Vinberg’s theorem on hyperbolic reflection groups

Chen Meiri
April 1, 2016
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 infinite index. As a consequence
of this theorem we will show that certain hypergeometric monodromy groups are thin.

Combinatorics and Geometry to Arithmetic of Circle Packings

Kei Nakamura
Rutgers University
April 1, 2016
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 infinite circle packing. For example, starting with the tetrahedron, we take a configuration
of four pairwise tangent circles and invert successively to obtain the classical Apollonian
Circle Packing. The latter, an object of much recent study, is ”arithmetic”, in that there

Monodromy of nFn−1 hypergeometric functions and arithmetic groups II

T. N. Venkataramana
Tata Institute of Fundamental Research
April 4, 2016
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 obtained
in recent years towards this question.

Knot surgery and Heegaard Floer homology

Jennifer Hom
Member, School of Mathematics
April 4, 2016
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 Floer homology package of Ozsvath and Szabo. This is joint work with Cagri Karakurt and Tye Lidman.

The solution to the sphere packing problem in 24 dimensions via modular forms

Stephen Miller
Rutgers University
April 4, 2016
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 follows an approach suggested by Cohn-Elkies from 1999 via harmonic analysis.

Collapsing hyperbolic structures: from rigity to flexibility and back

Steve Kerckhoff
Stanford University
April 5, 2016
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 structures, leading to the question of which structures arise as limits of hyperbolic structures.

Quantum Yang-Mills theory in two dimensions: exact versus perturbative

Timothy Nguyen
Michigan State University
April 6, 2016
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 formulation, it makes sense to ask whether the two approaches are consistent (i.e., do perturbative computations yield asymptotic expansions for the nonperturbative ones?).

Picasso and Abstraction: Encounters and Avoidance

Yve-Alain Bois
Professor, School of Historical Studies
April 6, 2016
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 summer 1910, when abstraction suddenly appeared as the logical development of his previous work, a possibility at which he recoiled in horror. But though he swore to never go again near abstraction, he could not prevent himself from testing his resolve from time to time.