Georgia Institute of Technology College of Sciences

The emergent shape of a knitted fabric is highly sensitive to the underlying stitch pattern. Here, by a stitch pattern we mean a periodic array of symbols encoding a set of rules or instructions performed to produce a swatch or a piece of fabric. So, it is crucial to understand what exactly these instructions mean in terms of mechanical moves performed using a yarn (a smooth piece of string) and a set of knitting needles (oriented sticks).

We will describe several appearances of Milnor’s invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions.

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

Gromov revolutionized the study of finitely generated groups by showing that an intrinsic metric on a group is intimately connected with the algebra of the group. This point of view has produced deep applications not only in group theory, but also topology, geometry, logic, and dynamical systems.

The Torelli group is the subgroup of the mapping class group acting trivially on homology.  We will discuss some basic properties of the Torelli group and explain how to define it for surfaces with boundary.  We will also give some Torelli analogues of the Birman exact sequence.

The smallest volume cusped hyperbolic 3-manifolds, the figure-eight knot complement and its sister, contain many immersed but no embedded closed totally geodesic surfaces. In this talk we discuss the existence or lack thereof of codimension-1 closed embedded totally geodesic submanifolds in minimal volume cusped hyperbolic 4-manifolds. This talk is based on joint work with Alan Reid.

The satellite construction, which associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere the so-called satellite knot P(K), features prominently in knot theory and low-dimensional topology.  Besides the intuition that P(K) is “more complicated” than either P or K, one can attempt to quantify how the complexity of a knot changes under the satellite operation. In this talk, I’ll discuss how several notions of complexity based on the minimal genus of an embedded surface change under satelliting.