# Séminaire "Fables Géométriques".

Friday, Dec 8, 14h30, Salle 06-13

**Francesca Carocci (Genève)**

**Degenerations of Limit linear series**

Maps to projective space are given by basepoint-free linear series, thus these are key to understanding the extrinsic geometry of algebraic curves. How does a linear series degenerate when the underlying curve degenerates and becomes nodal? Eisenbud and Harris gave a satisfactory answer to this question when the nodal curve is of compact type. Eisenbud-Harris's theory of limit linear series gives proofs via degenerations of many foundational results in Brill–Noether theory, and it is powerful enough to answer several birational geometry questions on the moduli space of curves. I will report on a joint work in progress with Lucaq Battistella and Jonathan Wise, in which we review this question from a moduli-theoretic and logarithmic perspective. The logarithmic prospective helps understanding the rich polyhedral and combinatorial structures underlying degenerations of linear series. These are linked with matroids and Bruhat-Titts buildings.

Monday, Nov 13, 15h, Salle 06-13

**Francesca Carocci (Genève)**

**What can we do with the Logarithmic Hilbert Scheme?**

In 2020 Maulik-Ranganathan defined the Logarithmic Hilbert-Scheme, which is interesting for the enumerative geometry of 3-folds; for example, it gives access to degeneration techniques in sheaf-theoretic approaches to curve counting. If we go one step back and look at degree d curves in toric surfaces, the construction of the log Hilbert scheme has as a main ingredient the secondary fan of a toric fan, though as moduli space of tropical plane curves up to translation.

I will try to explain some of the ideas of the construction, trying to put emphasis on the tropical aspects of the theory.

The main goal of the talk would be to understand if this theory gives rise to some interesting questions and the relation of such questions with tropical geometry.

May 22, salle 6-13, 15h

**Oleg Viro (Stony Brook)**

**Simplest numerical invariants for some kinds of curves**

In the 90s, Arnold introduced several numerical characteristics of generic plane curves via axiomatic approach based on behavior of curves under “perestroikas”. Soon explicit formulas for the invariants have been invented. The formulas have disclosed unexpected aspects of nature of the invariants and suggested various new objects to study, like real algebraic curves or circles inscribed in a generic plane curve.

**FABLES GEOMETRIQUES MINICOURSE, April 24-27**

Lecture 1, Monday, April 24, 15h, room 6-13 Lecture 2, Tuesday, April 25, 13h, Room 1-07 Lecture 3, Thursday, April 27, 16h15, Room 1-15

**Sergey Finashin (METU Ankara)**

**Strong Invariants in Real Enumerative Geometry**

In the first lecture I will discuss a signed count of real lines on real projective hypersurfaces, which is independent of the choice of real structures and in that sense is “strong invariant”. The simplest examples: a signed count of real lines on a real cubic surface gives 3, while a similar count on a real quintic 3-fold gives 15. In the other lectures I will stick to the case of real del Pezzo surfaces and discuss a generalization of the signed count of lines to a signed count of rational curves (involving some combinations of the Welschinger numbers). All the results are joint with V.Kharlamov.

Monday, April 3, 2023 room 6-13

**15h00 — Alexander Bobenko (TU Berlin)**

**Discrete conformal mappings, ideal hyperbolic polyhedra, and Ronkin function**

The general idea of discrete differential geometry is to find and investigate discrete models that exhibit properties and structures characteristic for the corresponding smooth geometric objects. We focus on a discrete notion of conformal equivalence of polyhedral metrics. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory. We review connections between conformal geometry of triangulated surfaces, the geometry of ideal hyperbolic polyhedra and discrete uniformization of Riemann surfaces. Surprisingly, variational description of discrete conformal mappings is given by Ronkin function on amoeba with three ends. Applications in geometry processing and computer graphics will be demonstrated.

Monday, March 27, 2023 room 6-13

**16h00 — Sebastian Haney (Columbia U)**

**Mirror Lagrangians to lines in P^3**

We discuss work in progress in which we construct, for any tropical curve in $R^n$ with vertices of valence at most $4$, a Lagrangian submanifold of $(C^*)^n$ whose moment map projection is a tropical amoeba. These Lagrangians will have singular points modeled on the Harvey-Lawson cone over a $2$-torus. We also consider a certain $4$-valent tropical curve in $R^3$, for which we can modify the singular Lagrangian lift to obtain a cleanly immersed Lagrangian. The objects of the wrapped Fukaya category supported on this Lagrangian correspond, under mirror symmetry, to lines in $CP^3$. If time permits, we will explain how to use functors induced by Lagrangian correspondences to see this mirror relation.

Monday, March 20, 2023 room 6-13

**16h00 — Ilia Itenberg (Sorbonne)**

**Maximal real algebraic hypersurfaces of projective spaces**

The talk is devoted to a combinatorial patchworking construction of maximal (in the sense of the generalized Harnack inequality) real algebraic hypersurfaces in real projective spaces (joint work with Oleg Viro). During the talk, we will mainly concentrate on the construction of a maximal quintic hypersurface in the 4-dimensional real projective space.

Monday, March 6, 2023 room 6-13

**15h00 — Ali Ulaş Özgür Kişisel (METU, Ankara)**

**Expected measures of amoebas of random plane curves**

There are several natural measures that one can place on the amoeba of an algebraic curve in the complex projective plane. Passare and Rullgård prove that the total mass of the Lebesgue measure on the amoeba of a degree $d$ curve is bounded above by $π^{2} d^{2} / 2$, by comparing it to another Monge-Ampère type measure, which is dual to the usual measure on the Newton polytope of the defining polynomial via the Legendre transform. Mikhalkin generalizes this upper bound to half-dimensional complete intersections in higher dimensions, by considering another measure supported on their amoebas; their multivolume. The goal of this talk will be to discuss these measures in the setting of random plane curves. In particular, I’ll first present our results with Bayraktar, showing that the expected multiarea of the amoeba of a random Kostlan degree $d$ curve is equal to $π^2 d$. For Lebesgue measure, it turns out that the expected asymptotics are much lower: I’ll describe our results with Welschinger, showing that the expected Lebesgue area of the amoeba of a random Kostlan degree $d$ curve is of the order $(\log d)^2.$

Monday, February 27, 2023 room 6-13

**15h00 — Evgeni Abakoumov (Paris/Eiffel U)**

**Chui's conjecture аnd rational approximation**

C. K. Chui conjectured in 1971 that the average gravitaional field strength in the unit disk due to unit point masses on its boundary was the smallest when these point masses were equidistributed on the circle. We will present an elementary solution to some weighted versions of this problem, and discuss related questions concerning approximation of holomorphic functions by simple partial fractions. This is joint work with A. Borichev and K. Fedorovskiy.

