Georgia Institute of Technology College of Sciences

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.

The talk will discuss the relationship between topology and
geometry of Einstein 4-manifolds such as K3 surfaces.

One of the simplest and, at the same time, most prominent models for the discrete quasi-periodic Schrodinger operator is the almost Mathieu operator (also called the Harper's model). This simple-looking operator is known to present exotic spectral properties. Three (out of fifteen) of Barry Simon's problems on Schrodinger operators in the 21st century concerns the almost Mathieu operator. In 2014, Artur Avila won a Fields Medal for work including the solutions to these three problems. In this talk, I will concentrate on the one concerning the Lebesgue measure of the spectrum. I will also talk about the difficulties in generalizing this result to the extended Harper's model. Students with background in numerics are especially welcome to attend!