20.10.2021 - Caroline Turnage-Butterbaugh (Carleton College): Small gaps between zeros of the Riemann zeta-function
Abstract: Let g_1, g_2, ... denote the ordinates of the complex zeros of the Riemann zeta-function function in the upper half-plane in increasing order. The average distance between g_n and g_{n+1) is 2\pi / log g_n$ as n goes to infinity. An important goal is to prove unconditionally that these distances between consecutive zeros can be much, much smaller than the average for a positive proportion of zeros. We will discuss the motivation behind this endeavour, progress made assuming the Riemann Hypothesis, and recent work with A. Simonič and T. Trudgian to obtain an unconditional result that holds for a positive proportion of zeros.
05.10.2021 - Ziyang Gao (Leibniz University Hannover): A proof of the Uniform Mordell—Lang Conjecture
Abstract: Let A be an abelian variety and let X be a subvariety, both defined over the field of algebraic numbers. For any finite rank subgroup Gamma of the points of A over the algebraic numbers, the famous Mordell—Lang Conjecture predicts that each component of the intersection of X and Gamma is a coset of A. This conjecture is proved by Faltings and one also needs a result of Hindry to handle division points.
The Uniform Mordell—Lang Conjecture predicts that the number of irreducible components concerned above is bounded solely in terms of the dimension of A, the degree of X and the rank of Gamma. The question was posed by Mazur and David—Philippon. Recently this conjecture is proved in a series of work (Dimitrov—Gao—Habegger, Kühne; Gao—Ge—Kühne). In this talk I will report this proof. I will focus on the case of rational points on curves and then explain how to generalize this method to the general case. This is a joint project with Vesselin Dimitrov, Philipp Habegger; Tangli Ge, Lars Kühne.
28.09.2021 - Henrik Gustafsson (Chalmers/GU):
Abstract: Dirichlet series and L-functions in one complex variable have a long and fruitful history in number theory, but the theory of Dirichlet series in several complex variables (multiple Dirichlet series) is much less developed. In this talk I will review the historical development of so called Weyl group multiple Dirichlet series including their many different definitions that were worked out in parallel and how these were eventually all connected via metaplectic Whittaker functions. I will then discuss my recent research on metaplectic Whittaker functions which is joint with Ben Brubaker, Valentin Buciumas and Daniel Bump. If time permits, I will also briefly mention our related work on solvable lattice models describing these Whittaker functions.
15.09.2021 - Michael Griffin (Brigham Young University): Jensen polynomials of the Xi-function and other arithmetic sequences.
Abstract: Expanding on notes of Jensen, P\'olya proved that the Riemann Hypothesis is true if and only if a certain doubly-infinite family of polynomials associated to the Riemann Xi function are hyperbolic (i.e. all their roots are real). This property was proven for all Jensen polynomials of degree $d ≤ 3$ through careful analysis of the polynomials, and was later proven for all $d<2\cdot 10^{17}$ using the known height of zeros satisfying RH. In joint work with Ono, Rolen, and Zagier, we showed that for all positive $d$, the Jensen polynomials of fixed degree d are all eventually hyperbolic (as the second parameter tends to infinity). Moreover, in a follow up paper we, along with Thorner, Tripp and Wagner, made this statement effective. Our methods show that under a certain renormalization, the Jensen polynomials of degree d converge to the d-th hermite polynomial which are known to be hyperbolic and exhibit a regular distribution of their zeros. It turns out that the techniques we employed are far more general than originally thought. In particular, it is clear that knowledge of the limiting distributions of the Jensen polynomials of a given degree can tell us very little about the distribution of the zeros of the original function. Refinements to the theory must be made if we are to apply these tools effectively to such questions.
