Topology

Telescopic stable homotopy theory is an approach to studying the stable homotopy category by decomposing it into atomic periodic pieces, the T(n)-local categories. It is a variant of the approach of chromatic homotopy theory that has…

Motivated by an observation of Dehornoy, we study the roots of Alexander polynomials of knots and links that are closures of positive 3-strand braids. We give experimental data on random such braids and find that the roots exhibit marked patterns, which we refine into precise conjectures. We…

Equivariant knots are knots equipped with a specified symmetry. Extending this notion, one can define equivariant Seifert surfaces in S^3 and equivariant slice surfaces in D^4. However, even the equivariant Seifert genus is difficult to determine, as there is no analogue of the classical Seifert…

Any reasonable exotic phenomena in simply connected 4-manifolds are unstable. It is an open question if there is a universal upper bound to the number of stabilizations needed. The case of 1 stabilization was proven in works of Lin and Guth-K., but whether we need more than two stabilizations…

In 2019, Budney and Gabai introduced barbell diffeomorphisms and used them to construct knotted 3-balls in S^4. We will review the construction of barbell diffeomorphisms, and show how to use them to construct n-component Brunnian links of 3-balls in S^4 for all n \geq 2. Time permitting, we…

What do quantum invariants know about knot geometry? Gauge-theoretic invariants, such as knot Floer homology, detect geometric features. However, analogous results for quantum invariants, including the Jones polynomial and its categorification, Khovanov homology, are largely missing. In…

Recently introduced by Guth and Manolescu, real Heegaard Floer homology is an invariant associated to 3-manifolds equipped with an involution. In this talk, we will see how under certain assumptions, the real Heegaard Floer homology groups admit an absolute Z/2 grading. We then specialize to…

This talk focuses on applications of real Seiberg-Witten theory to knotted surfaces in S^4. It is divided into two parts. In the first, we discuss how to construct exotic unknotted RP^2s. In the second, we discuss current work in progress relating the real Bauer-Furuta invariant to cobordism…

One of the distinguishing features of higher algebra is the difficulty of constructing quotients. In this talk I will explain a new technique for constructing algebra structures on quotients. This allows us to prove for example that 𝕊/8 is an E1-algebra

Unknotting number is a fundamental measure of how complicated a knot is, measuring how `far' it is from the unknot via crossing changes. It is a challenging invariant to compute; a vast array of tools have been applied to its calculation, and many conjectureshave grown up…