Seminar Calendar
The calendar of seminar talks for spring 2020 may be found below.
1/21: Benjamin Linowitz (Oberlin)
Title: Counting central simple algebras over number fields
Abstract: This talk will begin with a friendly introduction to central simple algebras over number fields. After reviewing some non-commutative algebra we will discuss the elements of class field theory which are needed to classify central simple algebras in terms of their ramification data. We will then use a Tauberian theorem in order to count the number of central simple algebras with discriminant less than X. In the case that the central simple algebras we are counting have dimension 4 (i.e., are quaternion algebras), we can do much better and count the number of quaternion algebras over a number field k which admit embeddings of any finite number of fixed quadratic field extensions of k and which have discriminant at most X. We will conclude the talk by discussing some applications of this to hyperbolic geometry. In particular we will use the aforementioned results to count the number of arithmetic hyperbolic manifolds containing closed geodesics with prescribed lengths.
1/28: Subhadip Dey (UC Davis)
Title: Hausdorff dimension of the limit sets of Anosov subgroups
Abstract: Patterson-Sullivan measures were introduced by Patterson (1976) and Sullivan (1979) to study the limit sets of Kleinian groups, discrete isometry groups of hyperbolic spaces. Using these measures, they showed a close relationship between the critical exponent of a Kleinian group and the Hausdorff dimension of the limit set of that group. The critical exponent gives a geometric measurement of the exponential growth rate of orbits and, on the other hand, the Hausdorff dimension measures the size of the limit set. For convex-cocompact (or, more generally, geometrically finite) Kleinian groups, Sullivan proved that the critical exponent matches the Hausdorff dimension. Anosov subgroups, introduced by Labourie and further developed by Guichard-Wienhard and Kapovich-Leeb-Porti, extend the notion of convex-cocompactness to the higher-rank. In this talk, we discuss how one can similarly understand the Hausdorff dimension of the limit sets of Anosov subgroups in terms of their appropriate critical exponents. This is a joint work with my advisor Michael Kapovich.
2/4: Paul Nelson (ETH Zurich/IAS)
Title: Eisenstein Series and the Cubic Moment for PGL(2)
Abstract: We will discuss how to study the cubic moment of any family of automorphic L-functions on PGL(2) using regularized diagonal periods of Eisenstein series, following a strategy suggested by Michel–Venkatesh. Applications include generalizations to the setting of number fields of some results of Conrey–Iwaniec and Petrow–Young, improved estimates for representation numbers of ternary quadratic forms over number fields, and improvements to the prime geodesic theorem on arithmetic hyperbolic 3-folds.
2/11: Chen Meiri (Technion/IAS)
Title: First order rigidity of higher-rank arithmetic groups
Abstract: In many contexts, there is a dichotomy between lattices in Lie groups of rank one and lattices in Lie groups of higher-rank. I will talk about a manifestation of this dichotomy in Model Theory. Based on joint works with Nir Avni and Alex Lubotzky.
3/3: Naser Sardari (MPIM/IAS)
Title: Vanishing Fourier Coefficients of Hecke Eigenforms
Abstract: We prove that, for fixed tame level (N,p) = 1, there are only finitely many Hecke eigenforms f of level Gamma_1(N) and even weight with a_p(f) = 0 which are not CM.
3/10: Thomas Hille (Yale)
Title: Bounds for the least solutions of quadratic inequalities
Abstract: Let Q be a non-degenerate indefinite quadratic form in d variables. In the mid 80’s, Margulis proved the Oppenheim conjecture, which states that if d \geq 3 and Q is not proportional to a rational form then Q takes values arbitrarily close to zero at integral points. In this talk we will discuss the problem of obtaining bounds for the least integral solution of the Diophantine inequality |Q[x]|< \epsilon for any positive \epsilon if d \geq 5. We will review historical, as well as recent results in this direction and show how to obtain explicit bounds that are polynomial in \epsilon^{-1}, with exponents depending only on the signature of Q or if applicable, the Diophantine properties of Q. This talk is based on joint work with P. Buterus, F. Götze and G. Margulis.
3/31: Noah Stevens-Davidoff (Berkeley) — Cancelled
4/7: Yihang Zhu (Columbia) — Cancelled
4/14: Ayla Gafni (Ole Miss)
Title: Partitions into powers of primes
Abstract: Given a subset A of the integers, the restricted partition function p_A(n) counts the number of integer partitions of n with all parts in A. In this talk, we will explore the features of the restricted partitions function p_(P_k), where P_k is the set of k-th powers of primes. Powers of primes are both sparse and irregular, which makes p_(P_k) quite an elusive function to understand. We will discuss some of the challenges involved in studying restricted partition functions and what is known in the case of primes, k-th powers, and k-th powers of primes.
This seminar will be hosted online and will be open to the public. Those not on the seminar mailing list may request access by emailing alexander.walker@rutgers.edu .
4/28: Jan Vonk (IAS)
Title: TBD
Abstract: TBD