University of California, San Diego.
Academic year: 2016-2017
Room: APM 7421 (unless it is announced otherwise)
Monday 2:15-2:45. A pre-talk to prepare graduate students.
Monday 3:00-4:00. We continue the fun with more details.

Date Speaker Topic
Oct 17 Henry Tucker


Fusion categories appear in many areas of mathematics. They are realized by topological quantum field theories, representations of finite groups and Hopf algebras, and invariants for knots and Murray-von Neumann subfactors. An important numerical invariant of these categories are the Frobenius-Schur indicators, which are generalized versions of those for finite group representations. It is thought that these indicators should provide a complete invariant for a fairly wide class of fusion categories; in this talk we will discuss new families of so-called near-group fusion categories (i.e. those with only one non-invertible indecomposable object) which satisfy this property.

Oct 24 Keivan Mallahi Karai
Jacobs University
  Asymptotic distribution of values of isotropic quadratic forms at S-integral points


Let q be a non-degenerate indefinite quadratic form over R in n > 2 variables. Establishing a longstanding conjecture of Oppenheim, Margulis proved in 1986 that if q is not a multiple of a rational form, then the set of values q(Zn) is a dense subset of R. Quantifying this result, Eskin, Margulis, and Mozes proved in 1986 that unless q has signature (2,1) or (2,2), then the number N(a,b;r) of integral vectors v of norm at most r satisfying q(v) ∈ (a,b) has the asymptotic behavior

N(a,b;r) ∼ λ(q) (b-a) rn-2.

Now, let S is a finite set of places of Q containing the Archimedean one, and q=(qv)v ∈ S is an S-tuple of irrational isotropic quadratic forms over the completions Qv. In this talk I will discuss the question of distribution of values of q(v) as v runes over S-balls in ZS. This talk is based on a joint work with Seonhee Lim and Jiyoung Han.

Oct 31 Xin Zhang
University of Illinois at Urbana-Champaign
  Finding integers from orbits of thin subgroups of SL2(Z)  


Let Λ < SL(2, Z) be a finitely generated, non-elementary Fuchsian group of the second kind, and v,w be two primitive vectors in Z2-0. We consider the set

S={ < v γ, w >R2: γ ∈ Λ },

where < ., .>R2 is the standard inner product in R2. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if Λ has parabolic elements, and the critical exponent δ of Λ exceeds 0.995371, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in S, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in S). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when Λ is free, finitely generated, has no parabolics and has critical exponent δ>0.999950.

Nov 7 Name



Nov 14 Max Ehrman
Yale University
  Almost Prime Coordinates in Thin Pythagorean Triangles


The affine sieve is a technique first developed by Bourgain, Gamburd, and Sarnak in 2006 and later completed by Salehi Golsefidy and Sarnak in 2010 to study almost-primality in a broad class of affine linear actions. The beauty of this is that it gives us effective bounds on the saturation number for thin orbits coming from GLn - in particular, producing infinitely many R-almost primes for some R. However, in practice this value of R is often far from optimal. The case of thin Pythagorean triangles has been of particular interest since the outset of the affine sieve, and I will discuss recent progress on improving bounds for the saturation numbers for their hypotenuses and areas using Archimedean sieve theory.

Nov 21 Sue Sierra
University of Edinburgh
  Noncommutative minimal surfaces


In the classification of (commutative) projective surfaces, one first classifies minimal models for a given birational class, and then shows that any surface can be blown down at a finite number of curves to obtain a minimal model.

Artin has proposed a similar programme for noncommutative surfaces (that is, domains of GK-dimension 3). In the generic ``rational'' case of rings birational to a Sklyanin algebra, the likely candidates for minimal models are the Sklyanin algebra itself and Van den Bergh's quadric surfaces. We show, using our previously developed noncommutative version of blowing down, that these algebras are minimal in a very strong sense: given a Sklyanin algebra or quadric R, if S is a connected graded, noetherian overring of R with the same graded ring of fractions, then S=R.

This is a joint work with Rogalski and Stafford.