- You are here:
- GT Home
- Home
- News & Events

Series: Algebra Seminar

Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime (tropical) ideal is either empty or consists of a single point. This is joint work with D. Joó.

Monday, April 15, 2019 - 12:45 ,
Location: Skiles 006 ,
Patrick Orson ,
Boston College ,
patrick.orson@bc.edu ,
Organizer: JungHwan Park

I will give a brief survey of concordance in high-dimensional knot theory and how slice results have classically been obtained in this setting with the aid of surgery theory. Time permitting, I will then discuss an example of how some non-abelian slice obstructions come into the picture for 1-knots, as intuition for the seminar talk about L^2 invariants.

Series: CDSNS Colloquium

Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T-periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement. The stability conditions are developed in non-dimensional parameter space through two basic nonlinear maps based on switching manifolds that correspond to impacts with the top and bottom membranes of the energy harvesting device. The range for stable simple T-periodic behavior is reduced with increasing angle of incline β, since the influence of gravity increases the asymmetry of dynamics following impacts at the bottom and top. These asymmetric T-periodic solutions lose stability to period doubling solutions for β ≥ 0, which appear through increased asymmetry. The period doubling, symmetric and asymmetric periodic motion are illustrated by bifurcation diagrams, phase portraits and velocity time series.

Series: ACO Colloquium

Stability is a multivariate generalization for real-rootedness in univariate polynomials. Within the past ten years, the theory of stable polynomials has contributed to breakthroughs in combinatorics, convex optimization, and operator theory. I will introduce a generalization of stability, called complete log-concavity, that satisfies many of the same desirable properties. These polynomials were inspired by work of Adiprasito, Huh, and Katz on combinatorial Hodge theory, but can be defined and understood in elementary terms. The structure of these polynomials is closely tied with notions of discrete convexity, including matroids, submodular functions, and generalized permutohedra. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and applications to matroid theory, including a proof of Mason’s conjecture on numbers of independent sets. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.

(*Refreshments available at 2:30pm before the colloquium.*)

Friday, April 12, 2019 - 15:05 ,
Location: Skiles 246 ,
Adrian P. Bustamante ,
Georgia Tech ,
Organizer: Adrian Perez Bustamante

In this talk we will follow the paper titled "Aubry-Mather theory for homeomorphisms", in which it is developed a variational approach to study the dynamics of a homeomorphism on a compact metric space. In particular, they are described orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semidistance. This is work of Albert Fathi and Pierre Pageault.

Friday, April 12, 2019 - 12:00 ,
Location: Skiles 006 ,
Stephen McKean ,
Georgia Tech ,
mckean@math.gatech.edu ,
Organizer:

Milnor K-theory is a field invariant that originated as an attempt to study algebraic K-theory. Instead, Milnor K-theory has proved to have many other applications, including Galois cohomology computations, Voevodsky's proof of the Bloch-Kato conjecture, and Kato's higher class field theory. In this talk, we will go over the basic definitions and theorems of Milnor K-theory. We will also discuss some of these applications.

Series: Stochastics Seminar

Neural networks have led to new and state of the art approaches for image recovery. They provide a contrast to standard image processing methods based on the ideas of sparsity and wavelets. In this talk, we will study two different random neural networks. One acts as a model for a learned neural network that is trained to sample from the distribution of natural images. Another acts as an unlearned model which can be used to process natural images without any training data. In both cases we will use high dimensional concentration estimates to establish theory for the performance of random neural networks in imaging problems.

Series: Graph Theory Seminar

In a fractional coloring, vertices of a graph are assigned subsets of the [0, 1]-interval such that adjacent vertices receive disjoint subsets. The fractional chromatic number of a graph is at most k if it admits a fractional coloring in which the amount of "color" assigned to each vertex is at least 1/k. We investigate fractional colorings where vertices "demand" different amounts of color, determined by local parameters such as the degree of a vertex. Many well-known results concerning the fractional chromatic number and independence number have natural generalizations in this new paradigm. We discuss several such results as well as open problems. In particular, we will sketch a proof of a "local demands" version of Brooks' Theorem that considerably generalizes the Caro-Wei Theorem and implies new bounds on the independence number. Joint work with Luke Postle.

Series: High Dimensional Seminar

We discuss a general approach to a problem of estimation of a smooth function $f(\theta)$ of a high-dimensional parameter $\theta$

of statistical models. In particular, in the case of $n$ i.i.d. Gaussian observations $X_1,\doot, X_n$ with mean $\mu$ and covariance

matrix $\Sigma,$ the unknown parameter is $\theta = (\mu, \Sigma)$ and our approach yields an estimator of $f(\theta)$

for a function $f$ of smoothness $s>0$ with mean squared error of the order $(\frac{1}{n} \vee (\frac{d}{n})^s) \wedge 1$

(provided that the Euclidean norm of $\mu$ and operator norms of $\Sigma,\Sigma^{-1}$ are uniformly bounded),

with the error rate being minimax optimal up to a log factor (joint result with Mayya Zhilova). The construction of optimal estimators

crucially relies on a new bias reduction method in high-dimensional problems

and the bounds on the mean squared error are based on controlling finite differences of smooth functions along certain Markov chains

in high-dimensional parameter spaces as well as on concentration inequalities.

Wednesday, April 10, 2019 - 14:00 ,
Location: Skiles 006 ,
Hongyi Zhou ,
Georgia Institute of Technology ,
hzhou@gatech.edu ,
Organizer: Surena Hozoori

Casson invariant is defined for the class of oriented integral homology 3-spheres. It satisfies certain properties, and reduce to Rohlin invariant after mod 2. We will define Casson invariant as half of the algebraic intersection number of irreducible representation spaces (space consists of representations of fundamental group to SU(2)), and then prove this definition satisfies the expected properties.