AGNT seminar 2018

Organizers: Anders Södergren, Christian Johansson.


Past seminars​

18.12.2018 - Henrik Gustafsson (Stanford): Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker functions

Abstract: In this talk, based on joint work with Ben Brubaker, Valentin Buciumas and Daniel Bump, I will explain new connections relating metaplectic Whittaker functions and certain solvable lattice models with operators on a q-deformed fermionic Fock space. We will discuss the locality properties of these operators which agree with those of a q-deformed vertex operator algebra, and review the underlying quantum groups that are part of the above connections. In the process we also obtain a Fock space operator description of ribbon symmetric functions, or LLT polynomials, introduced by Lascoux, Leclerc and Thibon. 

11.12.2018 - Arash Ardehali (Uppsala): Microscopic structure of black holes from asymptotic analysis of special functions.

Abstract: After a concise introduction to quantum field theory from a combinatorial perspective, it will be explained how ideas from analytic combinatorics are helping to extract information about the microscopic structure of black holes via asymptotic analysis of various quantum field theory generating functions.


28.11.2018 - Federico Zerbini (IPhT): A class of non-holomorphic modular forms from string theory.

Abstract: Modular graph functions were discovered by physicists studying genus-one string-theory amplitudes. They generalize special values of non-holomorphic Eisenstein series, and they display striking properties, which led to conjecture that they should be related to iterated Eichler integrals of Eisenstein series. In this talk we will introduce modular graph functions and iterated Eichler integrals, and describe the state of the art.

21.11.2018 - Brandon Williams (TU Darmstadt): Hilbert modular forms and Borcherds products.

Abstract: I will talk about how Borcherds products can be used to compute graded rings of Hilbert modular forms.

14.11.2018 - Pär Kurlberg (KTH): Class numbers and class groups for definite binary quadratic forms.

Abstract: Gauss made the remarkable discovery that the set of integral binary quadratic forms of fixed discriminant carries a composition law, i.e., two forms can be "glued together" into a third form.  Moreover, as two quadratic forms related to each other via an integral linear change of variables can be viewed as equivalent, it is natural to consider equivalence classes of quadratic forms. Amazingly, Gauss' composition law makes these equivalence classes into a finite abelian group - in a sense it is the first abstract group "found in nature". Extensive calculations led Gauss and others to conjecture that the number h(d) of equivalence classes of such forms of negative discriminant d tends to infinity with |d|, and that the class number is h(d)=1 in exactly 13 cases: d is in { -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163 }.  While this was known assuming the Generalized Riemann Hypothesis, it was only in the 1960's that the problem was solved by Alan Baker and by Harold Stark. We will outline the resolution of Gauss' class number one problem and survey some known results regarding the growth of h(d).  We will also consider some recent conjectures regarding how often a fixed abelian group occur as a class group, and how often an integer occurs as a class number.  In particular: do all abelian groups occur, or are there "missing" class groups?

31.10.2018 - Martín Sombra (ICREA & Universitat de Barcelona): The zero set of the independence polynomial of a graph.

Abstract: In statistical mechanics, the independence polynomial of a graph $G$ arises as the partition function of the hard-core lattice gas model on $G$. The distribution of the zeros of these polynomials when $G \to \infty$ is relevant for the study of this model and, in particular, for the determination of its phase transitions.  In this talk, I will review the known results on the location of these zeros, with emphasis on the case of rooted regular trees of fixed degree and varying depth $k \ge 0$. Our main result states that for these graphs, the zero sets of their independence polynomials converge as $k \to \infty$ to the bifurcation measure of a certain family of dynamical systems on the Riemann sphere.​ This is ongoing work with Juan Rivera-Letelier (Rochester).

10.10.2018 - Jonathan Nilsson (Chalmers/GU): Representation theory for Lie algebras of vector fields.

Abstract: Let A be the algebra of polynomial functions on a fixed affine algebraic variety. The Lie algebra V of polynomial vector fields on the variety is isomorphic to the Lie algebra of derivations of A. This Lie algebra does generically not admit a Cartan subalgebra, so the classical root-weight approach to representation theory fails here. Instead I will discuss some new classes of modules that have the property that they admit compatible A- and V-module structures. To concretize the general theory I will also show explicitly what happens in a simple nontrivial case, namely when the variety is the 2-sphere. The talk is based on joint work with Yuly Billig and Vyacheslav Futorny.

26.09.2018 - Christian Johansson (Chalmers/GU): Mod p completed cohomology of locally symmetric varieties.

