Abstract: We prove that sufficiently large positive surgeries on negative alternating chainmail links are L-spaces. These give rise to further examples of asymmetric hyperbolic L-spaces.

Abstract: The work of Hodgson and Kerckhoff on cone-manifold deformations in the early 2000s ushered in an era of effective hyperbolic geometry. They took Thurston’s Dehn surgery theorem (which says that all but finitely many Dehn surgeries on a hyperbolic knot remain hyperbolic) and turned it into an explicit and quantitative statement. Their work turned the “finitely many” exceptions into a bound of 60, and provided strong geometric estimates for all but these 60 slopes.

The cosmetic surgery conjecture, posed by Gordon in 1990, can be paraphrased as saying that knots in arbitrary 3-manifolds are determined by their complements. This is nearly equivalent to the statement that two distinct Dehn surgeries on a knot produce distinct closed manifolds. I will describe a finiteness result that reduces the cosmetic surgery conjecture for a hyperbolic knot K to an explicit and finite verification problem. As an application, we check the cosmetic surgery conjecture on the first 350 million knots and on some other data sets. The Hodgson-Kerckhoff theory of cone deformations is central to making this all work. This is joint work with Jessica Purcell and Saul Schleimer.

Abstract: I will discuss recent joint work with Kupers and Randal-Williams (arXiv:2005.05620) on filtrations on algebraic K-theory spaces and spectra by rank.

Abstract:There is a long history in topology of extracting homotopical information about a space from sufficiently nice covers by subspaces, starting with the classical Nerve Theorem of Borsuk. In the last two decades, there have been a number of new results in this area, including Lurie’s higher Seifert-Van Kampen Theorem, Björner’s n-connected version of the Nerve Theorem for posets, and work of G. Ruzzi motivated by Algebraic Quantum Field Theory. I’ll explain a perspective relating these ideas to cofinality theorems for homotopy colimits, leading to new versions of Quillen’s poset fiber theorem, the Bousfield-Kan homotopy cofinality theorem, and Lurie’s theorem itself.

Abstract: Configuration spaces have played an important role in mathematics and its applications. In particular, the question of how their topology changes as the cardinality of the underlying configuration changes has been studied for some fifty years and has attracted renewed attention in the last decade.

While classically additional information is associated "locally" to the points of the configuration, there are interesting examples when this additional information is "non-local". With Martin Palmer we have studied homology stability in some of these cases, including Hurwitz space and moduli spaces of asymptotic monopoles.

Abstract: In filtration 1 of the Adams spectral sequence, using secondary cohomology operations, Adams computed the differentials on the classes \(h_j\), resolving the Hopf invariant one problem. In Adams filtration 2, using equivariant and chromatic homotopy theory, Hill–Hopkins–Ravenel proved that the classes \(h_j^2\) support non-trivial differentials for \( j \geq 7\), resolving the celebrated Kervaire invariant one problem.

I will talk about joint work with Robert Burklund: In Adams filtration 3, we prove an infinite family of non-trivial d_4-differentials on the classes \(h_j^3\) for \(j \geq 6\), confirming a conjecture of Mahowald. Our proof uses two different deformations of stable homotopy theory – C-motivic stable homotopy theory and \(F_2\)-synthetic homotopy theory – both in an essential way.