# Tropical working group seminar

2021, February 26, Friday, 15:00-16:00 Rue du Conseil-Général 7 Weronika Czerniawska (UNIGE)

**Tropical geometry and Newton polygons for p-adic numbers**

2020, Thursday, December 17, 14:30-15:30 Gleb Smirnov (ETH Zürich) Meeting ID: 971 8147 2717 Passcode: (the Euler characteristic of the K3-surface)

**Running through Seiberg-Witten invariants**

This is a mini-course on four-dimensional gauge theory.

In the first lecture, we will introduce spin and complex spin structures and, time permitting, the Dirac operator in four dimensions.

2020, Friday, December 4, 14:00-15:00 Gleb Smirnov (ETH Zürich) Meeting ID: 971 8147 2717 Passcode: (the Euler characteristic of the K3-surface)

**Running through Seiberg-Witten invariants**

This is a mini-course on four-dimensional gauge theory.

In the first lecture, we will introduce spin and complex spin structures and, time permitting, the Dirac operator in four dimensions.

2020, Friday, 20 November, Thomas Blomme, 14:30

**Refined count of rational tropical curves in arbitrary dimension**

In this talk we will introduce a refined multiplicity for rational tropical curves in any dimension. This multiplicity generalizes the multiplicity of Block-Göttsche for planar tropical curves. We also show that the count of solutions to some general tropical enumerative problem using this new multiplicity leads tropical refined invariants, hinting toward the existence of classical refined invariants for classical rational curves.

2020, Thursday, November 19, 14:30-15:30 Gleb Smirnov (ETH Zürich) Meeting ID: 971 8147 2717 Passcode: (the Euler characteristic of the K3-surface)

**Running through Seiberg-Witten invariants**

This is a mini-course on four-dimensional gauge theory.

In the first lecture, we will introduce spin and complex spin structures and, time permitting, the Dirac operator in four dimensions.

2020, Thursday, November 13, 14:00-15:00 Gleb Smirnov (ETH Zürich) Meeting ID: 971 8147 2717 Passcode: (the Euler characteristic of the K3-surface)

**Running through Seiberg-Witten invariants**

This is a mini-course on four-dimensional gauge theory.

2020, Thursday, November 5, 14:30-15:30 2020, Friday, November 6, 14:00-15:00 Gleb Smirnov (ETH Zürich) Meeting ID: 971 8147 2717 Passcode: (the Euler characteristic of the K3-surface)

**Running through Seiberg-Witten invariants**

This is a mini-course on four-dimensional gauge theory.

2020, Thursday, 29 October, Mikhail Pirogov, 14:30

**Lagrangian embedding of Klein bottle to $\mathbb{R^4}$**

The Stefan Nemirovski’s article (arXiv:math/0106122) will be discussed.

2019, Monday, November 11, 16:30, Battelle, Weronika Czerniawska

**Beilinson's Conjectures **

2019, Wednesday, November 6, 11:00, Battelle, Alina Pavlikova

**Critical values of coamoebas of codimension two affine planes**

Despite some general results about (co)amoebas of half-dimensional varieties and linear spaces, very little is known about topological structure of the corresponding critical loci. In this talk, I will give a discription of critical values for the rolled coamoeba of a generic affine plane in four dimensional space.

2019, Friday, October 18, 15:00, Battelle, Weronika Czerniawska

**Beilinson's Conjectures **

2019, Friday, September 20, 15:00 Battelle, Misha Shkolnikov

**Introduction to Persistent Homology**

2016, Wednesday, May 31, 16:00, Battelle, Nikita Kalinin, CINVESTAV (Mexico), HSE(St.-Petersburg)

**Power law in the (tropical) linearized sandpile model**

I will define tropical linearized sandpile model and will speak about various experimental results obtained on Xiuhcoatl supercomputer in Mexico City.

2016, Monday, December 5, 16:15, Battelle, Johannes Josi

**An introduction to Heegaard Floer homology.**

I will try to define Heegaard-Floer homology in the talk on Monday. I will follow the introduction by Oszvath-Szabo (math.mit.edu/~petero/Introduction.pdf).

2016, Friday, November 25, 14:30, Battelle, Nikon Kurnosov (HSE, IUM)

**Cohomological constraints for hyperkaehler manifolds.**

In my talk I will explain cohomological constraints for the Betti numbers of hyperkahler manifolds arising from Rozansky-Witten invariants and so(b_2+2)-action on cohomology. In dimension four boundary conditions were obtained by Guan. Recently Sawon got boundness of b_2 in dimension six. I will talk about generalization of Guan's results in higher dimensions.

2016, Friday, November 18, 14:30, Battelle, Alina Pavlikova

**An introduction to Heegaard Floer homology.**

(Continuation)

Monday, November 14, 2016, 16:30, Villa Battelle, Yuto Yamamoto (The University of Tokyo)

**Tropical homologies of hypersurfaces in tori**

Tropical homology is a homology theory for tropical varieties. In this talk, we calculate dimensions of tropical homologies of smooth hypersurfaces in tori. We also show that they coincide with the limit mixed Hodge numbers of corresponding degenerating families.