**16h00 — Ferit Ozturk (Istanbul/Bosphorus U and Budapest/Renyi Inst)**

**Every real 3-manifold admits a real contact structure**

We survey our results regarding real contact 3-manifolds and present our result in the title. A real 3-manifold is a smooth 3-manifold together with an orientation preserving smooth involution, called a real structure. A real contact 3-manifold is a real 3-manifold with a contact distribution that is antisymmetric with respect to the real structure. The standard examples of real contact 3-manifolds are link manifolds of isolated, real analytic surface singularities. We show that every real contact 3-manifold can be obtained via contact surgery along invariant knots starting from the standard real contact 3-sphere. As a corollary we show that any oriented overtwisted contact structure on an integer homology real 3-sphere can be isotoped to be real.

Monday, February 6, 2023 16:00, room 6-13

**Sergey Finashin (Ankara)**

**“Affine Real Cubic Surfaces”**

Abstract: (A joint work with V.Kharlamov) We prove that the space of affine, transversal at infinity, non-singular real cubic surfaces has 15 connected components. We give a topological criterion to distinguish them and show also how these 15 components are adjacent to each other via wall-crossing.

Thursday, June 16, 2022 16:00, room 1-07

**Prof. Yakov Eliashberg (Stanford)**

**“Topology of spaces of Legendrian knots via Algebraic K-theory”**

Abstract: The highly non-trivial stable homotopy groups of the Waldhausen’s h-cobordism space inject into the homotopy groups of spaces of appropriate Legendrian submanifolds. For instance, there is a homotopically non-trivial 2-parametric family of Legendrian unknots in ${\mathbb R}^{2n+1}$ for a sufficiently large $n$. This is a joint work with Thomas Kragh.

Fri 17.12.2021, 13h30, room 6-13 Andras Szenes

**Diagonal bases and wall-crossings in moduli spaces of vector bundle**

The idea of the calculation of the Hilbert function of the moduli spaces of vector bundles on Riemann surfaces goes back to the works of Michael Thaddeus in the early 90’s. I describe joint work with Olga Trapeznikova where this plan is carried out in detail, which uses only basic tools of Geometric Invariant Theory and a combinatorial/analytic device introduced by myself: the diagonal basis of hyperplane arrangements.

Nov 1, 16h15. Room 06-13 Vasily Golyshev (Moscow, Bures-sur-Yvette)

**Markov numbers in number theory, topology, algebraic geometry, and differential equations**

I will explain how the Markov numbers arise in different mathematical disciplines, and sketch the links. A recent contribution will be discussed, too.

2020, Wednesday, May 20, 16:00 (CEST), Virtual seminar, Lionel Lang (Stockholm University)

https://unige.zoom.us/j/5928729514 Meeting ID: 592 872 9514 Password: (the number of lines on a cubic surface)

**Co-amoebas, dimers and vanishing cycles**

In this joint work in progress with J. Forsgård, we study the topology of maps P:(\C*)^2 \to \C given by Laurent polynomials P(z,w). For specific P, we observed that the topology of the corresponding map can be described in terms of the co-amoeba of a generic fiber. When the latter co-amoeba is maximal, it contains a dimer (a particularly nice graph) whose fundamental cycles corresponds to the vanishing cycles of the map P. For general P, the existence of maximal co-amoebas is widely open. In the meantime, we can bypass co-amoebas, going directly to dimers using a construction of Goncharov-Kenyon and obtain a virtual correspondence between fundamental cycles and vanishing cycles. In this talk, we will discuss how this (virtual) correspondence can be used to compute the monodromy of the map P.

2020, Tuesday, April 7, 17:00, Virtual seminar (EDGE seminar) Grigory Mikhalkin (Geneva)

https://zoom.us/j/870554816?pwd=bERmR0ZQTitYNXJ1aFZLckxzeXZJZz09 Meeting ID: 870 554 816 Password: 014504

**Area in real K3-surfaces**

Real locus of a K3-surfaces is a multicomponent topological surface. The canonical class provides an area form on these components (well defined up to multiplication by a scalar). In the talk we'll explore inequalities on total areas of different components as well a link between such inequalities and a class of real algebraic curves called simple Harnack curves. Based on a joint work with Ilia Itenberg.

2020, Monday, March 31, 17:00, Virtual seminar, Vladimir Fock (Strasbourg)

https://unige.zoom.us/j/737573471 Meeting ID: 737 573 471

**Higher measured laminations and tropical curves**

We shall define a notion of a higher lamination - a graph embedded into a Riemann surface with edges coloured by generators of an affine Weyl group. This notion generalises the notion of the ordinary integral measured lamination and on the other hand of a tropical curve and can be constructed out of a integral Lagrangian submanifold of the cotangent bundle.

2020, Monday, March 16, 16:30, Battelle, Alexander Veselov (Loughborough University)[POSTPONED]

**On integrability, geometrization and knots**

I will start with a short review of Liouville integrability in relation with Thurston’s geometrization programme, using as the main example the geodesic flows on the 3-folds with SL(2,R)-geometry.

A particular case of such 3-folds the modular quotient SL(2,R)/SL(2,Z), which is known, after Quillen, to be equivalent to the complement in 3-sphere of the trefoil knot. I will show that remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil.

The talk is partly based on a recent joint work with Alexey Bolsinov and Yiru Ye.

2020, Monday, February 17, 16:30, Battelle, Karim Adiprasito (University of Copenhagen, Hebrew University of Jerusalem)

** Algebraic geometry of the sphere at infinity, polyhedral de Rham theory and L^2 vanishing conjectures **

I will discuss a conjecture of Singer concerning the vanishing of L^2 cohomology on non-positively curved manifolds, and relate it to Hodge theory on a Hilbert space that arises as the limit of Chow rings of certain complex varieties.

2019, Friday, December 6, 15:00, Battelle, Tomasz Pelka (UniBe)

** Q-homology planes satisfying the Negativity Conjecture **

A smooth complex algebraic surface S is called a Q-homology plane if H_i(S,Q)=0 for i>0. This holds for example if S is a complement of a rational cuspidal curve in P^2. The geometry of such S is understood unless S is of log general type, in which case the log MMP applied to the log smooth completion (X,D) of S is insufficient. The idea of K. Palka was to study the pair (X,(1/2)D) instead. This approach gives much stronger constraints on the shape of D, and leads to the Negativity Conjecture, which asserts that the Kodaira dimension of K_X+(1/2)D is negative. It is a natural generalization e.g. of the Coolidge-Nagata conjecture about rational cuspidal curves, which was recently proved using these methods by M. Koras and K. Palka.

