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...
Combinatorics and Geometry to Arithmetic of Circle Packings
Abstract: The Koebe-Andreev-Thurston/Schramm theorem assigns a
conformally rigid fi- nite circle packing to a convex polyhedron,
and then successive inversions yield a conformally rigid...
In this talk we will expalin the main ideas of the proof of the
following theorem of Vinberg: Let f be an integral quadratic form
of signature (n, 1). If n ≥ 30 then the subgroup of SO(n, 1)(Z)
which is generated by all hyperbolic reflections has...
In these two talks we will discuss situations in which geometric
input can be used as a method to certify that a group is thin. This
involves a mix of theory and computation.
Monodromy of nFn−1 hypergeometric functions and arithmetic groups I
Abstract: 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...
Using available results on the strong approximation property for
the set of Markoff triples together with an extension of Zagier’s
counting result, we show that most Markoff numbers are composite.
Non-commutative super approximation and the product replacement
algorithm Abstract: Let A be the free abelian group on n generators
and C a finite simple abelian group. The action of Aut(A) on E =
Epi(A, C) ( = the set of epimorphisms from A to C)...