We will discuss some natural problems in arithmetic that can be
reformulated in terms of orbits of certain "thin" (semi)groups of
integer matrix groups.
We present several classification results for Lagrangian tori, all
proven using the splitting construction from symplectic field
theory. Notably, we classify Lagrangian tori in the symplectic
vector space up to Hamiltonian isotopy; they are either...
I'll outline recent results with Steven Sivek classifying the Stein
fillings, up to topological homotopy equivalence, of the canonical
contact structure on the unit cotangent bundle of a surface. The
proof begins with Li, Mak and Yasui's technology...
Pablo Picasso did not speak often about abstraction, but when he
did, it was either to dismiss it as complacent decoration or to
declare its very notion an oxymoron. The root of this hostility is
to be found in the impasse that the artist reached in...
The conventional perturbative approach and the nonperturbative
lattice approach are the two standard yet very distinct
formulations of quantum gauge theories. Since in dimension two
Yang-Mills theory has a rigorous continuum limit of the lattice...
This talk will be about some phenomena that occur as (singular)
hyperbolic structures on 3-manifolds collapse to and transition
through other geometric structures. Typically, the collapsed
structures are much more flexible than the hyperbolic...
Maryna Viazovska recently made a stunning breakthrough on sphere
packing by showing the E8 root lattice gives the densest packing of
spheres in 8 dimensional space [arxiv:1603.04246]. This is the
first result of its kind for dimensions $> 3$, and...
One way to construct new 3-manifolds is by surgery on a knot in the
3-sphere; that is, we remove a neighborhood of a knot, and reglue
it in a different way. What 3-manifolds can be obtained in this
manner? We provide obstructions using the Heegaard...
We prove an average-case depth hierarchy theorem for Boolean
circuits over the standard basis of AND, OR, and NOT gates. Our
hierarchy theorem says that for every $d \geq 2$, there is an
explicit $n$-variable Boolean function $f$, computed by a...
We describe results of Levelt and Beukers-Heckman on the explicit
computation of monodromy for generalised hypergeometric functions
of one variable. We then discuss the question of arithmeticity of
these monodromy groups and describe various results...