If this conjecture holds, all Q-homology planes of log general type can be classified. It turns out that, as expected by tom Dieck and Petrie, they are arranged in finitely many discrete families, each obtainable in a uniform way from certain arrangements of lines and conics on P^2. As a consequence, they all satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and their automorphism groups are subgroups of S_3. To illustrate this surprising rigidity, I will show how to construct all rational cuspidal curves (with complements of log general type, satisfying the Negativity Conjecture) inductively, by iterating quadratic Cremona maps. This construction in particular shows that any such curve is uniquely determined, up to a projective equivalence, by the topology of its singular points.

2019, Monday, November 25, 16:30, Battelle, Felix Schlenk (UniNe)

** (Real) Lagrangian submanifolds **

We start with describing how Lagrangian submanifolds of symplectic manifolds naturally appear in many ways: In celestial mechanics, integrable systems, symplectic geometry, and algebraic geometry. We then look at real Lagrangians, namely those which are the fixed point set of an anti-symplectic involution. How special is the property of being real? While many of the examples discussed above are real, we explain why the central fibres in toric symplectic manifolds are real only if the moment polytope is centrally symmetric. The talk is based on work of and with Joé Brendel, Yuri Chekanov, and Joontae Kim.

2019, Friday, November 8, 14:00, Battelle, Johannes Rau (University of Tübingen)

** The dimension of an amoeba **

Amoebas are projections of algebraic varieties in logarithmic coordinates and were originally introduced by Gelfand, Kapranov and Zelevinsky in their influential book. Based on some computation, Nisse and Sottile formulated some questions concerning the dimension of amoebas. In a joint work with Jan Draisma and Chi Ho Yuen, we answer these questions by providing a general formula that computes the dimension of amoebas. If time permits, we also discuss the consequences of this formula for matroidal fans.

2019, Monday, November 4, 16.30, Battelle, Pierrick Bousseau (ETH Zurich)

** Quasimodular forms from Betti numbers**

This talk will be about refined curve counting on local P2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of dimension 1 coherent sheaves on P2. This gives a proof of some stringy predictions about the refined topological string theory of local P2 in the Nekrasov-Shatashvili limit. Partly based on work in progress with Honglu Fan, Shuai Guo, and Longting Wu.

2019, Monday, October 28, 16.30, Battelle, Ilia Itenberg, (Sorbonne University)

** Planes in four-dimensional cubics **

We discuss possible numbers of 2-planes in a smooth cubic hypersurface in the 5-dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357.

The proofs deal with the period spaces of cubic hypersurfaces in the 5-dimensional complex projective space and are based on the global Torelli theorem and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin's theory of discriminant forms.

Joint work with Alex Degtyarev and John Christian Ottem.

2019, Monday, October 14, 16:30, Battelle, Igor Krichever (Columbia University)

** Degenerations of real normalized differentials **

The behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration is studied. In particular, it is proved that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. It is further shown that the limits of zeroes of RN differentials are the divisor of zeroes of a twisted differential — an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials on smooth Riemann surfaces, in a plumbing neighborhood of a given stable curve.

—

2019, Monday, October 7, 16:30, Battelle, Jérémy Blanc (University of Basel)

** Quotients of higher dimensional Cremona groups **

We study large groups of birational transformations $\mathrm{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\mathbb{C}$ or a subfield of $\mathbb{C}$.Two prominent cases are when $X$ is the projective space $\mathbb{P}^n$, in which case $\Bir(X)$ is the Cremona group of rank~$n$, or when $X \subset \mathbb{P}^{n+1}$ is a smooth cubic hypersurface.In both cases, and more generally when $X$ is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from $\mathrm{Bir}(X)$ to $\mathbb{Z}/2$.As a consequence we also obtain that the Cremona group of rank ~$n \ge 3$ is not generated by linear and Jonquières elements.

Joint work with Stéphane Lamy and Susanna Zimmermann

2019, Monday, September 30,16:30, Battelle, Roman Golovko (Charles University in Prague)

** The wrapped Fukaya category of a Weinstein manifold is generated by the Lagrangian cocore discs**

In a joint work with B. Chantraine, G. Dimitroglou Rizell and P. Ghiggini, we decompose any object in the wrapped Fukaya category as a twisted complex built from the cocores of the critical (i.e. half-dimensional) handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories.

2019, Wednesday, September 25, 11:00, Battelle, Ivan Fesenko, (University of Nottingham)

** Two-dimensional local fields and integration on them**

Two-dimensional local fields include formal loop objects such as $R((t))$, $C((t))$, $Q_p((t))$ and also fields such as $F_p((t_1))((t_2))$, $Q_p\{\{t\}\}$. They play the fundamental role in two-dimensional number theory, arithmetic geometry, representation theory, algebraic topology and math physics. I will explain basic things about such fields, including their unusual topology and the theory of measure and integration on such fields and Fourier transform which can be viewed as a (rigorous) arithmetic version of the Feynman integral. While one-dimensional local fields show up in tropical geometry of curves, one may expect that two-dimensional local fields should be involved in tropical geometry of surfaces.

2019, Monday, September 16, 16:30, Battelle, Gleb Smirnov, (ETH Zurich)

**From flops to diffeomorphism groups**

Following a short introduction to the flop surgery, I will explain how this surgery can be used to detect non-contractible loops of diffeomorphisms for many algebraic surfaces.

2019, Monday, May 20, 14:00, Battelle, Ziming Ma (Chinese University of Hong Kong)

**Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties**

In this talk, we construct a dgBV algebra PV^∗,∗(X) associated to a possibly degenerate Calabi-Yau variety X equipped with local thickening data. This gives a singular version of the (extended) Kodaira-Spencer dgLa which is applicable to both log smooth and maximally degenerated Calabi-Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi-Yau varieties X satisfying Hodge-deRham degeneracy property for cohomology of X, in the spirit of Kontsevich-Katzarkov-Pantev. We also demonstrate how our construction can be applied to produce a log Frobenius manifold structure on a formal neighborhood of the extended moduli space using Barannikov’s technique. This is a joint work with Kwokwai Chan and Naichung Conan Leung.

2019, Monday, April 8, 16:00, Battelle, Michele Ancona (Institut Camille Jordan)

**Random sections of line bundles over real Riemann surfaces**

Given a line bundle L over a real Riemann surface, we study the number of real zeros of a random section of L. We prove a rarefaction result for sections whose number of real zeros deviates from the expected one.

2019, Monday, April 1, 16:00, Battelle, Mikhail Shkolnikov (IST Austria)

**PSL-tropical limits**

The classical tropical limit is defined for families of varieties in the algebraic torus. One of the ways to generalize this framework is to consider non-commutative groups instead of algebraic tori. We describe tropical limits for subvarieties in PSL(2,C):the result is spelled in terms of floor diagrams and has parallels with symplectic field theory. The talk is based on the work in progress with Grigory Mikhalkin.

