Adela (YiYu) Zhang
I'm a fifth year graduate student at MIT, working with Jeremy Hahn and Haynes Miller.
I expect to graduate in May 2023. Here is my CV.
Inertia groups of (n-1)-connected 2n-manifolds with
Andrew Senger. (2022)
We compute the inertia groups of (n-1)-connected, smooth, closed, oriented 2n-manifolds where n>2. As a consequence, we complete the diffeomorphism classification of such manifolds, finishing a program initiated by Wall sixty years ago, with the exception of the 126-dimensional case of the Kervaire invariant one problem.
In particular, we find that the inertia group always vanishes for n not equal to 4,8,9 -- for large enoough n, this was known by the work of several previous authors, including Wall, Stolz, and Burklund and Hahn with the first named author. When n=4,8,9, we apply Kreck's modified surgery and a special case of Crowley's Q-form conjecture, proven by Nagy, to compute the inertia groups of these manifolds. In the cases n=4,8, our results recover unpublished work of Crowley--Nagy and Crowley--Olbermann. In contrast, we show that the the homotopy and concordance inertia groups of (n-1)-connected, smooth, closed, oriented 2n-manifolds with n>2 always vanish.
Operations on spectral partition Lie algebras and TAQ cohomology. (2022)
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on spectral partition Lie algebras, we deduce the structure of F_p-linear TAQ cohomology operations, thereby subsuming unpublished results of Kriz and Basterra-Mandell.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.
Quillen homology of spectral Lie algebras, with application to mod p homology of labeled configuration spaces. (2021)
We provide a general method computing the mod p Quillen homology of algebras over a monad that parametrizes the structure of mod p homology of spectral Lie algebras. This is the E^2-page of the bar spectral sequence converging to the mod p topological Quillen homology of spectral Lie algebras. The computation of the Quillen homology of the trivial algebra allows us to deduce that the F_p-linear spectral Lie operad is not formal. As an application, we study the mod p homology of labeled configuration spaces B_k(M;X) in a manifold M with labels in a spectrum X, which is the mod p topological Quillen homology of certain spectral Lie algebras by a result of Knudsen. We obtain general upper bounds for the mod p homology of B_k(M;X), as well as explicit computations for small k. When p is odd, we observe that the mod p homology of B_k(M^n;S^r) for small k depends on and only on the cohomology ring of the one-point compactification of M when n+r is even. This supplements and contrasts with the result of Bodigheimer-Cohen-Taylor when n+r is odd.
Mod p homology of unordered configuration spaces of surfaces with Matthew Chen. (2022)
As a follow-up, we provide a short proof that the dimensions of the mod p homology of the unordered configuration space of k points in a closed torus or punctured genus g>0 surface agree with its Betti numbers for p>2 and k at most p. Therefore its integral homology has no p-power torsion.
Diagonal form of the Verchanko matrices with Yibo Gao. Journal of Algebraic Combinatorics 48 (3), 351-368, 2018.
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.