Abstract: This talk is about singular cohomology of (certain) manifolds, with coefficients in F_p (p some fixed prime). More precisely, the manifolds in question are certain locally symmetric varieties. I will discuss how one can prove that the direct limit of such cohomology groups vanish above the middle degree as one goes up a certain tower of locally symmetric varieties. This confirms many cases of a conjecture of Calegari and Emerton, which is of central importance in the p-adic part of Langlands program. This is joint work in progress with David Hansen.

14.09.2018, Rainer Dietmann (Royal Holloway, University of London): Lines on cubic hypersurfaces.

Abstract: Wooley has shown that every rational cubic hypersurface defined by a cubic form in at least 37 variables contains a rational line. In joint work with Julia Brandes we were able to improve this in the generic case of non-singular cubic forms, reducing the required number of variables to 31, applying recent work of Browning, Dietmann and Heath-Brown on intersections of cubic and quadric hypersurfaces. Time permitting, we also briefly want to discuss the related problem of finding lines on cubic hypersurfaces defined over p-adic fields.​

12.09.2018 - Daniele Casazza (ICMAT, Spain): On the factorization of p-adic L-series.

Abstract: In this talk we present the basic ideas concerning p-adic L-functions and how they can be used to deduce interesting arithmetic information. After a general overview and the presentation of some notable example, we will focus more in detail on the case of interest for us. We will explain how one can construct a new p-adic L-function using Saito-Kurokawa lifts and use it to deduce some interesting arithmetic information.​

16.08.2018 - Yoonbok Lee (Incheon National University): The a-values of the Riemann zeta function near the critical line.

Abstract: We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $, $T \leq \Im s \leq 2T$ for fixed $h_1, h_2>0$ and $ 0 < \theta <1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwill's recent results on the discrepancy between the distribution of $\zeta(s)$ and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line $\Re s = 1/2 + 1/(\log T)^\theta $. This is a joint work with Junsoo Ha in KIAS.

31.05.2018 - Chunhui Liu (Kyoto University): Counting rational points in arithmetic varieties by the determinant method​.

Abstract: By the slope method in Arakelov geometry, we can construct a family of hypersurfaces which cover the rational points of bounded height on an arithmetic variety but don't contain the generic point of this variety. By estimating some invariants of Arakelov geometry, we can control the number and the maximal degree of this family of auxiliary hypersurfaces explicitly. In this talk, I will explain the method of studying the problem of counting rational points by the approach of Arakelov geometry.

29.05.2018 - Manish M. Patnaik (University of Alberta):
 Metaplectic covers of Kac-Moody groups and Whittaker functions.

Abstract: We will describe how to construct metaplectic covers of Kac-Moody groups, generalizing a classical construction of Matsumoto. In the case of non-archimedean local fields, we will then explain how to formulate Whittaker functions on such groups, and compute them via a Casselman-Shalika type formula. Joint work with A. Puskas.

- Will Sawin (ETH-ITS):
The circle method and free rational curves on hypersurfaces.

Abstract: In joint work in progress with Tim Browning, we study a system of two of Diophantine equations over F_q[t], and use the circle method to show a relationship between their numbers of solutions. As a consequence, we bound the dimension of the singular locus of the moduli space of rational curves on a smooth projective hypersurface. I will explain how these problems are related and what techniques we use to get the best bound.


26.04.2018 - Sofia Tirabassi (University of Bergen): Fourier--Mukai partners of Enriques and bielliptic surfaces in positive characteristic.

Abstract: We show that Enriques and bielliptic surfaces in positive characteristic do not have any non-trivial Fourier--Mukai partners. This is joint work with K. Honigs and M. Lieblich.


24.04.2018 - Wanmin Liu (IBS Center for Geometry and Physics): Classification of full exceptional collections of line bundles on three blow-ups of P^3.

Abstract: A fullness conjecture of Kuznetsov says that if a smooth projective variety X admits a full exceptional collection of line bundles of length l, then any exceptional collection of line bundles of length l is full. In this talk, we show that this conjecture holds for X as the blow-up of P^3 at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on X is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such X. This is a joint work with Song Yang and Xun Yu at Tianjin University.​

12.04.2018 - Dennis Eriksson (Chalmers/GU): Transfinite diameters in the Berkovich setting.

Abstract: The classical transfinite diameter is a notion of size of a compact set in the complex plane, given by a well-defined limiting process. It generalizes the notion of diameter of a disc. The definition and existence of the limit were further generalized and studied on more general complex manifolds by Berman-Boucksom. In this talk, I will explain how these results can be extended to the non-archimedean/Berkovich setting. A key ingredient is given by Knudsen-Mumfords version of the Grothendieck-Riemann-Roch theorem. This is joint work with Sébastien Boucksom.

