Workshop "Real algebraic geometry in Saint Petersburg"

11 Oct 2021, 17:00 → 15 Oct 2021, 20:00 Europe/Moscow

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Workshop "Real algebraic geometry in Saint Petersburg"

October 11 - 15, 2021

This workshop will focus on real algebraic geometry, particularly in regards to applications to enumerative geometry and topology of real algebraic varieties. Additional related topics will include topology of tropical varieties, amoebas of algebraic varieties and extremal problems in real algebraic geometry.

The main intertwined themes we plan to address during the workshop are:

• (tropical) generalizations of Viro’s patchworking; applications to constructions of real algebraic varieties with a rich topology of the real part, of parabolic locus of affine real hypersurfaces;
• relations between combinatorial patchworking and tropical homology;
• real linear series on real algebraic curves and their Weierstrass points: upper bounds on their number, tropical constructions, special case of real inflection points of plane algebraic curves;
• totally real morphisms of varieties and curves, choreography of divisors on real algebraic curves after Viro, the separating semi-group of a real curve, generalisations of simple Harnack curves to higher dimensions;
• refined invariants of algebraic surfaces; relation between Göttsche-Shende invariants, Block-Göttsche invariants, and Mikhalkin’s quantum indices of real algebraic curves; generalizations to non-toric surfaces; relation of these invariants with Berkovich geometry and motivic integration;
• development of tropical techniques to compute recently discovered real and open enumerative invariants;
• geometry of amoebas of complex algebraic and analytic varieties;
• topology of tropical varieties; tropical intersection theory; tropical homology.

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Scientific Committee

• Ilia Itenberg • Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Université
• Hannah Markwig • Eberhard Karls Universität Tübingen

Organisers:

• Nikita Kalinin • SPbU and HSE
• Kris Shaw • University of Oslo

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Local institutions participating in the organization of the event

• St. Petersburg Department of Steklov Mathematical Institute of the Russian Academy of Sciences
• Leonhard Euler International Mathematical Institute in St. Petersburg

The conference is financially supported by a grant from the Government of the Russian Federation, agreements 075-15-2019-1619 and 075-15-2019-1620 and by a grant from Simons Foundation.

Monday, 11 October

17:00 → 18:00 A version of the mixed volume taking values in Z/2Z, Alexandr Esterov

I will introduce a version of the mixed volume of lattice polytopes taking values in Z/2Z (very differently from the usual mixed volume modulo 2). It comes from arithmetic geometry, is constructed using tropical techniques, and appears in real algebraic geometry whenever the usual mixed volume appears over the complex numbers. For instance, it governs the parity of roots with the negative first coordinate for generic real sparse systems of equations (while the total number of roots is governed by the usual mixed volume) and defines the signs of the leading coefficients of sparse resultants.

18:10 → 19:10 Enumerative geometry and the quantum torus. Michael Polyak

A number of classical enumerative problems can be easily solved by methods of tropical
geometry. Tropical curves are planar metric graphs with certain requirements of balancing,
rationality of slopes and integrality.
We consider a generalization of tropical curves, removing requirements of rationality of
slopes and integrality. One of the classical enumerative problems is that of counting of (either
complex or real) rational curves through a collection of points in a toric variety. We explain this
counting procedure as a construction of some simple top-dimensional cycles on certain moduli
of (pseudo)tropical curves.
Cycles on these moduli turn out to be closely related to maps of spheres and associative and
Lie algebras. In particular, counting of both complex and real curves is related to the quantum
torus algebra. More complicated counting invariants (the so-called Gromov-Witten descendants)
are similarly related to the super-Lie structure on the quantum torus.

Tuesday, 12 October

17:00 → 18:00 Viro's writhe and Ulrich sheaves. Mario Kummer

Viro's writhe is an invariant of rigid isotopy for real algebraic curves in projective three-space. We show that it agrees with the topological degree of a natural map from a certain projective bundle over the second symmetric product of the curve to projective three-space. This map admits a relative Ulrich line bundle. The space of global sections of this line bundle carries a symmetric bilinear form in a natural way. The signature of this bilinear form again agrees with Viro's writhe. This is a joint work in progress with Daniele Agostini.

Wednesday, 13 October

17:00 → 18:00 Tropical hypersurfaces with mild singularities in sandpile models. Nikita Kalinin

A small perturbation of the maximal stable state in sandpile models on big subsets of the standard grid exhibits tropical curves. A proof of the planar case uses certain dynamics on tropical curves, while curves appearing in this dynamic are smooth or nodal. A direct generalisation of the problem for higher dimensions leads to tropical hypersurfaces which have only mild singularities. This means that the dual lattice polyhedron for each face (of any dimension) of such a hypersurface contains no lattice points except vertices. I explain the motivation from sandpiles and give an overview of the aforementioned dynamics.

18:10 → 19:10 800 conics on a smooth quartic surface. Alex Degtyarev

I will discuss the recent progress concerning the problem on the maximal number of smooth rational curves of a fixed degree (most notably, conics) on polarized K3-surfaces. The most intriguing open question concerns conics on quartics. I will explain several constructions of the current champion, one with 800 conics, and try to substantiate the conjecture that 800 is indeed the maximum.

Thursday, 14 October

15:25 → 16:25 A tropical version of jacobian conjecture. Dima Grigoriev

We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.

18:00 → 19:00 Real loci of generic complex varieties. Oleg Viro

The set of real points of a generic non-real complex algebraic variety
is a real algebraic variety of special type. It is co-oriented in the
real part of the ambient variety, realize there an integer
cohomology class and has integer intersection and linking numbers with
oriented submanifolds (even if the real part of the ambient space is not
orientable). If the codimension of the original complex variety is one,
the intersection is a base for a real pencil of hypersurfaces.