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 describe general fixed points, Young restrictions, and strict Young quotients of the
partition complex |Π_{n}| with its natural Σ_{n}-action:
We express the fixed point space |Π_{n}|^{G} in terms of subgroup posets for general
G⊆ Σ_{n} . Moreover, we give a formula for the restriction of |Π_{n}| to
Young subgroups Σ_{n1}x...x Σ_{nk}.
Both results follow by applying a new general method for producing equivariant branching
rules on lattices with group actions. This technique is proven with discrete Morse theory.
We uncover surprising links between strict Young quotients of |Π_{n}|, strictly
commutative monoid spaces, and the cotangent complex and establish an EHP sequence
for strictly commutative monoid spaces. Combining all our results, we decompose strict
Young quotients of |Π_{n}| in terms of ``nuclear pieces"
|Π_{d}|^{◊}∧ _{Σd } (S^{j})^{∧d} for j odd, compute their
cohomology, and give surprising combinatorial proofs of results about the cotangent
complex in derived algebraic geometry.

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: material learnt and article written within 3 weeks. Written before the groundbreaking work of Bhatt, Morrow, and Scholze.