07.09.2021 - Oliver Leigh (Uppsala University): Automorphic forms from the banana threefold
Abstract: Gromov-Witten/Donaldson-Thomas invariants can provide fundamental information about Calabi-Yau threefolds. Moreover, the partition functions of these invariants have remarkable properties and relationships to other areas of mathematics. A key example of this is the link between automorphic forms and the GW/DT theory of elliptically-fibred threefolds. A specific highlight of this is the relationship between the Schoen threefold and χ₁₀, the Igusa cusp form of weight 10. In this talk I will discuss recent developments in the GW/DT theory of Bryan's banana threefold. A key point of this will be how the banana threefold leads to a natural refinement of χ₁₀.
02.06.2021 - Wushi Goldring (Stockholm University): Propagating algebraicity via functoriality
Abstract: We attempt -- and almost entirely succeed -- to classify the automorphic representations \pi of reductive groups over number fields, for which the algebraicity of the Hecke eigenvalues (or Satake parameters) is reducible via Langlands functoriality to cases already known for algebro-geometric reasons (if \pi is singular, the only current option being via the coherent cohomology of Shimura varieties). Historically, an important case -- for which algebraicity is still wide open -- is that of Maass forms of Galois-type: non-holomorphic eigenfunctions of the hyperbolic Laplacian on the upper half-plane with eigenvalue 1/4. Also, using functoriality to reduce to the cohomology of Shimura varieties has been applied successfully by many people to the construction of automorphic Galois representations, in cases where the algebraicity of the Hecke eigenvalues was already known (usually via the Betti cohomology of a locally symmetric space). However, as far as we know, no new case of algebraicity of Hecke eigenvalues had previously been established by reducing via functoriality to Shimura varieties (where we transfer \pi
"forward" to a larger group; looking at "special" \pi which descend to a smaller group "going backward" is a different matter).
In the positive direction, we give several examples of non-holomorphic automorphic forms which are superficially similar to Maass forms, but whose algebraicity does reduce to the coherent cohomology of Shimura varieties via either known or open cases of functoriality; in the known cases of functoriality we are also able to attach Galois
representations. We also introduce new notions of "D" and "M-algebraic", which generalize/refine the "L" and "C" of Buzzard-Gee and the "W-algebraic" of Patrikis, and we give examples of cuspidal, M-algebraic \pi which are "farther" from L-algebraic than previously considered yet still have algebraic Hecke eigenvalues, answering a question of
Patrikis. In the negative direction, we give a conceptual, group-theoretic explanation for why Maass forms and many other forms are not reducible to known cases via functoriality (so the sign error found by Taylor in a previous attempt on Maass forms was not coincidental, but rather necessary). There remains only a sliver of cases where it is still perhaps unclear whether or not reduction via functoriality is possible.
26.05.2021 - Ana Ros Camacho (Cardiff University): Computational aspects of orbifold equivalence
Abstract: Landau-Ginzburg models are a family of quantum field theories characterized by a polynomial (satisfying some conditions) usually called ‘potential’. Often appearing in mirror-symmetric phenomena, they can be collected in categories with nice properties that allow direct computations. In this context, it is possible to introduce an equivalence relation between two different potentials called `orbifold equivalence’. We will present some recent examples of this equivalence, and discuss the computational challenges posed by the search of new ones. Joint work with Timo Kluck.
25.05.2021 - Roger Howe (Yale University): Ranks and representations of classical groups over local and finite fields
(Time: 13.15-14.15, joint with Analysis and Probability)
Abstract: The relationship between harmonic analysis on a group and on it subgroups is a natural issue to study in representation theory. In representation theory of reductive algebraic groups, a key example of this is the study of representations induced from parabolic subgroups, which leads to the philosophy of cusp forms, to Harish-Chandra's Plancherel Formula, and to the classification of admissible representations.
Parabolic subgroups are relatively large and have complicated structure. It is worth asking whether fruitful relationships can be found between harmonic analysis on a reductive group and that on relatively simple minded subgroups. A case that has shed considerable light on representations of classical groups is to study the restriction of representations to abelian unipotent radicals. This leads by simple considerations to the idea of rank of representations, which has provided substantial information that is complementary to the approach via parabolic induction. This talk will review results about rank of representations for groups over local fields, and explain how some results can be extended to groups over finite fields.
