Midwest Topology Seminar, Fall 2024

Schedule (under construction!)

Saturday, Nov. 2

• 9:30-10:30
  Jeremy Hahn
  Crystallinity for reduced p-adic cohomology theories
  Show Abstract

  Crystalline cohomology witnesses the (p-completed, derived) deRham cohomology of a smooth, p-adic formal Z_p scheme X as a functor of its (derived) mod p reduction X_{p=0}. There are now many p-adic cohomology theories containing more refined information than deRham cohomology, such as prismatic cohomology, syntomic cohomology, and p-completed topological cyclic homology. How much of these theories can be recovered from X_{p=0}, X_{p^2=0}, etc.? I will explain work, joint with Ishan Levy and Andrew Senger, answering this question in terms of the element v_1 of chromatic homotopy theory. It turns out that, as n grows, X_{p^n=0} sees increasingly much of the v_1-adic filtration on p-adic cohomology theories of X. As one application, I will explain how to completely and explicitly compute the mod (p,v_1) algebraic K-theory of Z/p^n. As another, I will explain how the etale cohomology of the adic generic fiber of X can be calculated in terms of X_{p^b=0}, where b is a quantity depending on the dimension of X.
• 11:00-12:00
  Agnes Beaudry
  On chromatic vanishing in characteristic p
  Show Abstract

  The chromatic vanishing conjecture is a beautifully simplifying statement about the continuous cohomology of the Morava stabilizer group with coefficients in the Lubin-Tate deformation ring. This year, there has been an incredible amount of progress on this problem, due to work of Barthel-Schlank-Stapleton-Weinstein and of Kang-Salch. This gives me hope that the tools necessary to prove solve this problem may finally, or almost, be at our disposal. In this mostly expository talk, my goal will be to tell you all about chromatic vanishing, and to explain a reduction based on work of Kohlhaase of the one-parameter problem in characteristic p to a computation about the cohomology of a certain Lie algebra.
• 2:30-3:30
  Jacob Lurie
  Prismatic Stable Homotopy Theory
  Show Abstract

  One of the most powerful approaches to the study of algebraic K-theory is the use of {\it trace methods}: that is, approximations of K-theory by more computable invariants (such as Hochschild homology). In this talk, I'll describe a (conjectural) extension of this methodology to other cohomological invariants in algebraic geometry.
• 4:00-5:00
  Zhouli Xu
  Computing Differentials in the Adams Spectral Sequence
  Show Abstract

  This talk will provide an overview of methods for computing differentials in the Adams spectral sequence. Additionally, I will discuss recent collaborative work with Weinan Lin and Guozhen Wang, focusing on the resolution of the Last Kervaire Invariant Problem in dimension 126.

Sunday, Nov. 3 (note time change!)

• 9:00-10:00
  Hana Jia Kong
  Motivic Milnor formulas
  Show Abstract

  In the latest research, Lin, Wang, and Xu have advanced the Adams computation of stable homotopy groups using computer programs. Developing these algorithms requires a deep understanding of the Steenrod algebra. While Milnor's work provides a foundation in the classical case, the motivic case is much more complex. In this talk, I'll share my work with Weinan Lin on the motivic formulas. This is part of a larger project with Lin, Wang, and Xu, aiming to compute real motivic stable stems over a large range with the assistance of computers.
• 10:30-11:30
  Mike Hill
  Slices and transfers and bears, oh my!
  Show Abstract

  Index or transfer systems give a way to parameterize compatibility relations that show up when you have some collection of transfers or norms. Using the multiplicative form, we can build versions of the slice filtration which are adapted to that collection of norms. Rewriting this in terms of connectivity leads us to unexpected structure results about transfer systems and they way they decompose.
• 11:45-12:45
  Tomer Schlank
  Title TBA
  Show Abstract

  Abstract TBA