If k is a field of characteristic zero, a theorem of Lurie and Pridham
establishes an equivalence between formal moduli problems and differential
graded Lie algebras over k. We generalise this equivalence in two different
ways to arbitrary ground fields by using “partition Lie algebras”.
These mysterious new gadgets are intimately related to the genuine
equivariant topology of the partition complex, which allows us to access
the operations acting on their homotopy groups (relying on earlier work of
Dyer-Lashof, Priddy, Goerss, and Arone-B.)

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 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 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.

My
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.