Abstracts of the talks
Margaret Bilu (Institut de Mathématiques de Bordeaux)
Title: A motivic circle method
Abstract: The Hardy–Littlewood circle method is a well-known technique of analytic number theory that has successfully solved several major number theory problems. In particular, it has been instrumental in the study of rational points on hypersurfaces of low degree. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces. I will report on joint work with Tim Browning on implementing a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties. We establish analogues for the key steps of the method, enabling us to approximate the classes of moduli spaces of rational curves on hypersurfaces directly, without recourse to point counting, leading thus to more precise results about their geometry.
Nero Budur (KU Leuven)
Title: Contact loci of arcs
Abstract: Contact loci are sets of arcs on a smooth variety with prescribed contact order along a fixed hypersurface. They can be regarded as the building blocks of motivic integration. I give an overview of recent results about the topology of the contact loci.
Francesca Carocci (EPFL)
Title: BPS invariant from non Archimedean integrals
Abstract: We consider moduli spaces of one-dimensional semistable sheaves on del Pezzo and K3 surfaces supported on ample curve classes.
Working over a non-archimedean local field F, we define a natural measure on the F-points of such moduli spaces. We prove that the integral of a certain naturally defined gerbe on the moduli spaces with respect to this measure is independent of the Euler characteristic.
Analogous statements hold for (meromorphic or not) Higgs bundles.
Recent results of Maulik-Shen and Kinjo-Coseki imply that these integrals compute the BPS invariants for the del Pezzo case and for Higgs bundles.
This is a joint work with Giulio Orecchia and Dimitri Wyss.
Antoine Ducros (Sorbonne Université)
Title: Stratification of the image of a map between analytic spaces
Abstract: Let f : Y—>X be a morphism between compact Berkovich spaces over an arbitrary non-Archimedean field. In general, the structure of the image f(Y) appears to be rather mysterious (there is no Chevalley theorem in this setting). Nevertheless, one can use recent flattening results in non-Archimedean geometry to show that f(Y) admits a nice finite stratification with reasonable pieces (each of them is a Zariski-closed subset of an analytic domain of X).
Itay Glazer (Northwestern)
Title: Small ball estimates for polynomial maps over local fields, and some applications
Abstract: Given a local field F, and a polynomial map f:F^n--->F^m, it is natural to ask, for each small \delta>0:
Question 1: What is the probability that f(x) is contained in a ball of radius \delta around y in F^m, where x is a random vector in the unit ball in F^n?
Question 1 and its variants are studied in various mathematical fields, such as singularity theory, group theory, random matrix theory, high-dimensional convex geometry, and algebraic combinatorics.
Here is a group-theoretic variant of Question 1; consider, for each n, the polynomial map w:SL_n^2--->SL_n which takes a pair of n by n matrices A,B of determinant 1, and sends them to their commutator ABA^(-1)B^(-1). We can then ask:
Question 2: What is the probability that for two random matrices A and B in SL_n(Z_p), their commutator ABA^(-1)B^(-1) lies in a p-adic ball of radius p^(-k) around a matrix C? Can we give estimates which are uniform in n and p?
In this talk I will discuss the two questions above and their variants, as well the connection to the singularities of those polynomial maps.
Based on joint projects with Yotam Hendel and with Yotam Hendel and Raf Cluckers.
Julia Gordon (University of British Columbia)
Title: Coefficients of the Harish-Chandra expansion are motivic (Online)
Abstract: Let G be a connected reductive group over a non-Archimedean local field.
Harish-Chandras local character expansion expresses the character of a representation of G, restricted to the neighbourhood of the identity element, as a linear combination of Fourier transforms of the nilpotent orbital integrals. The coefficients of this expansion are known to be rational, in many cases. We prove that in fact in these cases they are motivic functions of the parameters defining the `germs at the identity of the irreducible representations; this allows us to get some uniform in p results for families of complex representations of p-adic groups. This is joint work with L.Spice and T.C. Hales.
Yotam Hendel (Université de Lille)
Title: Uniform number theoretic estimates for fibers of algebraic morphisms over finite rings of the form Z/p^kZ
Abstract: Let f:X \to Y be a morphism between smooth, geometrically irreducible Z-schemes of finite type.
We study the number of solutions N_{p,k,y}:=#{x \in X(Z/p^kZ) :f(x)=y mod p^k} for prime p, positive number k, and y \in Y(Z/p^kZ), and show that the geometry and singularities of the fibers of f determine the asymptotic behavior of this quantity as p, k and y vary.
In particular, we show that f:X \to Y is flat with fibers of rational singularities, a property abbreviated (FRS), if and only if p^{-k(\dim X -\dim Y)}N_{p,k,y} is uniformly bounded in p, k and y. We then consider a natural family of singularity properties, which are variants of the (FRS) property, and provide for each member of this family a number theoretic characterization using the asymptotics of p^{-k(\dim X -\dim Y)} N_{p,k,y}.
