I’ll try and give some background on the definition of knot Floer homology, and perhaps also bordered Heegaard Floer homology if time permits.
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.).
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.
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.
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.
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 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.
The talk will discuss the relationship between topology and
geometry of Einstein 4-manifolds such as K3 surfaces.
That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No background in geometry will be assumed. Joint work with Ramon van Handel.
