# AGNT seminar 2022

Organizers: Anders Södergren, Christian Johansson.

Note​: For the time being, all upcoming seminars will be online. Please check the calendar

for information on how to join, or email one of the organizers. The standard time for the seminars in Spring 2022 is Tuesdays 15.15-16.15, but there might be exceptions to the "standard" time to accommodate speakers. Please check the calendar above or email one of the organizers for information about the time.

## Upcoming seminars​

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18.05.2022 - Alexander Stolin (Chalmers/GU)40 years of Lie bialgebras: From definition to classification

Abstract: The history of Lie bialgebras began with the paper where the Lie bialgebras were defined: V. G. Drinfeld, "Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations", Dokl. Akad. Nauk SSSR, 268:2 (1983) Presented: L.D. Faddeev. Received: 04.06.1982.

The aim of my talk is to celebrate 40 years of Lie bialgebras in mathematics and to explain how these important algebraic structures can be classified. This classification goes "hand in hand" with the classification of the so-called Manin triples, Drinfeld doubles also introduced in Drinfeld's paper cited above, and certain solutions of the classical Yang-Baxter equation (CYBE).

The ingenious idea how to classify Drinfeld doubles associated with Lie algebras possessing a root system is due to F. Montaner and E. Zelmanov. In particular, using their approach the speaker classified Lie bialgebras, Manin triples and Drinfeld doubles associated with a simple finite dimensional Lie algebra g (the paper was based on a private communication by E. Zelmanov and it was published in Comm. Alg. in 1999).

Further, in 2010, F. Montaner, E. Zelmanov and the speaker published a paper in Selecta Math., where they classified Drinfeld doubles on the Lie algebra of the formal Taylor power series gu and all Lie bialgebra structures on the polynomial Lie algebra g[u].

Finally, in March 2022 S. Maximov, E. Zelmanov and the speaker published an Arxive preprint, where they made a crucial progress towards a complete classification of Manin triples and Lie bialgebra structures on gu. The involved technique includes many different algebraic tools: from the theory of alternative algebras to  deep relations between Lie bialgebras, Manin triples and solutions of the CYBE with torsion free sheaves of Lie algebras on Weierstraß cubic curves (the latter is due to a new co-author, Raschid Abedin).

Of course, it is impossible to compress a 40 years history of the subject in one talk but the speaker will try his best to do this.

25.05.2022 - Andrés Chirre (NTNU Trondheim)Fourier analysis, Hilbert spaces, and non-vanishing of L-functions

Abstract: Given a family of L-functions, there has been a great deal of interest in estimating the proportion of the family that does not vanish at special points on the critical line. In particular, the case of the central point $s=1/2$ has been relevant in the last years. But, what happens with the zeros near the real axis? Generalizing a problem of Iwaniec, Luo, and Sarnak (2000), we address the problem of estimating the proportion of non-vanishing in a family of L-functions at a low-lying height on the critical line (measured by the analytic conductor). In this context, a new Fourier optimization problem emerges which we solve using the theory of reproducing kernel Hilbert spaces of entire functions. This is joint work with Emanuel Carneiro (ICTP) and Micah Milinovich (Mississippi).

02.06.2022 - Olivier Robert (Université Jean Monnet)Rational points on a intersection of diagonal forms

Abstract : We consider intersections of  diagonal forms with integer coefficients, of different degrees, and we prove an asymptotic formula for the number of rational points of  bounded height on these varieties. The proof uses the Hardy-Littlewood Method and recent breakthroughs on the Vinogradov system. We also give a sharper result for a specific set of degrees, using a technique due to Wooley and an estimate on exponential sums derived from a recent approach in the van der Corput method.  The results presented here are a joint work with Simon Boyer.

## Past seminars

10.05.2022 - Dan Petersen (Stockholm University)Hyperelliptic curves, the scanning map, and moments of quadratic L-functions

Abstract: I will explain a calculation of the stable cohomology of the hyperelliptic mapping class group with coefficients in an arbitrary symplectic representation. The result is closely related to, and provides a geometric interpretation of, a series of conjectures on asymptotics of moments of families of quadratic L-functions. (Joint with J. Bergström, A. Diaconu and C. Westerland).