23.03.2018 - Sebastián Herrero (Chalmers/GU): A Jensen–Rohrlich type formula for the hyperbolic 3-space.

Abstract: The classical Jensen’s formula is a well-known theorem of complex analysis which characterizes, for a meromorphic function f on the unit disc, the value of the integral of log|f(z)| on the unit circle in terms of the zeros and poles of f inside the unit disc. An important theorem of Rohrlich establishes a version of Jensen’s formula for modular functions f with respect to the full modular group PSL_2(Z) and expresses the integral of log|f(z)| over the corresponding modular curve in terms of special values of Dedekind’s eta function. 
In this talk I will present a Jensen–Rohrlich type formula for certain family of functions defined in the hyperbolic 3-space which are automorphic for the group PSL_2(O_K) where O_K denotes the ring of integers of an imaginary quadratic field. This is joint work with Ö. Imamoglu (ETH Zurich), A.-M. von Pippich (TU Darmstadt) and Á. Tóth (Eotvos Lorand Univ.).​

09.03.2018 - Julia Brandes (Chalmers/GU): Optimal mean value estimates beyond Vinogradov's mean value theorem.

Abstract: Mean values for exponential sums play a central role in the study of diophantine equations. In particular, strong upper bounds for such mean values control the number of integer solutions of the corresponding systems of diagonal equations. Since the groundbreaking resolution of Vinogradov's mean value theorem by Wooley and Bourgain, Demeter and Guth, we can now prove optimal upper bounds for mean values connected to translation-dilation-invariant systems. This has inspired Wooley's call for a "Big Theory of Everything", a challenge to establish optimal mean value estimates for any mean values associated with systems of diagonal equations.
We establish optimal bounds for a family of mean values that are not of Vinogradov type. This is the first time bounds of this quality have been obtained for non-translation-dilation-invariant systems. As a consequence, we establish the analytic Hasse principle for the number of solutions of certain systems of quadratic and cubic equations in fewer variables than hitherto thought necessary. This is joint work with Trevor Wooley.​

23.02.2018 - Jakob Palmkvist (Chalmers/GU): Generators and relations for (generalized) Cartan superalgebras.

Abstract: In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(0,n-1)=sl(1|n) can be constructed by adding a "gray'' node to the Dynkin diagram of A_{n-1}=sl(n), corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W(n), the derivation algebra of the Grassmann algebra on n generators. I will in my talk present a novel construction of W(n), from the same Dynkin diagram as A(0,n-1), but with additional generators and relations. I will then generalize this result to the exceptional Lie algebras E_n, which can be extended to infinite-dimensional Borcherds superalgebras, in the same way as A_{n-1} can be extended to A(0,n-1). In this case, the construction leads to so called tensor hierarchy algebras, which seem to provide an underlying algebraic structure of certain supergravity models.
08.02.2018 - Håkan Granath (Stockholms universitet): Heun functions and quaternionic modular forms.

Abstract: The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic (2,2,2,3) quadrilateral group.  Hence its uniformization is related to a class of special functions called Heun functions.  In the talk I will give an introduction to these concepts, present the differential structure of the ring of modular forms for the Shimura curve, and show how one can relate the ring generators to explicit Heun functions for the quadrilateral group.  Furthermore I will describe how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. As an application of these explicit identifications, I will describe how the exceptional sets of the associated Heun functions can be determined, and, time permitting, say a few words on how exceptional values can be computed. The talk is based on joint work with Srinath Baba.

- Martin Raum (Chalmers/GU): Unifying relaxed notions of modular forms.

Abstract: Elliptic modular forms are functions on the complex upper half plane that are invariant under a certain action of the special linear group with integer entries. Their history comprises close to two centuries of amazing discoveries and application: The proof of Fermat's Last Theorem is probably the most famous; The theory of theta functions is among its most frequently employed parts. During the past decade it has been à la mode to study relaxed notions of modularity. Relevant keywords that we will discuss are mock modular forms and higher order modular forms. We have witnessed their application, equally stunning as surprising, to conformal field theory, string theory, combinatorics, and many more areas. In this talk, we suggest a change of perspective on such generalizations. Most of the novel variants of modular forms (with one prominent exception) can be viewed as components of vector-valued modular forms. This unification draws its charm from the past and the future. On the one hand, we integrate results by Kuga and Shimura that hitherto seemed almost forgotten. On the other hand, we can point out connections, for example, between mock modular forms and so-called iterated integrals that have not yet been noticed. This is joint work with Michael Mertens.

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