2019,Tuesday, March 26, 14:00, Battelle, Enrica Mazzon (Imperial College London)

**Berkovich approach to degenerations of hyper-Kähler varieties**

To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety, hence the dual complex can be embedded in the Berkovich space of the variety. In this talk I will explain how this interpretation gives an insight in the study of the dual complexes. I will focus on some degenerations of hyper-Kähler varieties and show that we are able to determine the homeomorphism type of their dual complex using techniques of Berkovich geometry. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.

2019, Monday, March 18, 16:00, Battelle, Danilo Lewanski (Max Planck Institut für Mathematik)

**Refreshing Tropical Jucys curves**

We derive explicit formulae for the generating series of Grothendieck dessins d’enfant and monotone Hurwitz numbers via the semi-infinite wedge formalism, and from it we obtain bosonic Fock space expressions. This yields to a tropical geometric interpretation involving Gromov-Witten invariants as local multiplicities.

2019, Monday, March 11, 16:00, Battelle, Anton Mellit (University of Vienna)

**Five-term relations**

I will review how the five term relation for the Fadeev-Kashaev's quantum dilogarithm arises in the Hall algebra context, and sketch a simple proof. Then I will explain how this proof can be transported to the elliptic Hall algebra situation, where the five term relation implies identities between Macdonald polynomials conjectured by Bergeron and Haiman. This is a joint work with Adriano Garsia.

2019, Monday, March 4, 16:00, Battelle, Andras Stipsicz (Budapest University)

**Knot Floer homology and double branched covers**

We will review the basic constructions of (various versions of) knot Floer homologies, show some applications and extensions of the definitions to the double branched cover, also using the covering transformation.

2019, Monday, February 25, 16:00, Battelle, Erwan Brugallé (Université de Nantes)

**On the invariance of Welschinger invariants**

Welshinger invariants are real analogs of Gromov-Witten invariants for symplectic 4-manifolds X. In this talk, I will strengthen the original Welschinger's invariance result. Our main result is that when X is a real rational algebraic surface, Welschinger invariants eventually only depend on the number of real interpolated points, and some homological data associated to X. This result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds differing by a surgery along a real Lagrangian sphere. As an application, we complete the computation of Welschinger invariants of real rational algebraic surfaces, and obtain vanishing, sign, and sharpness results generalizing previously known statements. If time permits, we will also discuss some hypothetical relations with tropical refined invariants defined by Block-Göttsche and Göttsche-Schroeter.

2018, Monday, December 10, 16:00, Battelle, Arthur Renaudineau (Lille)

**Lefschetz hyperplane section theorem for tropical hypersurfaces**

We will discuss variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of non-singular tropical hypersurfaces of toric varieties. As an application, we get that the integral tropical homology groups of non-singular tropical hypersurfaces are torsion free. This is a joint work with Charles Arnal and Kristin Shaw.

2018, Monday, November 26, 16:00, Battelle, Vladimir Fock (Strasbourg)

**Higher complex structures on surfaces**

We suggest a definition of a differential geometric structures on surfaces generalizing the notion of complex structure and discuss its properties. The moduli space of such structures share many common features and conjecturally coincide with higher Teichmüller space - the space of positive representations of the fundamental group of the surface into PGL(N) (like moduli of ordinary complex structure give a representation of the fundamental group to PGL(2)). Joint work with A.Thomas.

2018, Monday, November 19, 16:15, Battelle, Stepan Orevkov (Moscow, Toulouse)

**Orthogonal polynomials in two variables**

A natural generalization of classical systems of (one-variable) orthogonal polynomials is as follows. Let $D$ be a domain in $R^n$ endowed with a Riemannian metric and a mesure. Suppose that the Laplace-Beltrami operator (for the given metric) is symmetric (for the given mesure) and leave invariant the set of polynomials of a given degree. Then its eigenfunctions is a system of orthogonal polynomials.

I present a complete classification of domains in $R2$ for which this construction can be applied. The talk is based on a joint work with D. Bakry and M. Zani.

2018, Monday, October 8, 16:30, Battelle, Sione Ma'u (Auckland)

**Polynomial degree via pluripotential theory**

Given a complex polynomial $p$ in one variable, $\log|p|$ is a subharmonic function that grows like $(deg p)\log|z|$ as $|z|\to\infty$. Such functions are studied using complex potential theory, based on the Laplace operator in the complex plane.

Multivariable polynomials can also be studied using potential theory (more precisely, a non-linear version called pluripotential theory, which is based on the complex Monge-Ampere operator). In this talk I will motivate and define a notion of degree of a polynomial on an affine variety using pluripotential theory (Lelong degree). Using this notion, a straightforward calculation yields a version of Bezout's theorem. I will present some examples and describe how to compute Lelong degree explicitly on an algebraic curve. This is joint work with Jesse Hart.

2018, Monday, October 1, 16:00, Battelle, Mikhail Shkolnikov (Klosterneuburg)

**Extended sandpile group and its scaling limit**

Since its invention, the sandpile model is believed to be renormalizable due to the presence of power-laws. It appears that, the sandpile group, made of recurrent configurations of the model, approximates a continuous object that we call the extended sandpile group. In fact, this is a tropical abelian variety defined over Z and the subgroup of its integer points is exactly the usual sandpile group. Moreover, the extended sandpile group is naturally a sheaf on discrete domains and, thus, brings an explicit scale renormalization procedure for recurrent configurations. We compute the (projective) scaling limit of sandpile groups along growing convex domains: its is equal to the quotient of real-valued discrete harmonic functions by the subgroup of integer-valued ones. This is a joint work with Moritz Lang.

2018, Wednesday, July 18, 16:30, Battelle, Kristin Shaw (University of Oslo)

**Chern-Schwartz-MacPherson classes of matroids. Part II**

Chern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes to singular and non-complete varieties. Matroids are an abstraction of the notion of independence in mathematics. In this talk, I will provide a combinatorial analogue of CSM classes for matroids, motivated by the geometry of hyperplane arrangements. In this setting, CSM classes are polyhedral fans which are Minkowski weights. One goal for defining these classes is to express matroid invariants as invariants from algebraic geometry. The CSM classes can be used to study the complexity of more general objects such as subdivisions of matroid polytopes and tropical manifolds. This is based on joint work with Lucia López de Medrano and Felipe Rincón.

2018, Monday, July 16, 16:00, Battelle, Kristin Shaw (University of Oslo)

**Chern-Schwartz-MacPherson classes of matroids**

Chern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes to singular and non-complete varieties. Matroids are an abstraction of the notion of independence in mathematics. In this talk, I will provide a combinatorial analogue of CSM classes for matroids, motivated by the geometry of hyperplane arrangements. In this setting, CSM classes are polyhedral fans which are Minkowski weights. One goal for defining these classes is to express matroid invariants as invariants from algebraic geometry. The CSM classes can be used to study the complexity of more general objects such as subdivisions of matroid polytopes and tropical manifolds. This is based on joint work with Lucia López de Medrano and Felipe Rincón.