2016, Monday, October 31, 16:15, Battelle, Alina Pavlikova

**An introduction to Heegaard Floer homology.**

The aim of this talk is to give an introduction to Heegaard Floer homology for closed oriented 3-manifolds. This homology is defined using Heegaard diagrams and Lagrangian Floer homology. We will also discuss a related Floer homology invariant for knots in S3.

2016, Friday, October 21, 14:30, Battelle, Misha Shkolnikov

**Tropical formula for pi.**

I will explain a new formula expressing the number pi in terms of a natural series over Sl(2,Z). I will also sketch a geometric proof which is based on an argument localizing the area of a convex domain (the unit disk in our application) to the vertices of a certain tropical analytic curve inscribed in the domain.

2016, Monday, October 10, 16.30, Battelle, Arthur Renaudineau

**Tropical limit of log-inflection points. **

We will describe the tropical limit of log-inflection points for families of planar curves with smooth tropical limit (joint work with Grisha Mikhalkin). In a second part, we will discuss a slight generalization of Haas theorem describing maximal real algebraic curves close to the tropical limit (joint work with Benoît Bertrand and Erwan Brugallé).

2016, Monday, September 26, 16.30, Battelle, Misha Shkolnikov

**Introduction to Chern-Simons theory.**

Classical Chern-Simons theory appeared in the framework of differential geometry in seventies. Recognized as a filed theory in eighties, it was successfully quantized by Witten leading to appearance of new topological invariants and rediscovery of old ones such as Jones polynomial of links. In my talk, I will give a basic overview of some ideas behind those constructions.

2016, Saturday, July 2, 15.00, Battelle, Ilia Zharkov (Manhattan)

**Phase tropical hypersurfaces.**

To a smooth complex hypersurface in ${(C^*)}^N$ we associate a phase tropical hypersurface, which is the fibration over the tropical limit by coamoebas (tori over the facets). I will show that the phase tropical hypersurface is homeomorphic to the complex one.

2016, Monday 13 June, 15.00, Battelle, Arthur Renaudineau

**Parabolic points of tropical hypersurfaces.**

2016, Wednesday, 25 mai, 11.15, Battelle, Alina Pavlikova.

continuation

2016, Friday, 20 mai, 10.30, Battelle. Ivan Cherednik.

**DAHA and plane curve singularities**

2016, Wednesday, 18 mai, 11.15, Battelle. Misha Shkolnikov.

continuation.

2016, Friday, 6 mai, 14.30, Battelle. Misha Shkolnikov.

**Refined curve counting**

The concept of curve counting appear in a number of situations under different names. Among them are Severi degrees, Gromov-Witten and Welschinger invariants computation of which in some cases can be reduced to a tropical count with appropriate choice of multiplicities. This particular multiplicities can be included to a one parametric family of (quantized) multiplicities. This defines a new refined tropical count. I will review an approach of Goettsche and Shende to algebro-geometric counterpart of the refined count.

2016, Monday, 2 mai, 16.30, Battelle. Alina Pavlikova.

**Lagrangian embeddings of Klein bottle,** Сontinuation.

2016, Friday, 29 avril, 16.00, Battelle, Nikita Kalinin.

**Geometric definition of a tropical limit**

2016, Wednesday, 27 avril, 11.15, Battelle, G. Mikhalkin,

**Real algebraic knots.**

2016, Monday, 25 avril, 16.30, Battelle, Nikita Kalinin.

**Short exposé autour de Conformal Field Theory.**

I give a short review about CFT (conformal field theory) that I learned on lectures in Lausanne. It would be nice if you read something about Ising model in advance (definition and notion of critical temperature and critical exponents will be enough).

Then, 17.10 — 17.30 we will discuss the exercise about correct definition of Luttinger surgery from Alina’s talk. Usually people use another definition (see https://math.berkeley.edu/~auroux/papers/lagrsurg.pdf) which is topologically equivalent to the initial definition of Luttinger (http://projecteuclid.org/euclid.jdg/1214457232) but it would be nice to untangle the Luttinger’s argument.

2016, Friday, 22 avril, 15.30, Battelle. Misha Khristoforov

Introduction to SLE

2016, Wednesday, 20 avril, 10.30, Battelle. Alina Pavlikova.

Сontinuation.

2016, Monday, 18 avril, 16.30, Battelle. Alina Pavlikova.

Continuation.

2016, Wednesday, 13 avril, 10.30, Battelle. Misha Shkolnikov.

Continuation.

2016, Wednesday, 6 avril, 11.15, Battelle. Misha Shkolnikov

**First localization formulas**

I will give an overview of equivariant cohomology following the classical “The moment map and equivariant cohomology” by Atiyah and Bott. In particular, we will see how the result of Duistermaat and Heckman on the push-forward of the Liouville measure under the moment map can be viewed as particular case of the localization formula.