19.05.2021 - Carl Wang Erickson (University of Pittsburgh): Small non-Gorenstein residually Eisenstein Hecke algebras
Abstract: In Mazur's work proving the torsion theorem for rational elliptic curves, he studied congruences between cusp forms and Eisenstein series in weight two and prime level. One of his innovations was to measure such congruences using a residually Eisenstein Hecke algebra. He asked for generalizations of his theory to squarefree levels. The speaker made progress toward such generalizations in joint work with Preston Wake; however, a crucial condition in their work was that the Hecke algebra be Gorenstein, which is often but by no means always true. We present joint work with Catherine Hsu and Preston Wake in which we study the smallest possible non-Gorenstein cases and leverage this smallness to draw an explicit link between its size and an invariant from algebraic number theory.
28.04.2021 - Chantal David (Concordia University): One-Level density for cubic characters over the Eisenstein field.
Abstract: We show that the one-level density for L-functions associated with the cubic residue symbols χn, with n ∈ Z[ω] square-free, satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in (−13/11, 13/11), under GRH. This is the first result extending the support outside the trivial range (−1, 1) for a family of cubic L-functions. This implies that a positive proportion of the L-functions associated with these characters do not vanish at the central point s = 1/2. A key ingredient is a bound on an average of generalized cubic Gauss sums at prime arguments, whose proof is based on the work of Heath-Brown and Patterson. Joint work with Ahmet M. Güloglu.
21.04.2021 - Wei Ho (University of Michigan): Splitting Brauer classes (Time: 16.30-17.30)
Abstract: Given a Brauer class over a field, what types of varieties split it? Or more geometrically, can we say anything about the varieties that map to a given Brauer-Severi variety? In this talk, we will discuss some open questions related to splitting Brauer classes. For example, we will review some classical algebro-geometric constructions that produce genus one curves splitting low index Brauer classes ((old) joint work with J. de Jong), and we will explain why a Brauer class of any index is split by a torsor under an abelian variety (joint work with M. Lieblich).
14.04.2021 - David Loeffler (University of Warwick): Euler systems and the Bloch--Kato conjecture
Abstract: The Bloch--Kato conjecture is a very general conjecture relating the properties of arithmetic objects over number fields to special values of L-functions, generalising a wealth of earlier theorems and conjectures such as the Birch--Swinnerton-Dyer conjecture for elliptic curves. I will explain a little about how this conjecture arises, and how it can be attacked using tools known as 'Euler systems'. In recent years there has been dramatic progress in the theory of Euler systems, and I'll explain recent work of Sarah Zerbes and myself from 2020, in which we use Euler systems to prove new cases of the Bloch--Kato conjecture for L-functions arising from the symplectic group GSp(4).
07.04.2021 - Kristian Holm (Chalmers/GU): A Central Limit Theorem for Symplectic Lattice Point Counting Functions
Abstract: We study a sequence of normalized counting functions on the space of 2d-dimensional symplectic lattices where d is at least 3. Using a combinatorial device introduced by Björklund and Gorodnik in order to estimate cumulants (alternating sums of moments), we prove that our sequence satisfies a central limit theorem. This is work in progress.
24.03.2021 - David Rydh (KTH): Stacky weighted blowups
Abstract: Many problems, such as resolution of singularities, are solved by blowing up smooth varieties along smooth centers. This is a very useful and explicit construction but sometimes a little restrictive. If we replace (smooth) varieties with (smooth) stacks/orbifolds, we obtain several new similar operations (root stacks, stacky blowups, weighted stacky blowups) that we can add to our toolbox. I will describe these new operations and some of their applications such as easier and more efficient algorithms for resolution of singularities (Abramovich--Temkin--Wlodarczyk) and weak factorization, as well as other applications such as \'etalification and destackification. There are also close relations to toric and logarithmic geometry.
This is partly joint work with Daniel Bergh and Ming Hao Quek.
