Seminars Sorted by Series

How many algebraic integers of bounded height have a minimal polynomial with a given Galois group? One approach to this problem is via Malle's conjecture. In this talk we will discuss an alternative approach using a construction with the Galois...

In recent work with Hajir, Larsen and Maire, we have proved that a large class of finitely generated pro-p groups G can be realized as tamely ramified extensions of a number field K, though ramification at an infinite number of primes is required...

For an elliptic curve E defined over Q, the Mordell-Weil group E(Q) is a finitely generated abelian group. We prove that there are infinitely many elliptic curves E over Q for which E(Q) has rank 2. Our elliptic curves will be given by explicit...

Large sieve inequalities are a fundamental tool used to investigate prime numbers and exponential sums. In this lecture I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis)...

The Kronecker limit formula is an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at s=0) of the Dirichlet L-function of an imaginary quadratic character. Colmez conjectured a generalization relating the...

Central predictions of arithmetic quantum chaos such as the Quantum Unique Ergodicity conjecture and the sup-norm problem ask about the mass distribution of automorphic forms, most classically in terms of their weight or Laplace eigenvalue (for...

I will give an overview of recent progress in random multiplicative functions (random models for multiplicative functions) and their connection to the study of the Fyodorov-Hiary-Keating conjecture and Polya's question on nonnegative character sums...

We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical “quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines. Building on Xue’s result, that this point is...

Borel proved that every holomorphic map from a product of punctured unit discs to a complex Shimura variety extends to a map from a product of discs to its Bailey-Borel compactification. In joint work with Oswal, Zhu, and Patel, we proved a p-adic...

It is possible to form "theta series" in other geometric contexts, e.g. for counting problems involving abelian varieties. It is expected that these theta series again have additional automorphic symmetry.


I will explain some “near-central”...

In this talk, I will present recent work on the intersection theory of modular correspondences. Specifically, we study the intersection numbers of modular correspondences of square-free level, focusing both on the nonsingular intersections and the...

In this talk, we continue to discuss the intersection of modular correspondences but on the self-product of two Shimura curves. This is partially motivated by a classical work of Kudla-Rapoport for the single Shimura curve case and by the self...

I will report on joint work with Aurel Page on a number/representation theoretic approach to the question "Can you hear the shape of a drum". We use quaternion algebras over number fields to construct pairs of manifolds that "sound the same", but...

The set of positive integer points of the celebrated Markov surface admits the structure of a 3-regular tree. My objective in this talk is to unveil a similar phenomenon for Mordell-Schinzel surfaces; namely that the set of positive integer points...

I will report on joint work in progress with Tim Browning and Ritabrata Munshi, concerning the use of the circle method to count rational points on a particular cubic hypersurface in six variables. This problem lies at the square-root barrier, and I...

Ellenberg-Jain-Venkatesh (2011) proposed a p-adic version of the Katz-Sarnak philosophy for usual L-functions: the statistics of zeroes of p-adic L-functions, averaged over families, should agree with the limiting eigenvalue statistics of p-adic...

Ostmann’s problem asks if there are sets $A1$ and $A2$ with $|A1|, |A2| > 1$ so that the sumset $A1 + A2$ differs from the set of primes by only finitely many elements. It is believed that no such $A1$ and $A2$ exist, but to date the problem remains...

The Chebotarev density theorem is a powerful tool in number theory, in part because it guarantees the existence of primes whose Frobenius lies in a given conjugacy class in a fixed Galois extension of number fields.  However, for some applications...

I will talk about my work in progress with Teruhisa Koshikawa defining a category of cuspidal sheaves on the moduli space of L-parameters for a p-adic group. Such a category is motivated by the categorical Langlands conjecture of Fargues and Scholze...

This is the opening talk of a learning seminar on geometric representation theory. The final goal of the seminar is to understand Bezrukavnikov's paper giving geometric realizations of the affine Hecke algebra, but we shall work through other topics...

We briefly review combinatorics and algebraic representation of reductive groups: root systems, Weyl groups and (extended) affine Weyl groups, weight spaces, and classification of irreducible representations in terms of highest weights. We define...

The main source for the works of Archimedes is a medieval manuscript, written originally around in 975 CE and then erased and covered with a prayerbook in the year 1229. Briefly studied in 1906, the full contents of the manuscripts were finally...