2016, Wednesday, 23 mars, 11.15, Battelle. Grigory Mikhalkin

**Some first tropical constructions of Lagrangian submanifolds, 2**

2016, Wednesday, 16 mars, 11.15, Battelle. Grigory Mikhalkin

**Some first tropical constructions of Lagrangian submanifolds**

I continue informal description (started in February) of known non-trivial cases of tropical-to-Lagrangian correspondences. After finishing a review of constructions and properties of certain exotic Lagrangian tori corresponding to tropical points (a joint work with Sergey Galkin) we reprove Givental's theorem on Lagrangian embeddability of non-orientable surfaces into R^4 in a tropical way.

2016, 10 February, Villa Battelle 11.15, Nikita Kalinin

**Tropical differential forms in number theory.**

A survey of http://arxiv.org/abs/1504.00694. About uniform bounds for number of rational points on curves of small Mordell-Weil rank.

2016, 1 February, Villa Battelle 16.15, Nikita Kalinin

**Emergence of tropical solitons (self-reproducing patterns) in sandpile models**

Based on joint work with M. Shkolnikov. I will prove that for each rational direction there exists a pattern, which translates under the action of waves. These patterns can be named solitons. We start with definitions, then consider a situation when we sand waves towards the infinite boundary of a given slope. We define slicings and smoothings of discrete integer valued superharmonic functions.

2016, 3 February, Villa Battelle 11.15, Nikita Kalinin

**Emergence of tropical triads of solitons in sandpile models, I**

Based on joint work with M. Shkolnikov. I will prove formulate the theorem about triads of sand solitons coming in a point and start the proof. We prove than while slicing the minimum of three linear functions the area of slicings grows at most linear and at least linear.

2016, 4 February, Villa Battelle 16.15, Nikita Kalinin

**Emergence of tropical triads of solitons in sandpile models, II**

Based on joint work with M. Shkolnikov. We continue the proof. We show that in the area of slicing there exists a big square near the boundary of slicing, where the function is harmonic. Using Poisson kernel estimates we prove that this function is linear. Then, using monotonicity of the results of slicings, we will prove the main theorem.

2015, 21 December, Villa Battelle 16.15, Mikhail Shkolnikov

**Lagrangians on complex plane**

I will explain some basics of symplectic and Lagrangian geometry, in particular, we will discuss the problem of describing Lagrangian surfaces in the complex projective plane and the definition of Chekanov tori.

2015, 16 November, Villa Battelle, 16.15, Alina Pavlikova, Unige.

**Descartes pairs.**

I will speak about 7'th chapter of the paper “Tropical aspects of real polynomials and hypergeometric functions” (Jens Forsgård).

In what follows we consider real univariate polynomials with non-vanishing coefficients. The famous Descartes’ rule of signs claims that the number of positive roots of such a polynomial does not exceed the number of sign changes in its sequence of coefficients.

An arbitrary ordered sequence $\sigma = (\sigma_0,\sigma_1,...,\sigma_d ) \in \{\pm 1\}^d+1$ is called a sign pattern. Given a sign pattern $\sigma$, we call by its Descartes pair $(p\sigma,n\sigma)$ the pair of non-negative integers counting sign changes and sign preservations of $\sigma$.

For any sign pattern $\sigma$ we call pairs $(p,n)$ satisfying some condition admissible for $\sigma$.

In general, given a sign pattern $\sigma$, not all of its admissible pairs are realizable by polynomials $f$ with sign pattern $sgn(f) = \sigma$.

The interesting problem is — For a given sign pattern $\sigma$, which admissible pairs $(p,n)$ are realizable by polynomials $f$ such that $sgn(f ) = \sigma$?

2015, 28 October, Wednesday 11.00, Ilia Itenberg.

**“Courbes de degré 6 dans le plan projectif réel“.**

2015, 21 October, Villa Battelle, 11.00, Mikhail Shkolnikov,

**Some proofs in tropical sand, II.**

We define waves and consider the case where $\Omega$ is a half-plane with the boundary of rational slope. If time permits, we define *smoothings* and *slicings*.

2015, 15 October, Villa Battelle, 09.00, Nikita Kalinin

**Tropical modifications, II.**

We continue the study of tropical modifications and review tropical momentum theorem, tropical Weil theorem, and obstructions for realizability of tropically non-transversla intersections.

2015, 13 October, Villa Battelle, 14.15, Mikhail Basok (SPbSU),

**Moduli spaces.**

The goal of my talk is to rewiev the construction of the moduli space of compact Riemann surfaces. We will discuss the Deligne-Mumford compactification of the moduli space and compute the relative dualizing sheaf of the universal family over this compactification. I will try to cover both algebraic and analytical point of view. If time permits, we will discuss some examples and properties.

2015, 12 October, Villa Battelle, 16.15, Misha Shkolnikov,

**Some proofs in tropical sand.**

I will review the main steps in the proof of our main theorem from the paper on tropical sandpiles. And then we will discuss one of these steps in details.

2015, 7 October, Villa Battelle, 11.00

**Introduction to the tropical modifications**

Nikita Kalinin, UniGe.

2015, 31 September, Villa Battelle, 16.15, Nikita Kalinin, Unige.

**Tropical differential forms in legendrian geometry**

I will show some applications of tropical differential forms for the legendrian curves. Also we discuss some subtleties in the definitions.

2015, 23 September, Villa Battelle, 11.00, Johannes Rau.

**Abstract tropical varieties**.