GEOMETRY AT LARGE

December 10-18 2018 



December 10 


Fabien Morel Motivic Matsumuto theorem FM1

Boris Zilber Model-theoretic version of anabelian geometry BZ1

Ivan Cheltsov Rationality and K-stability of Fano varieties IC 



December 11 


Olivia Caramello Grothendieck toposes as unifying 'bridges' in mathematics (90 min) OC

Christopher Deninger Dynamical systems for arithmetic schemes (90 min) CD

Fedor Bogomolov On geometry over small fields FB1



December 12 


Ilia Itenberg Tropical geometry and its applications II1

Discussions



December 13 


Caucher Birkar Spaces and singularities in birational geometry CB

Olivia Caramello Grothendieck toposes as unifying 'bridges' in mathematics (90 min) OC

Grigory Mikhalkin Tropical view in Symplectic Field Theory: planimetry GM 



December 14 


Discussions

Laurent Lafforgue Langlands' transfer, non-linear Fourier transforms and non-linear convolution operators (90 min) LL



December 15 


Ehud Hrushovski Model theory and the product formula (90 min) EH

Fabien Morel Motivic variations on homology and cohomology of smooth varieties FM2 

Grigory Mikhalkin Tropical view in Symplectic Field Theory:stereometry GM 

Wojciech Porowski Anabelian geometry in Mochizuki's IUT theory WP



December 16 


Yujiro Kawamata Non-commutative deformations of simple collections YK

Boris Zilber The model theory of structural approximation and quantum mechanics BZ2 

Ilia Itenberg Refined enumeration of algebraic curves II2 

Michael Spieß L-invariants associated to Hilbert modular forms MS



December 17 


Ivan Fesenko Adelification on elliptic surfaces IF 

Fedor Bogomolov Projective invariants of collections of torsion points of elliptic curves FB2



December 18 


Discussions






CB This talk concerns the evolution of the notion of "space" and their singularities in birational geometry. By way of examples I will describe the process of going from varieties to pairs to generalised pairs, and mention some fundamental applications. 


FB1 I would like to discuss properties of algebraic varieties defined over number fields or algebraic closure of a finite field (small fields). In particular I will discuss the existence of unramified correspondences between curves defined over small fields. I will also formulate several conjectures. 


FB2 The main object of the talk is a (complex) elliptic curve E with a standard degree 2 projection on P^1. Assuming that we fix one of the ramification points as a zero we obtain a subset PE_{tors} of the images of torsion points on E inside P^1. These sets are very different for different elliptic curves-they have finite intersection for any two non-isomorphic elliptic curves. However some subsets of the above PE_{tors} are PGL(2) equivalent. This holds for the images of points of order 3 and order 4. I will discuss generalization of these results to other sets of the images of torsion points. The talks is based on joint work with H. Fu and Y. Tschinkel. 


IC Fano varieties naturally appear in 2 seemingly unrelated problems. The first is rationality problem, which is historically inspired by Luroth problem. The second is the problem of existence of Kahler-Einstein metrics on complex manifolds, known as Calabi problem. I will present explicit results and examples that show the relation between the two problems.


OC After recalling the necessary topos-theoretic preliminaries, we will explain the sense in which Grothendieck toposes can act as unifying 'bridges' for relating different mathematical theories to each other and studying them from a multiplicity of different points of view. The theoretical presentation will be accompanied by the discussion of a number of selected applications of the use of toposes as ‘bridges’ in different mathematical fields. 


CD We construct infinite-dimensional continuous-time dynamical systems attached to integral normal schemes which are flat and of finite type over the spectrum of the integers. We study the periodic orbits and connectedness properties of these systems and ask several questions. 


IF There are two different types of class field theory: special and general. General class field theory is functorial and works over any global field. It started with Takagi’s existence theorem 100 years ago. All conventional existing achievements in the Langlands program and the study of special values of zeta and L-functions are not parallel to class field theory of general type. 2d class field theory generalises general 1d class field theory for surfaces, by using topological Milnor K_2-theory. The program of adelification for elliptic surfaces, the number theoretical analogue of categorification, views two ranks of elliptic surfaces, the analytic and geometric ones, as invariants associated to the multiplicative groups of two distinct adelic structures on the surfaces: analytic and geometric 2d adelic structures. At the level of their additive groups these two structures are fundamentally different, locally they correspond to the integral structures of rank 1 and of rank 2. However, at the multiplicative level they are related in explicit 2d class field theory and its existence theorem, thus providing a new unconventional approach to the BSD conjecture. The two adelic structures have also a number of deep analogies with the two fundamental symmetries in the IUT theory of Sh. Mochizuki. 