09.03.2021 - Dmitry Gourevitch (Weizmann Institute of Science): Relations between Fourier coefficients of automorphic forms, with applications to vanishing and to Eulerianity
Abstract: In recent works with H. P. A. Gustafsson, A. Kleinschmidt, D. Persson, and S. Sahi, we found a way to express any automorphic form through its Fourier coefficients, using adelic integrals, period integrals and discrete summation – generalizing the Piatetski-Shapiro – Shalika decomposition for GL(n). I will explain the general idea behind our formulas, and illustrate it on examples. I will also show applications to vanishing and Eulerianity of Fourier coefficients.
03.03.2021 - Per Salberger (Chalmers/GU): On n-torsion in class groups of number fields
Abstract: It is well known that the class group of a number field is of size bounded above by roughly the square root of its discriminant. But one expects by conjectures of Cohen-Lenstra that the n-torsion part of this group should be much smaller and there have recently been several papers on this by prominent mathematicians. We present in our talk some new bounds which are sharper than those in the literature.
17.02.2021 - Olof Sisask (Stockholm University): Breaking the logarithmic barrier in Roth's theorem
Abstract: We present an improvement to Roth's theorem on arithmetic progressions, implying the first non-trivial case of a conjecture of Erdős: if a subset A of {1,2,3,...} is not too sparse, in that the sum of its reciprocals diverges, then A must contain infinitely many three-term arithmetic progressions. Although a problem in number theory and combinatorics on the surface, it turns out to have fascinating links with geometry, harmonic analysis and probability, and we shall aim to give something of a flavour of this.
10.02.2021 - Jiacheng Xia (Chalmers/GU): The unitary Kudla conjecture in the cases of norm-Euclidean imaginary quadratic fields
Abstract: Modularity conjectures lie at the heart of Kudla's program, which is a historic development of the earlier work of Hirzebruch--Zagier, Gross--Zagier, Gross--Keating, and Kudla--Millson.
The modularity conjecture for orthogonal Shimura varieties over Q was fully resolved by Bruinier--Raum in 2015, thanks to earlier work of Richard Borcherds and Wei Zhang. In this talk, I will explain how to use the method of Bruinier--Raum to show the modularity conjecture for certain unitary Shimura varieties based on a result of Yifeng Liu.
If time allows, I'll also talk about recent development of certain arithmetic applications, including the work of Li--Liu on arithmetic inner product formula, which is a natural analogue of the Gross--Zagier formula in higher dimension.
27.01.2021 - James Maynard (University of Oxford): Primes in arithmetic progressions beyond the Riemann Hypothesis
Abstract: Questions about the distribution of primes in an arithmetic progression are closely linked to the Generalized Riemann Hypothesis (GRH), which unfortunately appears out of reach. A very useful unconditional substitute for the GRH is the Bombieri-Vinogradov Theorem, which shows that the GRH is true 'on average'.
I'll talk about some recent results on primes in arithmetic progressions which goes beyond the Bombieri-Vinogradov Theorem, and corresponds to proving something stronger than the Riemann Hypothesis holds 'on average'.
20.01.2021 - Christian johansson (Chalmers/GU): Langlands correspondences in families
Abstract: I will survey some old and new ideas on "geometrizing" the Langlands correspondence, by viewing it as a sheaf of representations over a moduli space of Galois representations (or other "Langlands parameters"). Towards the end, I hope to briefly discuss some work in progress on p-adic endoscopy for SL(2) over totally real fields utilizing this perspective, which is joint with Judith Ludwig.
16.12.2020 - Julia Brandes (Chalmers/GU): Diophantine systems containing both diagonal and non-diagonal equations
Abstract: We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree k and one general form of degree d, where d is smaller than k. By employing a hybrid method that combines ideas from the study of general forms with techniques adapted to the diagonal case, we are able to obtain bounds that grow exponentially in d but only quadratically in k, reflecting the growth rates typically obtained for both problems separately. Time permitting, we may also discuss some of the most interesting generalisations of our approach. This is joint work with Scott T. Parsell.
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 Néron heights and the Mordell-Weil theorem we can count rational points of bounded height on elliptic curves. Faltings'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.