2018, Monday, July 9, 16:30, Battelle, Ernesto Lupercio (CINVESTAV)

**Convex geometry, complex systems and quantum physics**

I will speak about our work on sandpiles and quantum integrable systems. Just as in classical mechanics toric manifolds correspond to rational convex polytopes, the irrational case in informed by the theory of sandpiles. Joint with Kalinin, Shkolnikov, Katzarkov, Meersseman and Verjovsky.

Workshop "Fables Géométriques", 2018, June Friday 15 and Saturday 16, Battelle

**Friday June 15th**

11:00-12:00 Yakov Eliashberg (Stanford)

12:30 lunch

14:30-15:30 Sergey Finashin (METU)

16:00-17:00 Viatcheslav Kharlamov (Strasbourg)

**Saturday June 16th**

14:30-15:30 Askold Khovansky (Toronto)

16:00-17:00 Stepan Orevkov (Toulouse)

17:30-18:30 Oleg Viro (Stony Brook)

19:30 dinner

2018, Monday, May 28, 15:00, Battelle, Alexander Esterov (Higher School of Economics - Moscow)

**Tropical characteristic classes and Plücker formulas**

Given a proper generic map of manifolds, the Thom polynomial counts (in terms of characteristic classes of the manifolds), how many fibers of the map have a prescribed singularity. However, this tool cannot be directly applied to the study of generic polynomial maps $C^m \to C^n$, because they are not proper. An attempt to extend Thom polynomials in this natural direction leads to what can be called tropical Thom polynomials and tropical characteristic classes.

I will introduce tropical characteristic classes of (very) affine algebraic varieties, compute the tropical version of the simplest Thom polynomials (the Plücker formulas for the number of cusps and nodes of a projectively dual curve), and outline their relation to tropical correspondence theorems and some other possible applications.

2018, Friday, May 25, 10:30, Battelle, Dimitry Kaledin (Steklov & NRU HSE - Moscow)

**Witt vectors, commutative and non-commutative, II**

Witt vectors were first introduced eighty years ago, but they still come up in different questions of commutative and homological algebra, algebraic geometry, and even algebraic topology. I will try to give a general introduction to this remarkable subject, and show both its classical parts and some recent discoveries. The first talk will be quite elementary, first-year algebra should be enough. In the somewhat more advanced second talk, I will try to explain how the simple constructions of the first talk lead to non-commutative generalization of Grothendieck's cristalline cohomology of smooth algebraic varieties over a finite field.

2018, Tuesday, May 22, 15:30, Battelle, Dimitry Kaledin (Steklov & NRU HSE - Moscow)

**Witt vectors, commutative and non-commutative, I**

Witt vectors were first introduced eighty years ago, but they still come up in different questions of commutative and homological algebra, algebraic geometry, and even algebraic topology. I will try to give a general introduction to this remarkable subject, and show both its classical parts and some recent discoveries. The first talk will be quite elementary, first-year algebra should be enough. In the somewhat more advanced second talk, I will try to explain how the simple constructions of the first talk lead to non-commutative generalization of Grothendieck's cristalline cohomology of smooth algebraic varieties over a finite field.

2018, Monday, May 14, 16:30, Battelle, Ilia Itenberg (Paris VI - ENS)

**Finite real algebraic curves**

The talk is devoted to real plane algebraic curves with finitely many real points. We study the following question: what is the maximal possible number of real points of such a curve provided that it has given (even) degree and given geometric genus? We obtain a complete answer in the case where the degree is sufficiently large with respect to the genus, and prove certain lower and upper bounds for the number in question in the general case. This is a joint work with E. Brugallé, A. Degtyarev and F. Mangolte.

2018, Monday, April 16, 16:30, Battelle, Ludmil Katzarkov (Vienna)

**Homological mirror symmetry and the P=W conjecture**

2018, Monday, March 5, 16:00, Battelle, Rahul Pandharipande (ETH)

**On Lehn's conjecture for Segre classes on Hilbert schemes of points of surfaces and
generalizations**

Let L→S be a line bundle on a nonsingular projective surface. I will discuss recent progress concerning the formula conjectured by Lehn in 1999 for the top Segre class of the tautological bundle L^[n] on Hilb(S,n) and the parallel question for vector bundles V→S. Results of Voisin play a crucial role. The talk represents joint work with A. Marian and D. Oprea.

2018, Monday, February 26, 16:30, Battelle, Anton Fonarev (Higher School of Economics)

**Embedding derived categories of curves into derived categories of moduli of stable vector bundles**

One out of many interesting questions about derived categories is the following conjecture by A. Bondal: the bounded derived category of coherent sheaves of a smooth projective variety can be embedded into the bounded derived category of coherent sheaves of a smooth Fano variety. This conjecture is rather nontrivial even for curves. We will show how to embed the derived category of a generic curve of genus g > 1 into the derived category of rank 2 stable vector bundles with a fixed determinant of odd degree. The proof is a nice interplay of algebraic geometry, representation theory and categorical methods. The talk is based on a joint work with A. Kuznetsov.

2017, Monday, November 6, 16:30, Battelle, Jeffrey Giansiracusa (Swansea University)

**Tropical geometry as a scheme theory**

Tropical geometry has become a powerful tool set for tackling problems in algebraic geometry, combinatorics, and number theory. The basic objects have traditionally been considered as certain polyhedral sets and heuristically thought of as algebraic objects defined over the real numbers with the max-plus semiring structure. I will explain how to realize this within an extension of scheme theory and describe the particular form of the equations of tropical varieties in terms of matroids.

2017, Monday, October 30, 16:30, Battelle, Diego Matessi (Università degli Studi di Milano)

**From tropical hypersurfaces to Lagrangian submanifolds**

I will explain a construction of Lagrangian submanifolds of $(\mathbb{C}^*)^2$ or $(\mathbb{C}^*)^3$ which lift tropical hypersurfaces in $\mathbb{R}^2$ or $\mathbb{R}^3$. The building blocks are what I call Lagrangian pairs of pants. These can be constructed as graphs of the differential of a smooth function defined on a Lagrangian co-ameba. I will also explain some possible generalizations and applications to mirror symmetry.