To prove our results, we use the theory of motivic integration (in the sense of uniform p-adic integration).
Based on a joint work with Raf Cluckers and Itay Glazer.
François Loeser (Sorbonne Université) and Vincent Jinhe Ye (Sorbonne Université)
Title: A finiteness theorem for tropical functions on skeleta I and II
Speakers: F. Loeser (Talk I) and J. Ye (Talk II)
Abstract: Skeleta are piecewise-linear subsets of Berkovich spaces that occur naturally in a number of contexts. We will present a general finiteness result, obtained in collaboration with A. Ducros and E. Hrushovski, about the ordered abelian group of tropical functions on skeleta of Berkovich analytifications of algebraic varieties. The proof uses the stable completion of an algebraic variety, a model theoretic version of analytification previously developed by Hrushovski and Loeser. The first lecture by F. Loeser will provide a general introduction to the problem and a quick overview of stable completion. In the second lecture J. Ye will present the strategy of proof and outline the proof of the main steps.
Enrica Mazzon (University of Regensburg)
Title: (Non-archimedean) SYZ fibrations for Calabi-Yau hypersurfaces
Abstract: The SYZ conjecture is a conjectural geometric explanation of mirror symmetry. Based on this, Kontsevich and Soibelman proposed a non-archimedean approach, which led to the construction of non-archimedean SYZ fibrations by Nicaise-Xu-Yu. A recent result by Li relates explicitly the non-archimedean approach to the classical SYZ conjecture.
In this talk, I will focus on maximally degenerate families of Calabi-Yau hypersurfaces in P^n. For a large class of them, in collaboration with Jakob Hultgren, Mattias Jonsson and Nick McCleerey, we solve a non-archimedean conjecture proposed by Li and deduce that classical SYZ fibrations exist on open regions of CY hypersurfaces.
Vlerë Mehmeti (Sorbonne Université)
Title: Local-global principles over non-Archimedean analytic curves
Abstract: I will be speaking of an application of non-Archimedean analytic geometry to questions related to the existence of rational points on varieties. More precisely, several local-global principles applicable to quadratic forms will be presented, all obtained by working over Berkovich analytic curves. The main tool I employ is an adaptation of the so called patching technique, which has lately become an important method for the study of such questions.
Kien Huu Nguyen (Université de Caen)
Title: Motivic oscillation indexes of arbitrary ideals
Abstract
Johannes Nicaise (Imperial College London/ KU Leuven)
Title: Birational invariance of the motivic zeta function for K-trivial varieties
Abstract: This talk is based on joint work with Luigi Lunardon. To every smooth and proper variety X with trivial canonical bundle over the field of complex Laurent series C((t)), one can attach its motivic zeta function, which measures how the variety degenerates as t goes to 0. We will show that this motivic zeta function is a birational invariant of X and deduce the birational invariance of the monodromy conjecture for X (the main open problem about these zeta functions). If time permits, we will also discuss a recent example by Cynk and van Straten of a Calabi-Yau threefold over C((t)) with trivial monodromy but no good reduction.
Matthew Satriano (University of Waterloo)
Title: A motivic change of variables formula for stacks
Abstract: Motivic integration is a powerful tool first introduced by Kontsevich to prove that birational Calabi-Yau manifolds have equal Hodge numbers. Yasuda extended the theory to Deligne-Mumford stacks. By proving a motivic change of variables formula in this setting, he showed that for varieties with quotient singularities, the stringy Hodge numbers agree with the orbifold cohomology of a canonical DM stack. In this talk, I discuss joint work with Usatine, where we extend the theory to Artin stacks and prove a general motivic change of variables formula.
Takehiko Yasuda (Osaka University)
Title: Stringy invariants and finiteness of local fundamental groups
Abstract: In this talk, we will discuss an approach using stringy invariants to showing finiteness of the local etale fundamental group of a log terminal surface singularity in positive and mixed characteristics. The main difficulty in this problem is the presence of wild ramification. The key points of the argument is to show that the stringy invariant of a singularity decreases after taking a quasi-etale cover and that the invariant satisifies a DCC property. This is a joint work with Javier Carvajal-Rojas.
Tony Yue Yu (California Institute of Technology)
Title: Mirror structure constants via non-archimedean analytic disks
Abstract: For any smooth affine log Calabi-Yau variety U, we construct the structure constants of the mirror algebra to U via counts of non-archimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, I will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.
Paul Ziegler (Technische Universität München)
Title: Donaldson-Thomas invariants via p-adic integration
Abstract: I will talk about joint work in progress with Dimitri Wyss and Michael Groechenig describing Donaldson-Thomas invariants via p-adic integration.
