### The link surgery formula and plumbed 3-manifolds

## Abstract

Lattice homology is a combinatorial invariant of plumbed 3-manifolds due to Nemethi. The definition is a formalization of Ozsvath and Szabo's computation of the Heegaard Floer homology of plumbed 3-manifolds. Nemethi conjectured that lattice homology is isomorphic to Heegaard Floer homology. For a restricted class of plumbings, this isomorphism is known to hold, due to work of Ozsvath-Szabo, Nemethi, and Ozsvath-Stipsicz-Szabo. By using the Manolescu-Ozsvath link surgery formula for Heegaard Floer homology, we prove the conjectured isomorphism in general. In this talk, we will talk about aspects of the proof, and some related topics and extensions of the result.

### Unknot recognition in quasi-polynomial time

## Abstract

I will outline a new algorithm for unknot recognition that runs in quasi-polynomial time. The input is a diagram of a knot with n crossings, and the running time is n^{O(log n)}. The algorithm uses hierarchies, normal surfaces and Heegaard splittings.

### Unknotting number and satellites

## Abstract

The unknotting number of a knot is the minimum number of crossing changes needed to untie the knot. It is one of the simplest knot invariants to define, yet remains notoriously difficult to compute. We will survey some basic knot theory invariants and constructions, including the satellite knot construction, a straightforward way to build new families of knots. We will give a lower bound on the unknotting number of certain satellites using knot Floer homology. This is joint work in progress with Tye Lidman and JungHwan Park.

### Conformal blocks for vertex operator algebras, sewing and factorization.

## Abstract

In rational conformal field theory, the sewing and factorization properties are probably the most important properties that conformal blocks satisfy. For special examples such as Weiss-Zumino-Witten models and minimal models, these two properties were proved decades ago (assuming the genus is ≤1 for the sewing theorem). But for general (strongly) rational vertex operator algebras (VOAs), their proofs were finished only very recently. In this talk, I will first motivate the definition of conformal blocks and VOAs using Segal's picture of CFT. I will then explain the importance of Sewing and Factorization Theorem in the construction of full rational conformal field theory.

### Quasi-isometric rigidity of generic cyclic HNN extensions of free groups

## Abstract

Studying quasi-isometries between groups is a major theme in geometric group theory. Of particular interest are the situations where the existence of a quasi-isometry between two groups implies that the groups are equivalent in a stronger algebraic sense, such as being commensurable. I will survey some results of this type, and then talk about recent work with Daniel Woodhouse where we prove quasi-isometric rigidity for certain graphs of virtually free groups, which include "generic" cyclic HNN extensions of free groups.

### Chromatic homotopy theory and algebraic K-theory

## Abstract

I will give an overview of the interactions between chromatic homotopy theory and the algebraic K-theory of ring spectra, especially around the subject of Ausoni-Rognes's principle of "chromatic redshift," and some of the recent advances in this field.

### The singularity category of C^*(BG)

## Abstract

For an ordinary commutative Noetherian ring R we would define the singularity category to be the quotient of the (derived category of) finitely generated modules modulo the (derived category of) fg projective modules [``the bounded derived category modulo compact objects’’]. For a ring spectrum like C^*(BG) (coefficients in a field of characteristic p) it is easy to define the module category and the compact objects, but finitely generated objects need a new definition. The talk will describe the definition and show that the singularity category is trivial exactly when G is p-nilpotent. We will go on to describe the singularity category for groups with cyclic Sylow p-subgroup.

### Veering triangulations and related polynomial invariants

## Abstract

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. During the talk I will discuss how and why it is connected to the Alexander polynomial of the underlying manifold.

### Introduction to Hierarchically Hyperbolic Groups

## Abstract

Hierarchically Hyperbolic Groups (HHGs) were introduced by Behrstock—Hagen—Sisto to provide a common framework to study several groups of interest in geometric group theory, and have been an object of great interest in the area ever since. The goal of the talk is to provide an introduction to the theory of HHGs and discuss the advantages of the unified approach that they provide. If time permits, we will conclude with applications to growth and asymptotic cones of groups.