Titles and Abstracts
Titles and Abstracts
Hiro Lee Tanaka
Title: Generating family spectra and excellent Morse functions
Abstract: Lisa Traynor and I recently introduced new invariants -- generating family spectra -- of Legendrians equipped with generating families. These invariants are stable homotopy lifts of classical invariants, and are conjectured to model stable homotopy lifts of morphisms in infinitesimally wrapped Fukaya categories. As an application, we prove that one can detect infinitely many new connected components in the space of Thom's "excellent Morse functions" on Euclidean spaces. In this talk, I will introduce these topics and give a rough explanation of the proofs.
Sanath Devalapurkar
Title: Local Tate duality for ring spectra
Abstract: Tate duality is a form of Poincare duality for the absolute Galois group of a p-adic local field F, which says that from the perspective of 'etale cohomology, Spec(F) behaves like a 2-manifold. This was refined by Bhatt-Lurie to show that under this duality, the syntomic cohomology of O_F forms a Lagrangian subspace of H*_et(F). An equicharacteristic generalization was proved by Kato, where it was shown that "n-local fields" behave like closed (n+1)- manifolds. I will describe some joint work with Jeremy Hahn and John Rognes, where we define the notion of a "higher local number ring" and prove a version of Tate duality, which says that higher local number rings of height n-1 behave like closed (n+1)-manifolds. In particular, I will sketch two key ingredients of our argument, namely duality at the level of THH, and a motivically filtered lift of the transfer map on THH. Both use the theory of finite Hopf algebras over perfect fields.
Jen Hom
Title: Distinguishing exotic R^4's with Heegaard Floer homology
Abstract: Attaching a Casson handle to a slice disk complement yields a smooth 4-manifold that is homeomorphic to R^4. We show that if two slice knots have sufficiently different knot Floer homology, then the resulting exotic R^4’s made using the simplest positive Casson handle are not diffeomorphic, giving us a countably infinite family of pairwise non-diffeomorphic exotic R^4’s. Our main tool is Gadgil’s end Floer homology. This is joint work with Sean Eli and Tye Lidman.
Juan Moreno
Title: Real homological trace methods
Abstract: Trace methods is an approach to approximating the algebraic K-theory of a ring by simpler invariants, most notably topological Hochschild homology (THH). While THH itself is not a great approximation to K-theory, it is computable and naturally equipped with a circle action which can be used to obtain closer K-theory approximations. In classical work, Bruner and Rognes developed robust methods to study various circle fixed points of THH from a homological perspective. Such tools were then used in later work of Lunoe-Nielsen and Rognes to prove the Segal conjecture for THH of the complex bordism spectrum.
In this talk, I will present work in progress towards generalizing this story to a C_2-equivariant setting, where C_2 denotes the cyclic group of order two. We consider the Real (in the sense of Atiyah) topological Hochschild homology of rings with anti-involutions and develop tools to study the RO(C_2)-graded homology of its fixed points. This is all joint work with Myungsin Cho, Teena Gerhardt, Liam Keenan, and J.D. Quigley.
Sam Nariman
Title:The Milnor-Wood Inequality: Geometry, Topology, and Flat Bundles
Abstract: The Milnor-Wood inequality, introduced in two landmark papers by John Milnor (1958) and John W. Wood (1971), is a striking result at the intersection of geometry, topology, and dynamics. It establishes sharp bounds on the Euler number of flat S^1-bundles over surfaces, revealing deep connections between geometric curvature and topological invariants. Milnor’s original inequality highlights the boundedness of Euler invariants for flat bundles with "linear" structures, which Gromov later generalized using bounded cohomology.
Wood extended Milnor's result to "non-linear" flat circle bundles, offering a perspective rooted in 1-dimensional dynamics. In the 1980s, Étienne Ghys posed the intriguing question of whether Wood’s inequality could be extended to flat-oriented S^3-bundles. In this talk, we will also discuss the surprising ways in which the inequality fails in higher-dimensional non-linear cases, showcasing the new calculations in the bounded cohomology of diffeomorphism groups.
Christy Hazel
Title: Mackey functors and Bredon equivariant cohomology
Abstract: Let G be a finite group. For spaces with a G-action, Bredon RO(G)-graded cohomology plays the role of singular cohomology: these are the ordinary theories represented by equivariant Eilenberg-MacLane spectra. Despite this foundational role, many computations are difficult to carry out given the RO(G)-grading and the complicated algebraic structure of the coefficients. In this talk we give an introduction to Bredon cohomology and to working with Mackey functors. We then give some examples of recent cohomology computations.
Melissa Zhang
Title: Skein lasagna modules with 1-dimensional inputs
Abstract: Khovanov skein lasagna modules describe a smooth 4-manifold in terms of the surfaces embedded within; these surfaces have 0-dimensional singularities, thought of as `inputs', in the following way: the boundary of a B^4 neighborhood of each singularity is an S^3 which the surface intersects at a link; Khovanov homology is used here to label these links at these singularities. This invariant has been used to study questions related to the Smooth Poincare Conjecture in Dimension 4; the focus of these questions is on `exotic behavior’ in 4-manifolds, i.e. the difference between being homeomorphic and diffeomorphic. While the original skein lasagna modules with 0-dimensional inputs can in some cases be computed for 4-manifolds built with only 0-, 2-, and 4-handles, it is currently not computable for 4-manifolds with 1-handles. Nevertheless, many of our known examples of exotic pairs of 4-manifolds involve 1-handles, and so there has been much interest in either finding a `1-handle formula’ or developing a 1-handle-friendly version of skein lasagna modules.
In this talk I will describe joint work with Qiuyu Ren, Ian Sullivan, Paul Wedrich, and Michael Willis, where we define a new version of skein lasagna modules from gl_2 link homology, with 1-dimensional inputs, which is more amenable to 4-manifolds with 1-handles. The strategy is to use an isomorphism discovered in previous joint work with Ian Sullivan, where we related the skein lasagna module of S^2 \times D^2 to Rozansky-Willis homology, a version of Khovanov homology for links in connected sums of S^2 \times S^1. I will begin with introductions to the relevant ingredients, such as Khovanov homology, functoriality, skein lasagna modules, and the categorified projectors that are used in Rozansky-Willis homology.
Ben Antieau
Title: The infinitesimal cohomology of Fp relative to the sphere spectrum.
Abstract: I will describe a spherical analogue of the Hodge-to-de Rham spectral sequence and relate it to the Adams spectral sequence and a spectral sequence discovered by Hill and Lawson. It turns out that these are all different instances of the same construction.