03.05.2022 - Kathlén Kohn (KTH)Adjoint Hypersurfaces of Rational Polypols
Abstract: Eugene Wachspress introduced polypols as real bounded semialgebraic sets that generalize polytopes. He aimed to generalize barycentric coordinates from simplices to arbitrary polytopes and further to polypols. The central ingredient for his construction is the adjoint hypersurface of a rational polypol. We will discuss the geometric properties of adjoint hypersurfaces and their use in a variety of applications. For instance, adjoint curves of convex polygons are hyperbolic, and a general quartic plane curve is the adjoint of 864 heptagons. Adjoint hypersurfaces of polytopes play a key role in the computation of Segre classes of monomial schemes and when studying uniform probability distributions on polytopes. Finally, in the study of scattering amplitudes in physics, positive geometries are real semialgebraic sets together with a unique rational canonical form. We provide an explicit formula for the canonical form in terms of defining equations of the adjoint hypersurface and the facets of the positive geometry.
This talk is based on joint works with R. Piene, K. Ranestad, F. Rydell, B. Shapiro, R. Sinn,  M.-S. Sorea, B. Sturmfels, and S. Telen.

03.05.2022 - Orlando Marigliano (KTH)Classifying one-dimensional discrete statistical models with maximum
likelihood degree one

Abstract: A discrete statistical model with maximum likelihood degree one is a kind of subvariety of  the n-simplex of discrete probability distributions which is of special interest in algebraic statistics. This talk is about the problem of classifying these models in finite terms in the one-dimensional case. I describe an approach that gives such a finite description for n = 2,3,4 and makes progress toward solving the problem for greater n. I also introduce the combinatorial game of 'splitting chips on a grid' which crucially facilitates this strategy. This talk is based on joint work with Arthur Bik.

29.03.2022 - Dustin Clausen (University of Copenhagen)Hirzebruch-Riemann-Roch for rigid analytic varieties

Abstract: Given a compact complex manifold X and a holomorphic vector bundle V on X, the Hirzebruch-Riemann-Roch theorem, due in this generality to Atiyah-Singer, gives a formula for the holomorphic Euler characteristic of V in terms of topological invariants of the pair (X,V)  I will describe a new proof of this HRR theorem.  The proof is fairly simple and general, and in particular it also works in the context of rigid analytic varieties, where the statement was not known before.  This is joint work with Peter Scholze.

05.04.2022 - Jon Keating (University of Oxford)The Ratios Conjecture Over Function Fields

Abstract: I will talk about recent joint work with Hung Bui and Alexandra Florea in which we study the Ratios Conjecture for the family of quadratic L-functions over function fields. I will also discuss progress on the closely related problem of obtaining upper bounds for negative moments of L-functions, which allows us to obtain partial results towards the Ratios Conjecture in the case of one over one, two over two and three over three L-functions

15.03.2022 - Felix Janda (University of Notre Dame)Higher genus mirror symmetry for quintic threefolds

Abstract: Gromov-Witten invariants count maps from curves with specified genus and degree to a projective variety X. When X is a Calabi-Yau threefold, such as a quintic threefold, there are intriguing conjectures inspired from physics about the structure of the invariants.

In my talk, I will give an overview over the conjectures and the
algebraic geometry behind the (now standard) computation of the genus zero invariants of quintic threefolds, and explain why it does not easily extend to higher genus. I will then proceed to discuss a construction (joint with Q. Chen and Y. Ruan) of new moduli spaces that can control the failure of the naive approach. In joint work with S. Guo and Y. Ruan, we use them to prove some of the conjectures about the structure of Gromov-Witten invariants of quintic threefolds.

08.03.2022 - Magnus Carlson (Stockholm University)Arithmetic field theories with finite coefficients

Abstract: In this talk I will discuss arithmetic field theories with finite coefficients. Arithmetic field theories are analogous to field theories for 3-manifolds in the same sense number fields are analogous to 3-manifolds. I will start by describing the general idea behind what an arithmetic field theory is, first focusing on arithmetic Dijkgraaf-Witten theory. I will then proceed by defining arithmetic BF-theory and explain how this is a field theory that is non-trivial even in the non-orientable situation. I will give examples showing how path integrals can be calculated for these field theories and relate these path integrals to classical arithmetic invariants. I will also explain how one can define arithmetic field theories on number fields with a non-trivial boundary. This talk is based on joint work with Minhyong Kim.

01.03.2022 - Sarah Peluse (Princeton University/Institute for Advanced Study)Bounds for subsets of F_p^n \times F_p^n without L-shaped configurations

Abstract: I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemerédi’s theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting.

15.02.2022 - Kaisa Matomäki (University of Turku)Siegel zeros, twin primes, Goldbach's conjecture, and primes in short intervals

Abstract: The generalised Riemann hypothesis asserts that all non-trivial zeros of Dirichlet L-functions                    L(s, \chi) satisfy Re s = 1/2. However, for Dirichlet L-functions the known zero-free region is even weaker than for the Riemann zeta function. In particular we do not know how to rule out the possibility that, for a real character \chi, there exists a real zero \beta which is very close to 1.

