23.11.2022 - Pär Kurlberg (KTH): Poisson spacings for lattice points on circles
Abstract: We will investigate the distribution of Z^2-lattice points lying on circles. Along a density one subsequence the angles of lattice points on circles are known to be uniformly distributed as the radius tends to infinity; in fact the angles are "very well distributed" in the sense of the discrepancy being *lower* than that of a random collection of points. A refined question is how lattice points are spaced at the local scale, i.e., when rescaled so that the mean spacing is one. I will discuss recent joint work with Steve Lester in which we show that the local spacing statistics are Poissonian along a density one subsequence of admissible radii.
30.11.2022 - Daniel Persson (Chalmers/GU): Toward minimal automorphic representations of Kac-Moody groups
Abstract: Automorphic forms attached to certain “minimal” representations of Lie groups play a central role in many different contexts, both in mathematics and in physics. In particular, they appear in theta correspondences as well as in string amplitudes. Such minimal automorphic forms are characterized by having very few non-vanishing Fourier coefficients. In this talk I will discuss results and conjectures that suggest the existence of minimal automorphic representations on Kac-Moody groups (infinite-dimensional generalizations of Lie groups). Along the way I will provide the necessary background on automorphic forms and Kac-Moody groups.
09.11.2022 - Sofia Tirabassi (Stockholm University): Effective Characterization of Quasi-Abelian Surfaces
Abstract: We prove a quasi-projective analogue of Enriques's characteriation theorem for abelian surfaces. This is a joint work with M. Mendes Lopes and R. Pardini
26.10.2022 - Daniel Fiorilli (Université Paris-Saclay): Omega results for cubic field counts via the Katz-Sarnak philosophy
Abstract: I will discuss recent joint work with P. Cho, Y. Lee and A. Södergren. Since the results of Davenport-Heilbronn, much work has been done to obtain a precise estimate for the number of cubic fields of discriminant at most X. This includes work of Belabas-Bhargava-Pomerance, Bhargava-Shankar-Tsimerman, Taniguchi-Thorne and Bhargava-Taniguchi-Thorne. In this talk I will present a negative result, which states that the GRH implies that the error term in this estimate cannot be too small. Our approach involves low-lying zeros of Dedekind zeta functions of cubic fields (first studied by Yang), and is strongly related to the Katz-Sarnak conjectures and the ratios conjecture of Conrey, Farmer and Zirnbauer. I will also present numerical support for our result.
12.10.2022 - Ioanna Motschan-Armen (Chalmers/GU): Classification of quartic plane Cremona maps with a maximum of two proper base points.
Abstract: A plane Cremona map is a birational map between two projective planes. Its base points can lie on the projective plane or in a surface obtained by blow-ups. The first ones are called proper base points. Birational maps can be classified in families such that they are, up to change of coordinates, not equivalent to another. The aim of my master thesis was to classify Cremona maps of degree four with one and two proper base points. The results were 3 types of quartic plane Cremona maps with one proper base point and 22 types with two proper base points. In this talk, I will give an introduction to Cremona maps and the procedure of finding the types of Cremona maps using Enriques diagram and associating Cremona maps to linear systems.
05.10.2022 - Valentijn Karemaker (Utrecht University): The Gauss problem for central leaves.
Abstract: For a family of finite sets whose cardinalities are naturally called class numbers, the Gauss problem asks to determine the subfamily in which every member has class number one. We study the Siegel moduli space of abelian varieties in characteristic p and solve the Gauss problem for the family of central leaves, which are the loci consisting of points whose associated p-divisible groups are isomorphic. Our solution involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus 4. This geometric Gauss problem is closely related to an arithmetic Gauss problem for genera of positive-definite quaternion Hermitian forms, which we also solve.
16.09.2022 - Kristian Holm (Chalmers/GU): 1-Level Density for Zeros of Hecke L-functions of Imaginary Quadratic Number Fields of Class Number 1
Abstract: If K is an imaginary quadratic number field of class number 1 and O its ring of integers, we study a natural family of Hecke L-functions associated to angular characters on the non-zero ideals of O. Using the powerful Ratios Conjecture due to Conrey, Farmer, and Zirnbauer, we compute a conditional asymptotic for the average 1-level density of the zeros of this family, including terms of lower order than the (typical) main term in the Katz-Sarnak Density Conjecture coming from random matrix theory. In the talk I will discuss this conditional result and an unconditional asymptotic for the mentioned 1-level density as well.
23.09.2022 - Adam Keilthy (Chalmers/GU): An introduction to motivic periods and applications to multiple zeta values
Abstract: Multiple zeta values are examples of a class of numbers called periods: integrals of rational functions over rationally defined domains. Period computations arise very naturally across mathematics, including as Feynman amplitudes and as statistics in decidability problems, but comparing them is challenging in general. However, by lifting to formal analogues known as motivic periods, we obtain a much more rigid algebraic structure, including a weight grading and a coproduct, that can be used to easily establish algebraic relations among periods. We briefly introduce this formalism and discuss how it can be applied to establish new families of relations among multiple zeta values. In particular, we hope to discuss a recent result showing that the period polynomial relations arise as a consequence of a natural symmetry of the system.
07.09.2022 - Patrick Meisner (Chalmers/GU): Lower Order Terms in the Katz-Sarnak Philosophy
Abstract: The Katz-Sarnak philosophy states that for nice families of L-functions defined over F_q[T], the Frobenii should become equidistributed in a compact matrix Lie group as q tends to infinity. However, in the case that the matrix Lie group is the unitary group, these statistics become uninteresting. In this talk I will discuss statistics of higher order dirichlet character and, in particular, find lower order terms which vanish as q tends to infinity. Further, I will discuss the implications of these lower order terms to moments of L functions away from the critical point.
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.
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).
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 g[u] 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 g[u]. 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.
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.