EH I'll begin with some reflections on the interaction of model theory with algebraic geometry and number theory. It tends to proceed via an addition to the field structure (derivation, automorphism, orderings, valuations) governed by a 'universal axiom' (e.g. Leibniz rule.) Model theoretic ideas (model companion, stability and generalisations) can go far in pointing out significant geometric structures. I will discuss in particular the case of an automorphism where the model theory suggests a possible difference algebraic geometry. Correspondences are brought in to the category of motives via the morphisms, to model theoretic difference algebra directly into the objects, leading to a more elementary, less linearised (and much less developed) approach. A similar model-theoretic analysis of the product formula is one of the current frontiers; it requires a well-understood extension of first-order logic (continuous logic) but so far classical stability appears to apply. Both cases incorporate appealing transfer principles. 


II1 In this introductory talk, I will discuss several tropical notions and applications of tropical geometry (mainly in algebraic geometry). 


II2 Refined enumerative invariants, introduced in the tropical setting by F. Block and L. Göttsche, give rise to a one-dimensional family of invariants that includes Gromov-Witten and Welschinger invariants (the latter can be seen as real analogs of Gromov-Witten invariants). We will discuss refined invariants in several situations and possible interpretations of these invariants. 


YK I will consider deformations over non-commutative base. The point is that there are more deformations than the commutative case. The deformations of simple collections behave especially well. I will explain examples including deformations of simple collections of coherent perverse sheaves. 


LL The purpose of this talk will be to raise the following question : what could it mean that functions or linear operators defined on groups with values in different local fields (possibly non-archimedean and archimedan) are the same? This talk about a recent work which connects functoriality issues in the Langlands program with non-linear Fourier transform and convolution operators.


FM1 This is the computation of the A^1-fundamental group of split simply connected semi-simple algebraic groups over fields. This is a kind of "motivic Matsumuto theorem" and started with the work of Brylinski-Deligne. This is elementary, basically combinatoric using root systems and also elementary facts in A^1-homotopy theory. It explains (not using Matsumuto theorem) why there is a canonical central extension of any such G by Milnor K_2. Joint work with A. Sawant. 


FM2 I will give an overview of the different kinds of "singular" cohomology and homology theories showing up in motivic homotopy theory, explain there advantages and inconveniences, and introduce a new version of homology theory, which, contrary to the other "variants" like Suslin homology, is computable (for instance entirely for projective spaces) and can be used in many geometric problems. 


GM Given a smooth (differentiable) manifold M, its cotangent bundle T*M is a naturally a symplectic manifold. A while ago Vafa has popularized studying differential topology of M through symplectic geometry of T*M, especially in the 3-dimensional case. This approach perfectly agrees with the Symplectic Field Theory approach (as introduced by Eliashberg, Givental and Hofer), in particular for studying topology of Lagrangian submanifolds M inside a given symplectic manifold. Tropical geometry comes into play if T*M or T*M-M admits a toric variety structure (over C) compatible with its symplectic structure, so that studying holomorphic curves in T*M through some simple piecewise-linear graphs becomes especially easy. This is manifestly the case if M is a (real) torus of any dimension, but it also holds in a number of other cases, including the celebrated "conifold transition" case of M=S^3. We'll take a look at this phenomenon and some of its implications in the two-dimensional case first, and then in the three-dimensional case. 


WP I will give a short and elementary overview of several techniques of anabelian geometry used in Sh. Mochizuki's IUT theory. They mainly concern reconstructions of various geometric objects and isomorphisms between them from the data of certain etale fundamental groups. 


MS I will explain the construction of L-invariants associated to Hilbert modular forms f(z) defined over a totally real base field F. They are defined in terms of the cohomology of the group GL_2(F). If f(z) corresponds to an elliptic curve E, then we prove that the L-invariant agrees with an L-invariant defined in terms of a Tate period associated of E. 


BZ1 We work out a formal language in which one can consider analytic universal covers U_an(X) of complex algebraic varieties X as well as projective limits U_pr(X) of finite etale covers of algebraic varieties X over arbitrary fields k of characteristic 0. We prove that for a given smooth X over k the two structures are elementarily equivalent. We also prove that the etale fundamental group of X can be identified as the automorphism group of U_pr(X). Note that these two theorems allow to derive facts about the etale fundamental group from the study of U_an(X). We thus proceed to prove within the methods that any k-rational point of X gives rise to a Grothendieck section of the fundamental group. Finally we give an equivalent formulation of Grothendieck's section conjecture in terms of definability in U_an(X). Joint work with R.Abdolahzadi.


BZ2 We approach the formalism of quantum physics, in particular limits and integration, from the abstract point of view of model theory. This leads to the theory of approximation of geometric structures of quantum mechanics by certain class of finite structures. We then show how quadratic Gaussian integrals with arbitrary parameters can be calculated by reduction to Gaussian quadratic sums. A similar scheme calculates the evolution operator with quadratic potential by paths integration. We present a new result: the derivation of (an analogue of) the stationary phase formula for higher order potential by the method of structural approximation.