D. Lukas B. Brantner

We study deformations of Calabi-Yau varieties in characteristic p using techniques from derived algebraic geometry. We prove a mixed characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that Calabi-Yau varieties in characteristic 0 are unobstructed), and we show that ordinary Calabi-Yau varieties admit canonical lifts to characteristic 0, generalising the Serre-Tate theorem on ordinary abelian varieties.

Infinitesimal deformations are governed by partition Lie algebras. In characteristic 0, these higher categorical structures are modelled by differential graded Lie algebras, but in characteristic p, they are more subtle. We give explicit models for partition Lie algebras over general coherent rings, both in the setting of spectral and derived algebraic geometry. For the spectral case, we refine operadic Koszul duality to a functor from operads to divided power operads, by taking ‘refined linear duals’ of Σ_n-representations. The derived case requires a further refinement of Koszul duality to a more genuine setting.

We construct a Galois correspondence for finite purely inseparable field extensions F/K, generalising a classical result of Jacobson for extensions of exponent one (where x^p ∈ K for all x ∈ F).

We construct a spectral sequence converging to the Morava E-theory of unordered configuration spaces and identify its E²-page as the homology of a Chevalley-Eilenberg-like complex for Hecke Lie algebras. Based on this, we compute the E-theory of the weight p summands of iterated loop spaces of spheres (parametrising the weight p operations on E_n-algebras), as well as the E-theory of the configuration spaces of p points on a punctured surface. We read off the corresponding Morava K-theory groups, which appear in a conjecture by Ravenel. Finally, we compute the F_p-homology of the space of unordered configurations of p particles on a punctured surface.

A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base.

We study the restrictions, the strict fixed points, and the strict quotients of the partition complex |Π_n|, which is Σ_n-space attached to the poset of proper nontrivial partitions of the set {1,...,n}.
We express the space of fixed points |Π_n|^G in terms of subgroup posets for general G ⊆ Σ_n and prove a formula for the restriction of |Π_n| to Young subgroups Σ_n1x...x Σ_nk. Both results follow by applying a general method, proven with discrete Morse theory, for producing equivariant branching rules on lattices with group actions. We uncover surprising links between strict Young quotients of |Π_n|, commutative monoid spaces, and the cotangent fibre in derived algebraic geometry. These connections allow us to construct a cofibre sequence relating various strict quotients |Π_n|^◊∧ _Σn (S^l)^∧n and give a combinatorial proof of a splitting in derived algebraic geometry. Combining all our results, we decompose strict Young quotients of |Π_n| in terms of "atoms" |Π_d|^◊∧ _Σd (S^l)^∧d for l odd and compute their homology. We thereby also generalise Goerss' computation of the algebraic André-Quillen homology of trivial square-zero extensions from 𝔽₂ to 𝔽_p for p an odd prime.

We consider Lie algebras in complete O_D^x-equivariant module spectra over Lubin-Tate space as a modular generalisation of Quillen's d.g. Lie algebras in rational homotopy theory. We carry out a general study of the relation between monadic Koszul duality and unstable power operations and apply our techniques to compute the operations which act on the homotopy groups of the aforementioned spectral Lie algebras.

We introduce general methods to analyse the Goodwillie tower of the identity functor on a wedge X∨Y of spaces (using the Hilton-Milnor theorem) and on the cofibre cof(f) of a map f: X → Y. We deduce some consequences for v_n-periodic homotopy groups: whereas the Goodwillie tower is finite and converges in periodic homotopy when evaluated on spheres (Arone-Mahowald), we show that neither of these statements remains true for wedges and Moore spaces.

This paper analyses stable commutator length in groups Z^r * Z^s . We bound scl from above in terms of the reduced wordlength (sharply in the limit) and from below in terms of the answer to an associated subset-sum type problem. Combining both estimates, we prove that, as n tends to infinity, words of reduced length n generically have scl arbitrarily close to ⁿ⁄₄ - 1. We then show that, unless P=NP, there is no polynomial time algorithm to compute scl of efficiently encoded words in F₂. All these results are obtained by exploiting the fundamental connection between scl and the geometry of certain rational polyhedra. Their extremal rays have been classified concisely and completely. However, we prove that a similar classification for extremal points is impossible in a very strong sense.

An expository essay on classical Hodge theory, Simpson's nonabelian Hodge theory, and Serre's GAGA. Cambridge essay.

In this expository article, we will describe the equivalence between weakly admissible filtered (Φ,N)-modules and semistable p-adic Galois representations. After motivating and constructing the required period rings, we focus on Colmez-Fontaine's proof that "weak admissibility implies admissibility". Harvard Minor Thesis. Cave: material learnt and article written within 3 weeks. Written before the groundbreaking work of Bhatt, Morrow, and Scholze.