Such exceptional zeros are called Siegel zeros. There has been lot of research concerning what would follow if Siegel zeros existed. In the talk I will describe how Siegel zeros are related to twin primes, Goldbach's conjecture, and primes in almost all very short intervals. In particular I will be talking about my joint work with Jori Merikoski.

01.02.2022 - Frank Thorne (University of South Carolina)Fourier Analysis in Arithmetic Statistics

Abstract: As demonstrated by Bhargava and many others, many counting questions in arithmetic statistics can be reduced to lattice point counting problems, which can then be solved by the geometry of numbers. The root of this technique is "Davenport's lemma", an elementary estimate for counting lattice points in nice regions.

What happens if one replaces Davenport's lemma with more efficient tools from Fourier analysis? I'll discuss some approaches (with Takashi Taniguchi, and with Theresa Anderson and Manjul Bhargava) to using Fourier analysis to obtain stronger quantitative results.

18.01.2022 - Jan Stevens (Chalmers/GU)Pointed curves with prescribed Weierstrass semigroup

Abstract: On a smooth projective curve each point has a Weierstrass gap sequence, and the non-gaps form a numerical semigroup. The space parametrising smooth pointed curves with given Weierstrass semigroup at the marked point is a locally closed subspace of the moduli space of pointed curves.

We provide a new lower bound for the dimension of this space. It is based on the
connection with the deformation theory of the monomial curve with the same semigroup. For certain families of curves this space is computed explicitly, showing that the lower bound is attained,  and that the corresponding moduli spaces are non-empty
and of pure dimension.

15.12.2021 - Hanneke Wiersema (University of Cambridge)Serre's modularity conjecture and its generalisations

Abstract: Serre’s modularity conjecture, now a theorem of Khare and Wintenberger, asserts a connection between Galois representations and modular forms. Since weights are a key invariant of modular forms it is natural to ask for a description of the (minimal) weight of the modular forms that correspond to any given Galois representation. In this talk we will focus on this question, known as the weight part of Serre's conjecture. In Serre’s original setting this is resolved, but we will also discuss recent work on generalisations of Serre's conjecture where much less is known.​

08.12.2021 - Alex Kontorovich (Rutgers University)Asymptotic Length Saturation for Zariski Dense Surfaces

Abstract: The lengths of closed geodesics on a hyperbolic manifold are determined by the traces of its fundamental group. When the latter is a Zariski dense subgroup of an arithmetic group, the trace set is contained in the ring of integers of a number field, and may have some local obstructions. We say that the surface's length set "saturates" (resp. "asymptotically saturates") if every (resp. almost every) sufficiently large admissible trace appears. In joint work with Xin Zhang, we prove the first instance of asymptotic length saturation for punctured covers of the modular surface, in the full range of critical exponent exceeding one-half (below which saturation is impossible).

01.12.2021 - Jordan Ellenberg (University of Wisconsin-Madison)Sparsity of rational points on moduli spaces of varieties

Abstract:  I will talk about a recent result with Lawrence and Venkatesh

which shows, among other things, that the number of isomorphism classes of degree-n homogeneous forms over Z of some fixed discriminant k and with all coordinates at most B grows more slowly than any power of B.  This result is not uniform in k (i.e. we do not have a single bound which holds for all k) but uniform bounds on rational points (in this case, a result of Broberg generalizing a result of Heath-Brown) are critical for the proof; I will explain this, as a case study for why uniform bounds are very useful in practice.

23.11.2021 - Shucheng Yu (Uppsala University)The light cone Siegel transform, its moment formulas and applications

Abstract: The classical Siegel transform is a transform which takes functions on the Euclidean space to functions on the space of lattices. In this talk I will discuss a new type of Siegel transform where the role of the Euclidean space is replaced by the light cone of a certain indefinite integral quadratic form. In this setting one can use the spectral theory of incomplete Eisenstein series to prove explicit first and second moment formulas for this transform, generalizing the classical results of Siegel and Rogers. I'll also discuss some applications of our moment formula to various counting problems, including one on intrinsic Diophantine approximations on spheres. This is work in progress with Dubi Kelmer.​

16.11.2021 - Lucile Devin (Université du Littoral Côte d'Opale)Lemke Oliver and Soundararajan bias for sums of two squares

Abstract: Recently Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy--Littlewood conjecture containing a large secondary term, which fits the data very well. We discuss the analogous question for consecutive pairs of sums of squares in arithmetic progressions, a bias also appears in the experimental data, and we develop a similar heuristic model based on the Hardy--Littlewood conjecture for sums of squares to explain it. This is joint work with Chantal David, Jungbae Nam and Jeremy Schlitt.