2017, Monday, October 2, 16:30, Battelle, Dmitry Novikov (Weizmann Institute)

**Complex cellular parameterization**

(joint work with Gal Binyamini)

We introduce the notion of a *complex cell*, a complex analog
of the cell decompositions used in real algebraic and analytic
geometry. Complex cells defined using holomorphic data admit a
natural notion of analytic continuation called $\delta$-extension,
which gives rise to a rich hyperbolic geometric structure absent in
the real case. We use this structure to prove that complex cellular
decompositions share some interesting features with the classical
constructions in the theory of resolution of singularities. Restriction
of a complex cellular
decomposition to the reals recovers the preparation theorem for
subanalytic functions, and can be viewed as an analytic continuation
thereof.

A key difference in comparison to the classical resolution of
singularities is that the cellular decompositions are intrinsically
uniform over (sub)analytic families. We deduce a
subanalytic version of the Yomdin-Gromov theorem where $C^k$-smooth
maps are replaced by *mild* maps.

2016, Friday, June 23, 11:00, Battelle, Ernesto Lupercio (CINVESTAV)

**Quantum toric varieties**

I will describe the theory of quantum toric varieties that generalizes usual toric geometry. Joint with Meersseman, Katzarkov and Verjovsky.

2016, Thursday, June 22, 11:30, Battelle, Conan Leung (CUHK)

**Informal introduction to G_2-manifolds III**

2016, Wednesday, June 21, 11:30, Battelle, Conan Leung (CUHK)

**Informal introduction to G_2-manifolds II**

2016, Monday, June 19, 15:00, Battelle, Conan Leung (CUHK)

**Informal introduction to G_2-manifolds I**

Villa Battelle, May 2, 14:00-15:00; May 3 14:15-15:15; May 5, 14:30-15:30, Aaron Bertram (Utah)

Minicourse: “Moduli Spaces of Complexes in Algebraic Geometry ”

The ideal of the twisted cubic in projective three-space is completely described by a 2×3 matrix of linear forms in four variables. The space of such matrices (modulo the actions of GL(2) and GL(3)) is a smooth, projective variety compactifying the space of twisted cubics. But the objects parametrized by the points at the boundary of this moduli space are not ideals of curves. They are complexes of line bundles that are stable with respect to a “stability condition on the derived category.” What does this mean? Can this be used to systematically find nice models for moduli and relate them to moduli spaces of coherent sheaves?

Day 1) Introduction to Stability Conditions. Ordinary stability of vector bundles on a Riemann Surface relies on two invariants: the rank and degree (first chern class). A stability condition on the derived category of coherent sheaves on a complex manifold relies on a generalized rank and degree, and also on an exotic t-structure on the derived category, with an abelian category of complexes at its heart. On an algebraic surface, there are stability conditions whose underlying heart can be described by a tilting construction. However, finding a single stability condition on a projective Calabi-Yau threefold (e.g. the quintic in P4) remains open.

Day 2) Models of the Hilbert Schemes of Points on a Surface. As the stability condition varies, the moduli spaces of stable objects (with respect to the stability condition) undergo a series of birational transformations. The particular example of the Hilbert scheme of ideal sheaves on an algebraic surface has been studied for various classes of surfaces. We will survey some results.

Day 3) The Euler Stability Condition on Projective Space. An interesting stability condition on P^n has the Euler characteristic playing the role of the rank. We will use this stability condition to study stratifications of the spaces of symmetric tensors, generalizing the secant varieties to the Veronese embeddings of P^n. This is joint work with Brooke Ullery.

Villa Battelle, Monday, Apr 3, 16:30-17:30, Lionel Lang (Uppsala University)

**The vanishing cycles of curves in toric surfaces : the spin case **

If the interior polygon of a lattice polygon $\Delta$ is divisible by 2, any generic curve $C$ of the linear system associated to $\Delta$ admits a spin structure $q$. If a loop in $C$ is a vanishing cycle, then the Dehn twist along the loop has to preserve $q$. As a consequence, the image of the monodromy of the linear system is a subgroup of the mapping class group $MCG(C,q)$ that preserves $q$. The main goal of this talk is to compare the image of the monodromy with $MCG(C,q)$. To this aim, we will show on one side that $MCG(C,q)$ admits a very explicit set of generators. On the other, we will construct elements of the monodromy by tropical means. The conclusion will be that the image of the monodromy is the full group $MCG(C,q)$ if and only if the interior polygon admits no other divisors than 2. (joint with R. Crétois)

Villa Battelle, Wednesday, Mar 8, 12:00, Maksim Karev (PDMI)

**Monotone Hurwitz Numbers**

Usual Hurwitz numbers count the number of covers over CP^1 with a fixed ramification profile over point \infty and simply ramified over a specified set of points. They also can be treated as a weighted count of factorizations in the symmetric group. It is known, that Hurwitz numbers can be calculated via intersection indices on the moduli spaces of complex curves by so-called ELSV-formula.

In my talk, I will discuss monotone Hurwitz numbers, which also arise as factorizations count with restrictions. It turns out, that they also can be related to the intersection indices on the moduli spaces of complex curves. I will give a definition of monotone Hurwitz numbers, and try to explain the origin of the monotone ELSV. If time permits, I will speak about the further development of the subject.

The talk is based on the joint work with Norman Do (Monash University).

Villa Battelle, Tuesday, Feb 21, 15:30, Yang-Hui He (London, Nankai and Oxford)

**Calabi-Yau Varieties: From Quiver Representations to Dessins d'Enfants**

We discuss how bipartite graphs on Riemann surfaces encapture a wealth of information about the physics and the mathematics of gauge theories. The correspondence between the gauge theory, the underlying algebraic geometry of its space of vacua as a quiver variety, the combinatorics of dimers and toric varieties, as well as the number theory of dessin d'enfants becomes particularly intricate under this light.

Joint session of “Fables géométriques” and “Groupes de Lie et espaces des modules” seminars.

Villa Battelle, Monday, Feb 20, 16:30, Yang-Hui He (London, Nankai and Oxford)

**Sporadic and Exceptional**

There tends to be exceptional structures in classifications: in geometry, there are the Platonic solids; in algebra, there are the exceptional Lie algebras; in group theory, there are the sporadic groups, to name but a few. Could these exceptional structures be related in some way? A champion for such Correspondences is Prof. John McKay. We take a casual promenade in this land of exceptionology, reviewing some classic results and presenting some new ones based on joint work with Prof. McKay.

Special lecture for “Geometry, Topology and Physics” masterclass students.

Villa Battelle, Friday, December 9, 14:30-15:30, Ozgur Ceyhan (Luxembourg)

**Backpropagation, its geometry and tropicalisation**

The algorithms that make current artificial neural networks successes possible are decades old. They became applicable only recently as these algorithms demand huge computational power. Any technique which reduces the needs for computation have a potential to make great impact. In this talk, I am going to discuss the basics of backpropagation techniques and tropicalisation of the problem that promises to reduce the time complexity and accelerate computations.

2016,Monday, November 7, 16:30, Battelle, Vladimir Fock.

