds with negative α-sectional curvature) are universally tight, irrducible and do not admit a Liouville cobordism to tight 3-sphere.

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