School of Mathematics

Is randomness ever necessary for space-efficient computation? It is commonly conjectured that L = BPL, meaning that halting decision algorithms can always be derandomized without increasing their space complexity by more than a constant factor. In...

Consider a function on Rn that can be written as a sum of functions f=f1 + f2 + ... + fm, for m greater than n.

The question of approximating f by a reweighted sum using only a small number of summands has many applications in CS theory, mathematical...

The Higgs mechanism is a part of the Standard Model of quantum mechanics that allows certain kinds of particles to have nonzero mass. In spite of its great importance, there is no rigorous proof that the Higgs mechanism can indeed generate mass in...

Results concerning rigidity of Lagrangian submanifolds lie at the heart of symplectic topology, and have been intensively studied since the 1990s. An example for this phenomenon is the concept of Lagrangian Barriers, a form of symplectic rigidity...

Strong spatial mixing (SSM) is an important and widely studied quantitative notion of "correlation decay" for a variety of natural distributions arising in statistical physics, probability theory, and theoretical computer science. One of the most...

A smooth, oriented n-manifold is called a homotopy sphere if it is homeomorphic, but not necessarily diffeomorphic, to the standard n-sphere. In dimensions n>4

, one often studies the group Θn of homotopy spheres up to orientation-preserving...

In chromatic homotopy theory, an object like the sphere spectrum S0

 is studied by means of its "localizations", much as an abelian group can be localized at each prime p.  Remarkably, the "primes" K(n)

 in the homotopy setting correspond to formal...

Given a grid diagram for a knot or link in the three-sphere, we construct a spectrum whose homology is the knot Floer homology of . We conjecture that the homotopy type of the spectrum is an invariant of . Our construction does not use holomorphic...

In joint work in progress with Anschütz and Le Bras we aim to construct a 6-functor formalism for quasicoherent sheaves on the relative Fargues-Fontaine curve over rigid-analytic varieties (and even general v-stacks), providing new insights into the...

The recent work of Drinfeld and Bhatt-Lurie led to a new geometric approach to p-adic cohomology theories, analogously to what was done earlier in Hodge theory by Simpson. This stacky perspective gives in particular a new approach to p-adic non...