**Separation of variables in cluster integrable systems**

Cluster integrable systems can be viewed from five rather different points of view. 1. As a double Bruhat cell of an affine Lie -Poisson group; 2. As a space of pairs (planar algebraic curve, line bundle on it); 3. Space of Abelian connection on a bipartite graph on a torus; 4. Hilbert scheme of points on algebraic torus. 5. Collection of flags in an infinite space invariant under the action of two commuting operators. We will see the relation between all these descriptions and discuss its quantization and possible generalizations.

2016,Friday, Nov 4, 14:30-15:15 part I, 15:30-16:15 part II, Johannes Walcher (Heidelberg).

**Ideas of D-branes**

Abstract: I will give an introduction to D-branes from the point of view of their origin in the physics of string theory. I will discuss both world-sheet and space-time aspects.

2016, Monday, 23 mai, 16.30, Battelle, Frédéric Bihan.

**Une généralisation de la règle de Descartes pour les systèmes
polynomiaux dont le support est un circuit**

Résumé : La règle de Descartes borne le nombre de racines positives d'un polynôme réel en une variable par le nombre de changements de signe consécutifs de ses coordonnées dans la base monomiale (ordonnée suivant les puissances croissantes). La borne obtenue est optimale et généraliser la règle de Descartes aux systèmes polynomiaux en plusieurs variables est un problème très difficile. Dans un travail avec Alicia Dickenstein (Université de Buenos Aires), nous avons obtenu une généralisation partielle de la règle de Descartes en plusieurs variables. Notre règle s'applique aux systèmes polynomiaux en un nombre arbitraire n de variables dont le support consiste en n+2 monômes quelconques. Comme pour la règle de Descartes usuelle, notre borne est optimale et s'exprime comme un nombre de changement de signes d'une suite de nombres obtenus en considérant les mineurs maximaux de la matrice des coefficients ainsi que de celle des exposants du système.

(in English)**Descartes' Rule of Signs for Polynomial Systems Supported on Circuits**

Descartes’ rule of signs bounds the number of positive roots of an univariate polynomial by the number of sign changes between consecutive coefficients. In particular, this produces a sharp bound depending on the number of monomials. Generalizing Descartes’ rule of signs or the corresponding sharp bound to the multivariable case is a challenging problem. In this talk, I will present a generalization of Descartes’ rule of signs for the number of positive solutions of any system of n real polynomial equations in n variables with at most n+2 monomials. This is a joint work with Alicia Dickenstein (Buenos Aires University).

2016, Monday, 9 mai, Battelle, Eugenii Shustin, 18.30-19.15

**On refined tropical invariants of toric surfaces.**

We discuss two examples of refined count of plane tropical curves. One of them is the refined broccoli invariant. It was introduce by Goettsche and Schroeter for genus zero case, and it turns into some descendant invariant or the broccoli invariant according as the parameter takes value 1 or -1. A possible extension of broccoli invariant to positive genera appeared to be rather problematic. However, the refined version turns to be easier to treat. Jointly with F. Schroeter, we have defined a refined broccoli invariant, counting elliptic tropical curves. This can be done for higher genera as well (work in progress). Another example (joint work with L. Blechman) is the refined descendant tropical invariant (involving arbitrary powers of psi-classes). We discuss also the most interesting related question: What is the complex and real enumerative meaning of these invariants?

2016, Monday, 4 april, 16.30, Battelle.

Lionel Lang (Uppsala)

**The vanishing cycles of curves in toric surfaces (joint work with Rémi Crétois)**

In [Do], Donaldson addressed the following : Do all Lagrangian spheres in a complex projective manifold arise from the vanishing cycles of a deformation to singular varieties? The answer might depend on the choice of the moduli space in which we are allowed to deform our manifold. Already for curves, it leads to interesting questions. In the Deligne-Mumford moduli space M_g, any loop inside a smooth curve can be contracted along a deformation towards a nodal (stable) curve, provided that the genus g>1. What happens if one restricts to a chosen linear system on a toric surface? Degree d curves in the projective plane, for instance. In the latter, two obstructions occur: the loop should not be separating for d>2 (Bezout), the Dehn twist along the loop should preserve a certain spin structure on the curve for d odd (see [Beau]). In the latter, Beauville proves (in particular) that any non-obstructed loop is homologous to a vanishing cycle. In this talk, we suggest a tropical proof of Beauville's result as well as an extension to any (big enough) linear systems on any smooth toric surface. This problem is directly related to the monodromy group given by the complement of the discriminant in the considered linear system. The proof will involve simple Harnack curves, introduced by Mikhalkin, and monodromy given by partial tropical compactifications of the linear system. If time permits, we will also discuss this problem at the isotopic level, problem that is still open.

[Beau] : Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections complètes. A. Beauville, 1986. [Do] : Polynomials, vanishing cycles and Floer homology. S.K. Donaldson, 2000.

2016, Monday, 21 mars, 16.30, Battelle.

Boris Shapiro (Stockholm)

**On the Waring problem for polynomial rings**

We discuss a natural analog of the classical Waring problem for $C[x_1,…,x_n]$. Namely, we show that a general form p from $C[x_1,…,x_n]$ of degree kd where k>1 can be represented as a sum of at most $k^n$ k-th powers of forms of degree d. Noticeably, $k^n$ coincides with the number obtained by naive dimension count if d is sufficiently large.

2016, Friday, March 18, 14.15, villa Battelle.

Sergey Galkin (Moscow)

**Gamma conjectures and mirror symmetry**

I will speak about an exotic integral structure in cohomology of Fano manifolds that conjecturally can be expressed in terms of Euler's gamma-function, how one can observe it by computing asymptotics of a quantum differential equation, and how one can prove the conjectures using mirror symmetry. This is a joint work with Vasily Golyshev and Hiroshi Iritani (1404.6407, 1508.00719).

2016, Thursday, March 17 Colloquium, villa Battelle

Vassily Golyshev (Moscow), 16:15

**Around the gamma conjectures.**

Abstract: We will state the gamma conjectures for Fano manifolds and explain how quantum cohomology makes it possible to enhance the classical Riemann-Roch-Hirzebruch theorem by relating the curve count on a variety to its characteristic classes. We will indicate how the gamma conjectures are proved in the known cases.

2016, Monday, 14 mars, 16.30, Battelle.

E. Abakoumov (Paris-Est)

**Growth of proper holomorphic maps and tropical power series**

How fast a proper holomorphic map, say, from C to C^n can grow? It turns out that the tropical power series appear naturally in answering this question, as well as in some related approximation problems on the complex plane. The talk is based on joint work with E. Dubtsov.

