02.12.2020 - Kevin Buzzard (Imperial College London): Formalising algebra and geometry
Abstract: Formalising mathematics is the process of typing it into a computer proof verification system, which then checks that all proofs follow from the axioms of mathematics. I will spend some time talking about why one might want to do this. I will then attempt to dispel the old-fashioned idea that it takes 100 pages to prove 1+1=2 by showing that it is now possible for mathematics undergraduates to formalise MSc level algebra and geometry in modern systems. I will be using the Lean theorem prover, although other theorem provers are available.
18.11.2020 - Tim Browning (IST Austria): Poor bounds for a rich problem
Abstract: Thanks to Neron heights and the Mordell-Weil theorem we can count rational points of bounded height on elliptic curves. Falting's theorem tells us that there are only finitely many rational points (of any height) on hyperelliptic curves of genus at least 2. Unfortunately, these estimates depend intimately on the individual curve and it is hard to apply them to study the corresponding problem for surfaces that are fibred into elliptic or hyperelliptic curves. In this talk I shall discuss an alternative approach that uses sieves, though the bounds are far from optimal! This is joint work with Dante Bonolis.
04.11.2020 - John Christian Ottem (University of Oslo): Enriques surface fibrations of even index
Abstract: I will explain a geometric construction of an Enriques surface fibration over P1 of even index. This answers a question of Colliot-Thèlene and Voisin, and provides new counterexamples to the integral Hodge conjecture. This is joint work with Fumiaki Suzuki.
28.10.2020 - Scott Ahlgren (University of Illinois, Urbana-Champaign): OBS! 14.30-15.30
Title: Congruences for the partition function
Abstract: The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study for many years. Much of the interest (and the difficulty) in this problem arises from the fact that values of the partition function are given by coefficients of modular forms of half integral weight. I’ll briefly discuss the history of this problem, and focus mostly on some new joint work with Olivia Beckwith and Martin Raum which goes a long way towards explaining exactly when congruences can occur. The main tools are techniques from the theory of modular forms, Galois representations, and analytic number theory.
21.10.2020 - Peter Sarnak (Princeton University/Institute for Advanced Study): Applications of points on subvarieties of tori
Abstract: The intersection of the division group of a finitely generated subgroup of a torus with an algebraic subvariety has been understood for some time (Lang, Laurent,...). After a brief review of some of the tools in the analysis and their recent extensions (André-Oort conjectures), we give some old and new applications; in particular to the additive structure of the spectra of metric graphs and crystalline measures.
Joint work with P. Kurasov.
07.10.2020 - Jack Shotton (University of Durham): Moduli of local Galois representations and representation theory
Abstract: It is understood that there should be close connections between moduli spaces of representations of local Galois groups and modular representation theory of groups such as GL_n(Z_p). I will survey what is expected and what is known, and talk about some of my own work in the 'l \neq p' setting.
24.09.2020 - Jan Gerken (Chalmers/GU): Single-valued maps at genus zero and one
Abstract: The single-valued map for for multiple zeta values (MZVs) due to Francis Brown and Oliver Schnetz is an intriguing algebra homomorphism for MZVs. Surprisingly, it appears in the leading contribution to scattering amplitudes in string theory, relating open- and closed-string amplitudes. Since MZVs are periods of configuration spaces of punctured genus-zero Riemann surfaces, a natural next step is the generalization of the single-valued map to genus-one surfaces. In string theory, these correspond to the subleading contributions to the scattering amplitudes. Using the structures provided by string theory, we propose a genus-one generalization of the single-valued map for MZVs which acts on the level of generating functions of genus-one periods.
09.09.2020 - Lucile Devin (Chalmers/GU): Chebyshev’s bias and sums of two squares
Abstract: Studying the secondary terms of the Prime Number Theorem in Arithmetic Progressions, Chebyshev claimed that there are more prime numbers congruent to 3 modulo 4 than to 1 modulo 4. We will explain and qualify this claim following the framework of Rubinstein and Sarnak. Then we will see how this framework can be adapted to other questions on the distribution of prime numbers. This will be illustrated by a new Chebyshev-like claim : there are “more” prime numbers that can be written as a sum of two squares with the even square larger than the odd square than the other way around.
11.06.2020 - Jakob Palmkvist (Chalmers/GU): Tensor hierarchy algebras
Abstract: Tensor hierarchy algebras constitute a new class of non-contragredient Lie superalgebras, whose finite-dimensional members are the simple Lie superalgebras of Cartan type in Kac’s classification. They have proven useful in describing gauge structures in physical models related to string theory. I will review their construction by generators and relations and some of the remarkable features they exhibit.
20.05.2020 - Asbjörn Nordentoft (University of Copenhagen): Reciprocity Laws, Quantum Modular Forms and Additive Twists of Modular L-functions
Abstract: In an unpublished paper from 2007, Conrey discovered certain ‘reciprocity relations’ satisfied by twisted moments of Dirichlet L-functions, linking the arithmetics of the finite fields F_p, F_q for two different primes p,q (as is the case with quadratic reciprocity). In this talk I will discuss a generalization to twisted moments of twisted modular L-functions. This will lead to a discussion of the notion of quantum modular forms due to Zagier, and in particular we will explain that additive twists of modular L-functions define examples of quantum modular forms.
06.05.2020 - Martin Raum (Chalmers/GU): Divisibilities of class numbers and partition counts
Abstract: Hurwitz class numbers, class numbers of imaginary quadratic fields, and partition counts are among the most classic quantities in number theory, and for each of them their factorizations, i.e. divisibilities, are celebrated open questions. In the case of class numbers the Cohen-Lehnstra Heuristics provides predictions of of statistical nature. In the case of partition counts, Ramanujan congruences opened the door to a whole new research area in~1920.We survey recent progress on divisibilities of class numbers and partition counts on arithmetic progressions. These result rely on a two new methods exploiting the finer structure of Fourier coefficients of real-analytic and meromorphic modular forms.The project on class numbers is partially based on joint work with Olivia Beckwith and Olav Richter. The project on partition counts is partially based on joint work with Olivia Beckwith and Scott Ahlgren.
04.03.2020 - Jasmin Matz (University of Copenhagen): Distribution of Hecke eigenvalues
Abstract: There are many difficult conjectures about automorphic representations, many of which seem to be out of reach at the moment. It has therefore become increasingly popular to study instead families of automorphic representations and their statistical properties, which allows for additional analytic techniques to be used.In my talk I want to discuss the distribution of Hecke eigenvalues or, in other words, Satake parameters in the family of spherical unramified automorphic representations of split classical groups. We obtain an effective distribution of the Satake parameters, when we order the family according to the size of analytic conductor. This has applications to various questions in number theory, for example, low-lying zeros in families of automorphic L-functions, but also yields an effective Weyl law for the underlying locally symmetric space. This is joint work with T. Finis.
26.02.2020 - Kirsti Biggs (Chalmers/GU): Efficient congruencing in ellipsephic sets
Abstract: An ellipsephic set is a subset of the natural numbers whose elements have digital restrictions in some fixed prime base---for example, the set of positive integers whose digits in the given base are squares. Such sets have a fractal structure and can be viewed as p-adic Cantor sets analogous to those studied over the real numbers. The results of this talk can similarly be viewed from either a number theoretic or a harmonic analytic perspective: we bound the number of ellipsephic solutions to a system of diagonal equations, or, alternatively, we obtain discrete restriction estimates for the moment curve over ellipsephic sets. In this talk, I will outline the key ideas from the proof, which uses Wooley's efficient congruencing method, give motivating examples and highlight the importance of the additive structure of our ellipsephic sets.