Georgia Institute of Technology College of Sciences

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.).

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.

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.