2015, Tuesday, 8 December, 14.30, Battelle. (joint with Séminaire "Groupes de Lie et espaces des modules”)

Bernd Sturmfels (UC Berkeley)

**Exponential Varieties**

Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Another special class, including Gaussian graphical models, are inverses of symmetric matrices satisfying linear constraints. We present a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. This is joint work with Mateusz Michalek, Caroline Uhler, and Piotr Zwiernik.

2015, Tuesday, December 8, 11:15 -- 12:15, Battelle.

Renzo Cavalieri (Colorado State)

**Tropical geometry: a graphical interface for the GW/Hurwitz correspondence.**

In their study of the Gromov-Witten theory of curves [OP], Okounkov and Pandharipande used the degeneration formula to express stationary descendant invariants of curves in terms of Hurwitz numbers and one point descendant relative invariants. Then they use operator formalism to organize the combinatorics of the degeneration formula, and the one point invariants into completed cycles. In joint work with Paul Johnson, Hannah Markwig and Dhruv Ranganathan, we revisit their formalism and show that the Feynmann diagrams that are secretly behind the scenes in [OP] are in fact tropical curves. This yields some mild refinements of the Gromov-Witten/Hurwitz correspondence of [OP]. Time permitting we will describe how a generalization of these tecniques should lead to unveiling a similar structure in the stationary/descendant GW theory of sliceable surfaces.

2015, Monday, 7 December, 16.15, Villa Battelle

Israel Vainsencher (Universidade Federal de Minas Gerais, Brasil)

**Legendrian curves**

A twisted cubic curve in 3-space is known to define a (non-integrable) distribution of planes. The planes of the distribution osculate the original twc. We show how to define virtual numbers N_d which enumerate the rational curves of degree d which are tangent to that distribution and further meet 2d+1 general lines. (Based on Eden Amorim thesis)

The next lecture of the course

**“Imaginary time in Kaehler geometry, quantization and tropical amoebas”**
by José Mourão

will be on Monday 9 November, 17.00 Battelle.

2015, October 27, 15.15 and October 29, 16.15, and November 2, 17.00, Villa Battelle

**(Minicourse) Imaginary time in Kaehler geometry, quantization and tropical amoebas.**

José Mourão, Mathematics Department, Instituto Superior Tecnico, Portugal.

For a compact Kaehler manifold $M$ and a function $H$ on $M$ we give a simple definition of the continuation of the flow defined by $H$ to complex time, $\tau$, using the Groebner theory of Lie series. The resulting complexified (or complex time) symplectomorphisms are diffeomorphisms for some $|\tau|< R_H$. For larger values of $|\tau|$ they may correspond e.g. to the collapse of $M$ to a totally real submanifold. Simple examples will be discussed.

Kahler geometry applications: Imaginary time symplectomorphisms correspond to Mabuchi geodesics in the infinite dimensional space of Kaehler metrics with fixed cohomology class. We get thus a explicit way of constructing Mabuchi geodesics from Hamiltonian flows.

Quantum theory applications: By lifting the imaginary time symplectomorphisms to the quantum bundle we get generalized coerent state transforms and are able to study the unitary equivalence of quantizations corresponding to nonequivalent polarizations.

Tropical geometry applications: For toric varieties the toric geodesics of the Mabuchi metric are straight lines in the space of Guillemin-Abreu symplectic potentials. Taking a strictly convex function $H$ (as a function on the moment polytope) one has that, for large geodesic times s, there is a simple relation between the moment map $\mu_s$ and the $Log_t$ map of amoeba theory ($t=e^s$) . This relation further simplifies if one takes as $H$ the full symplectic potential, which is continuous but not smooth on $M$ and corresponds to a geodesic of Kaehler metrics with cone angle singularities. The tropical limit corresponds thus, in this setting, to the infinite geodesic time limit corresponding to convex hamiltonians.

Materials: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/2015_Geneva_University_Seminar.pdf

Index: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_index.pdf

Lecture 1 (introduction and different definitions of complex time evolution): http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L1.pdf

Lecture 2: Kahler tropicalization of C^*: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L2.pdf

Lecture 3: Kahler tropicalization of C and (strange) actions of G_C on Kahler structures: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L3.pdf

Lecture 4: C^infty Kahler tropicalization of toric varieties and of hypersurfaces in toric varieties: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L4.pdf

Lecture 5: C^0 Kahler tropicalization of toric varieties and of hypersurfaces in toric varieties: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L5.pdf

Overview seminar: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/2015_Geneva_University_Seminar.pdf

2015, October 27, Tuesday, 15.15, Villa Battelle ( together with Séminaire “Groupes de Lie et espaces des modules”)

**Imaginary time in Kaehler geometry, quantization and tropical amoebas.**

José Manuel Cidade Mourão, Mathematics Department, Instituto Superior Tecnico, Portugal.

For a compact Kaehler manifold $M$ and a function $H$ on $M$ we define a continuation of the Hamiltonian flow of $H$ to complex time $\tau$. The resulting complexified (or complex time) symplectomorphisms are diffeomorphisms for some $|\tau|< R_H$. For larger values of $|\tau|$ they may correspond e.g. to the collapse of $M$ to a totally real submanifold. We'll discuss some simple examples and applications to Kaehler geometry, quantization and tropical geometry. This talk is the first lecture of a mini-course to be given during October-November 2015.

Materials: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/2015_Geneva_University_Seminar.pdf

2015, October 5, Monday, 16.20, Villa Battelle

**Tropicalization of Poisson-Lie groups**

Anton Alexeev (UniGe)

In the first part of the talk, we recall the notion of Poisson-Lie groups and cluster coordinates for some simple examples.

In the second part, we use the notion of tropicalization to construct completely integrable systems, and for the Poisson-Lie group SU(n)^* match it with the Gelfand-Zeiltin integrable system.

The talk is based on joint works with I. Davydenkova, M. Podkopaeva and A. Szenes.

2015, September 28, Monday, 16.15, Villa Battelle

**What is moonshine?**

Sergey Galkin (HSE, Moscow)

I will describe a few instances of geometric moonshines: surprising appearance of modular forms and sporadic groups as the answers to seemingly unrelated geometric and topological questions.

2015, 21 September, Monday, 16.15, Villa Battelle.

**Cohomology of superforms on polyhedral complexes and Poincare duality for tropical manifolds**

Kristin Shaw.

Superforms introduced by Lagerberg are bigraded differential forms on $\mathbb R^n$ which can be restricted to polyhedral complexes. We extend these forms to $\mathbb T^n = [-\infty, \infty)^n$ and show that their de Rham cohomology is equivalent to tropical $(p, q)$ cohomology Furthermore, we establish Poincaré duality for cohomology of tropical manifolds. As in the classical theory, the Poincaré pairing can be formulated in terms of integration of superforms.

old page of the seminar http://www.unige.ch/math/folks/langl/fables/