Adela (YiYu) Zhang
I'm a fourth year graduate student at MIT, working with Jeremy Hahn and Haynes Miller. I expect to graduate in May 2023.
We determine all natural operations and their relations on the homotopy groups of spectral partition Lie algebras, as well as mod p TAQ cohomology operations at any prime. In the limiting case, the algebra of unary operations on a single class is Koszul dual to the mod p Dyer-Lashof algebra. Comparing with Brantner-Mathew's result on the ranks of the homotopy groups of free spectral partition Lie algebras, we deduce that these generate all natural operations and obtain the target category. Since F_p-linear TAQ cohomology operations coincide with operations on spectral partition Lie algebras, we deduce the structure of F_p-linear TAQ cohomology operations, thereby subsuming unpublished results of Kriz and Basterra-Mandell. As a corollary, we determine the structure of natural operations on mod p S-linear TAQ cohomology.
We provide a general method to compute the mod 2 Andre-Quillen homology of spectral Lie algebras. This is the E^2-page of Knudsen's spectral sequence, which converges to the mod 2 homology of configuration spaces of k points in a parallelizable manifold M with labels in a connected spectrum V. The existence of higher differentials implies that the spectral Lie operad is not formal over F_2. Then we study the mod p analog of this spectral sequence for p>2, and observe that the mod p holomogy of the configuration space of k points in M with dimension n and labels in S^r depends on the cup product structure on the mod p cohomology of the one point compactification of M when n+r is even. This supplements and contrasts with results of Bodigheimer-Cohen-Taylor when n+r is odd.
We provide a complete determination of inertia groups, homotopy inertia groups, and concordance inertia groups of (n-1)-connected 2n-manifolds for n>2. Our approach builds on works of Wall, Stolz, Burklund-Hahn-Senger, Burklund-Senger, Nagy, and makes heavy use of Pstragowski's category of synthetic spectra.
Varchenko defined the Varchenko matrix associated to any real hyperplane arrangement and computed its determinant. In this paper, we show that the Varchenko matrix of a hyperplane arrangement has a diagonal form if and only if it is semigeneral, i.e., without degeneracy. In the case of semigeneral arrangement, we present an explicit computation of the diagonal form via combinatorial arguments and matrix operations, thus giving a combinatorial interpretation of the diagonal entries.