Seminars and Colloquia by Series

Koszul duality and Knot Floer homology

Series
Geometry Topology Seminar
Time
Monday, November 4, 2019 - 14:00 for
Location
Skiles 006
Speaker
Tom HockenhullUniversity of Glasgow

‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.

Nonstationary signal analysis and decomposition via Fast Iterative Filtering and Adaptive Local Iterative Filtering techniques. State of the art and open problems

Series
Applied and Computational Mathematics Seminar
Time
Monday, November 4, 2019 - 13:55 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Antonio CiconeUniversity of L'Aquila

The analysis and decomposition of nonstationary and nonlinear signals in the quest for the identification
of hidden quasiperiodicities and trends is of high theoretical and applied interest nowadays.

Linear techniques like Fourier and Wavelet Transform, historically used in signal processing, cannot capture
completely nonlinear and non stationary phenomena.

For this reason in the last few years new nonlinear methods have been developed like the groundbreaking
Empirical Mode Decomposition algorithm, aka Hilbert--Huang Transform, and the Iterative Filtering technique.

In this seminar I will give an overview of this kind of methods and I will introduce two new algorithms,
the Fast Iterative Filtering and the Adaptive Local Iterative Filtering. I will review the main theoretical results
and outline the most intriguing open problems that still need to be tackled in the field.
Some examples of applications of these techniques to both artificial and real life signals
will be shown to give a foretaste of their potential and robustness.
 

Knot Floer homology

Series
Geometry Topology Seminar Pre-talk
Time
Monday, November 4, 2019 - 12:45 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Tom HockenhullUniversity of Glasgow

I’ll try and give some background on the definition of knot Floer homology, and perhaps also bordered Heegaard Floer homology if time permits.

Asymptotic normality of the $r\to p$ norm for random matrices with non-negative entries

Series
ACO Student Seminar
Time
Friday, November 1, 2019 - 13:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Debankur MukherjeeISyE, Georgia Tech

For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $\|A_n\|_{r \to p}= \sup_{\|x\|_r\leq 1 } \|A_n x\|_p$ for $r,p\geq 1$. The $r\to p$ operator norm puts a huge number of important quantities of interest in diverse disciplines under a single unified framework. The application of this norm spans a broad spectrum of areas including data-dimensionality reduction in machine learning, finding oblivious routing schemes in transportation network, and matrix condition number estimation.

 

In this talk, we will consider the $r\to p$ norm of a class of symmetric random matrices with nonnegative entries, which includes the adjacency matrices of the Erd\H{o}s-R\'enyi random graphs and matrices with sub-Gaussian entries. For $1< p\leq r< \infty$, we establish the asymptotic normality of the appropriately centered and scaled $\|A_n\|_{r \to p}$, as $n\to\infty$. The special case $r=p=2$, which corresponds to the largest singular value of matrices, was proved in a seminal paper by F\"uredi and Koml\'os (1981). Of independent interest, we further obtain a sharp $\ell_\infty$-approximation for the maximizer vector. The results also hold for sparse matrices and further the $\ell_\infty$-approximation for the maximizer vector also holds for a broad class of deterministic sequence of matrices with certain asymptotic `expansion' properties.

 

This is based on a joint work with Souvik Dhara (MIT) and Kavita Ramanan (Brown U.).

Renormalization for the almost Mathieu operator and related skew products.

Series
CDSNS Colloquium
Time
Friday, November 1, 2019 - 11:15 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Hans KochUniv. of Texas, Austin

Considering SL(2,R) skew-product maps over circle rotations,
we prove that a renormalization transformation
associated with the golden mean alpha
has a nontrivial periodic orbit of length 3.
We also present some numerical results,
including evidence that this period 3 describes
scaling properties of the Hofstadter butterfly
near the top of the spectrum at alpha,
and scaling properties of the generalized eigenfunction
for this energy.

Local limit theorems for combinatorial random variables

Series
Combinatorics Seminar
Time
Friday, November 1, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 249
Speaker
Ross BerkowitzYale University

Let X be the number of length 3 arithmetic progressions in a random subset of Z/101Z.  Does X take the values 630 and 640 with roughly the same probability?
Let Y denote the number of triangles in a random graph on n vertices.  Despite looking similar to X, the local distribution of Y is quite different, as Y obeys a local limit theorem.  
We will talk about a method for distinguishing when combinatorial random variables obey local limit theorems and when they do not.

Research proposal: Matchings in hypergraphs

Series
Other Talks
Time
Thursday, October 31, 2019 - 13:30 for 30 minutes
Location
Skiles 005
Speaker
Xiaofan YuanGeorgia Tech

I will introduce a minimum l-degree threshold for the existence of a nearly perfect (i.e., covering all but a constant number of vertices) matching in a k-graph where k ≥ 3 and k/2 < l ≤ k − 1. This is joint work with Hongliang Lu and Xingxing Yu.

This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l ≤ k − 1. In some cases, such a matching can in fact be near perfect (i.e., covering all but at most k vertices) and our bound on the minimum l-degree is best possible.

New invariants of homology cobordism

Series
School of Mathematics Colloquium
Time
Thursday, October 31, 2019 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kristen HendricksRutgers

This is a talk about 3-manifolds and knots. We will begin by reviewing some basic constructions and motivations in low-dimensional topology, and will then introduce the homology cobordism group, the group of 3-manifolds with the same homology as the 3-dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will introduce a new invariant of homology cobordism coming from an equivariant version of the computationally-friendly Floer-theoretic 3-manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.

The Ehrhard-Borell inequality and hypoelliptic diffusions

Series
High Dimensional Seminar
Time
Wednesday, October 30, 2019 - 15:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yair ShenfeldPrinceton University

The Ehrhard-Borell inequality stands at the top of the pyramid of Gaussian inequalities. It is a powerful and delicate statement about the convexity of the Gaussian measure. In this talk I will discuss the inequality and its beautiful proof by Borell. The delicate nature of the inequality however makes the characterization of the equality cases difficult and they were left unknown. I will explain how we solved this problem. Joint work with Ramon van Handel.

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