09.11.2021 - Gerard Freixas i Montplet (IMJ-PRG)Complex Chern-Simons and the first tautological class

Abstract: In this talk I will propose a construction of the complex Chern-Simons line bundle, in the context of a family of compact Riemann surfaces and  a relative moduli space of flat vector bundles on it. The construction is inspired by Deligne’s functorial interpretation of Arakelov geometry, where direct images of characteristic classes of hermitian vector bundles are lifted to the level of hermitian line bundles. In our setting, hermitian metrics are replaced by flat relative connections, and some ideas from non-abelian Hodge theory are fundamental in the approach. We will discuss some properties of the complex Chern-Simons line bundle, and as an application we will present a new proof of Wolpert’s result to the effect that the first tautological class of the moduli space of curves is represented by the Weil-Petersson form. This is joint work with Dennis Eriksson and Richard Wentworth.

26.10.2021 - Tobias Magnusson (Chalmers/GU)Numerical Evaluation of Holomorphic Eichler Integrals via Generalized Second Order Modular Forms

Abstract: Holomorphic Eichler integrals occur as the simplest case of iterated Eichler-Shimura integrals. Their numeric values appear in the experimental study of path integrals associated with Feynman diagrams. In this talk, we describe how to efficiently evaluate holomorphic Eichler integrals. We express them as linear combinations of products of generalized second order Eisenstein series, whose evaluation is a significantly simpler task. Generalized second order modular forms have their origins in earlier work by Mertens-Raum, Chinta-Diamantis-O'Sullivan, and Goldfeld. This project is joint work with Albin Ahlbäck and Martin Raum.

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.​

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.

12.02.2020 - Kevin Hughes (University of Bristol): Discrete restriction to the curve (x,x^3)

Abstract: In this talk I will motivate the problem of discrete restriction to the curve (x,x^3). This is one of the simplest cases outside the recently introduced and powerful machinery of decoupling and efficient congruencing. While the expected 10th decoupling inequality fails, we show that the nigh-optimal discrete restriction estimate holds. This is work with Trevor Wooley.

19.02.2020 - Nils Matthes (University of Oxford): Motivic periods

Abstract: A period is a complex number which can be written as the integral of an algebraic differential form over a semialgebraic set. This is a classical notion whose roots can be traced back at least to Euler and which conjecturally contains all special values of L-functions of algebraic varieties. Beginning in the 1960s it was realized that the study of periods may be viewed as part of Grothendieck's vision of motives which very recently lead to the notion of "motivic period". Although progress has been made, many fundamental questions about (motivic) periods remain.

05.02.2020 - Pankaj Vishe (Durham University): Rational points on complete intersections over global fields

Abstract: The quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics over the function field F_q(t) is obtained, under the assumption that q is odd and n9. The main ingredient here is the development of a Kloosterman refinement over global fields.

28.01.2020 - Jonas Bergström (Stockholm University): Traces of Hecke operators on spaces of Siegel modular forms modulo prime powers

Abstract: I will report on ongoing work, where I apply the Lefschetz fixed point theorem to local systems on the moduli space of abelian varieties of dimension at most 3, and use simple equalities in modular arithmetic, to study traces of Hecke operators on spaces of Siegel modular forms (of degree at most 3) modulo prime powers.

15.01.2020 - Dennis Eriksson (Chalmers/GU): Genus one mirror symmetry

Abstract: Mirror symmetry, in a crude formulation, is usually presented as a correspondence between curve counting on a Calabi--Yau variety X, and some invariants extracted from a mirror family of Calabi--Yau varieties. After the physicists Bershadsky--Cecotti--Ooguri--Vafa (henceforth BCOV), this is organised according to the genus of the curves in X we wish to enumerate, and gives rise to an infinite recurrence of differential equations. In this talk, I will give a general introduction to these problems, and present a rigorous mathematical formulation of the BCOV conjecture at genus one, in terms of a lifting of the Grothendieck--Riemann--Roch. I will explain a proof of the conjecture for Calabi--Yau hypersurfaces in projective space, based on the Riemann--Roch theorem in Arakelov geometry. Our results generalise from dimension 3 to arbitrary dimensions previous work of Fang--Lu--Yoshikawa.
This is joint work with G. Freixas and C. Mourougane.

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Page manager Published: Mon 16 May 2022.