Local Organisers : Erwan Brugallé, Anaïs Goulian

Nantes, November 21st-23rd 2018

Laboratoire de Mathématique Jean Leray, Nantes






Omid Amini: Hybrid topology as a bridge between string theory and quantum field theory

In this talk, I will formulate and discuss from a mathematical perspective some problems and results concerning Riemann surfaces, tropical curves and their moduli, which emerge from the idea of realizing Feynman amplitudes as the low-energy limit of string amplitudes, as suggested by Physicists. The link is provided by Hybrid topology which relates tropical non-Archimedean and Archimedean worlds. This is joint work with S. Bloch, J. Burgos and J. Fresan.

Pierrick Bousseau: Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting.

Gross-Hacking-Keel have given a construction of mirror families of log Calabi-Yau surfaces in terms of counts of rational curves. I will explain how to deform this construction by counts of higher genus curves to get non-commutative deformations of these mirror families. The proof involves a correspondence theorem between Block-Göttsche refined tropical curve counts and some higher genus Gromov-Witten invariants of toric surfaces.

Marco Golla: Singular symplectic curves: isotopy and symplectic fillings

I will be talking about symplectic curves (mostly in the projective plane) whose singularity are modelled over complex singularities. I will discuss the (existence and) uniqueness (up to isotopy) of these curves, phrasing it in terms of symplectic fillings; the focus will be on rational curves with irreducible singularities. This is joint work with Laura Starkston (in progress).

Andrés Jaramillo: Morsification of totally real singularities of type (3,3n)

A morsification of a real plane singularity is a real deformation with the maximal possible number of hyperbolic nodes. Morsifications are an important tool for the study of Dynkin diagrams, monodromy, topology of the singularity link and other characteristics of singularities. In this talk I will address the problem of isotopy classification of morfisications of totally real singularities of type $(3,3n)$. I will show how to obtain this classification by combinatorial means via dessins d'enfants and how it can be encoded by wiring diagrams. I will also described the classification of these morsifications up to Reidemeister moves.

Viatcheslav Kharlamov: A glimpse into comparative asymptotic behavior in real and complex enumerative problems.

One of the first striking applications of the integer-valued real enumerative geometry originated was a disclosing of an unexpected log-asymptotic-equivalence between the number of real and the number of complex solutions in certain enumerative geometry problems. Gradually, with resolving of other enumerative geometry problems, this phenomenon of log-asymptotic-equivalence continues to show its ubiquity, albeit up to rather diverse scale factors. I plan to discuss what we know and what we don’t know about the latter in examples coming from the classical Schubert calculus. A certain advantage of these, more down to earth, examples is that they enable us to count objects of higher dimensions as well.

Marc Levine: Quadratic Welschinger invariants

For $S$ a smooth del Pezzo surface over a field $k$, $D$ an effective (semi-ample) divisor on $S$ with $-D.K>0$, and $p_*:=\sum_ip_i$ a reduced effective 0-cycle of degree $-D.K-1$, we define an invariant $Wel(S,D, p_*)$ in the Grothendieck-Witt ring of $k$. For general $p_*$, the rank of $Wel(S,D, p_*)$ is the count of rational curves in $|D|$ containing $p_*$ and for $k$ a subfield of the reals, the signature of $Wel(S,D, p_*)$ recovers Welschinger’s invariant. We also prove an invariance property of $Wel(S,D, p_*)$, namely, that this quadratic form is constant on $A^1$-connected components of $k$-points in the open subscheme $Sym^{-D.K-1}(S)^0$ of $Sym^{-D.K-1}(S)$ parametrizing reduced 0-cycles, strengthening Welschinger’s invariance result. The method of proof closely follows Welschinger’s arguments, as rephrased in the algebraic setting by Itenberg-Kharlamov-Shustin.

Grigory Mikhalkin: Tropical geometry around Lagrangian submanifolds.

Given a closed symplectic manifold $X$, and a differentiable manifold $L$, can we embed $L$ into $X$ as a Lagrangian submanifold? This central question in Symplectic Geometry is far from being resolved. According to the Symplectic Field Theory approach (as proposed by Eliashberg, Givental and Hofer about 20 years ago) if $L$ is embeddable to $X$ then enumerative invariants of $T^*L-L$ and $X$ must be compatible. If $L$ is a real torus then $T^*L$ coincides with $(C^*)^n$ and its enumerative geometry is described by the tropical geometry in $R^n$. In this talk we'll look at some other examples of $L$ when tropical geometry comes into play, including the famous "conifold transition" case of $L=S^3$ and its finite quotients. We'll also consider some cases when $L$ is disconnected or singular.

Stepan Orevkov: Separating semigroup of hyperelliptic curves and curves of genus 3 and 4.