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...
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...
With every bounded prism Bhatt and Scholze associated a
cohomology theory of formal p-adic schemes. The prismatic
cohomology comes equipped with the Nygaard filtration and the
Frobenius endomorphism. The Bhatt-Scholze construction has
been advanced...
I will talk about (very much in progress) joint work with Mark
Kisin on a Hodge—Newton style inequality for the mod p Breuil—Kisin
modules arising from crystalline Galois representations.
The minimal model program for 3-folds has been developed only in
characteristics p greater than or equal to 5. A key difficulty at
small primes is that the singularities occurring in the minimal
model program need not be Cohen-Macaulay, as they are...
Let f:Y→X be a finite covering map of complex algebraic
varieties. The essential dimension of f is the smallest integer e
such that, birationally, f arises as the pullback of a covering
Y′→X′
of dimension e, via a map X→X′. This invariant goes
back...