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_{n1}x...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 𝔽_{2} 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 F_{2}.
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.