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symplectic [2022/09/16 15:26] – kalinin0symplectic [2025/05/04 18:57] (Version actuelle) – g.m
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 ===== GeNeSys: Geneva-Neuchâtel Symplectic geometry seminar ===== ===== GeNeSys: Geneva-Neuchâtel Symplectic geometry seminar =====
  
 +
 +**2025, May 6, Tuesday, Université de Neuchâtel**
 +
 +  Marco Golla (Université de Nantes, CNRS) 
 +  13h00, Salle B217
 +  Alexander polynomials and symplectic curves in CP^2
 +  
 +Libgober defined the Alexander polynomial of a (complex) plane projective curve and showed that it detects some Zariski pairs ofcurves: these are curves with the same degree and the same singularities but with non-homeomorphic complements. He also proved that the Alexander polynomial of a curve divides the Alexander polynomial of its link at infinity and the product of Alexander polynomials of the links of its singularities. We extend Libgober's definition to the symplectic case and prove that the divisibility relations also hold in this context. This is joint work with Hanine Awada. 
 +
 +  Conan Leung (The Chinese AV¶ÌÊÓƵ of Hong Kong)
 +  14h30, Salle B217
 +  3d Mirror Symmetry is 2d Mirror Symmetry
 +  
 +We introduce an approach to study 3d mirror symmetry via 2d mirror symmetry. The main observations are: (1) 3d brane transforms are given by SYZ-type transforms; (2) the exchange of symplectic and complex structures in 2d mirror symmetry induces the exchange of Kähler and equivariant parameters in 3d mirror symmetry; and (3) the functionalities of 2d mirror symmetry control the gluing of 3d mirrors. 
 +
 +  Sobhan Seyfaddini (ETH Zürich)
 +  14h30, Salle B217
 +  Closing Lemmas on Symplectic Manifolds
 +  
 +Given a diffeomorphism of a manifold, can one perturb it to create a periodic orbit passing through a specified region? This question, first raised in the 1960s, is known as the Closing Lemma. While the problem was resolved positively in C^1 regularity long ago, it remains largely open at higher levels of smoothness. Recent years have seen significant progress in the C^\infty setting, particularly for area-preserving maps on surfaces. In this talk, I will review these developments, highlighting works by Asaoka, Irie, Cristofaro-Gardiner, Edtmair, Hutchings, Prasad, and Zhang. I will also present some recent joint work with Cineli & Tanny and Mak & Smith, including partial results in higher dimensions.
 +
 +
 +
 +**2025, March 24, Monday, Université de Genève**
 +
 +  Lionel Lang (Gävle)
 +  14h00, Salle 01-15
 +  Logarithmic volumes of holes of hypersurfaces and tropicalization of periods
 +
 +Integrating the logaritmic volume form on well chosen discs bounded on hypersurfaces gives a local system of coordinates on the linear system of such hypersurfaces. Surprisingly, the same procedure gives global coordinates on the corresponding linear system of tropical hypersurfaces. In both the algebraic and tropical settings, the differential of these coordinate systems with respect to the coefficents of the defining equations are period matrices.  I want to discuss how this observation can be used to study the tropicalization of periods of hypersurfaces.
 +
 +  Johannes Rau (Universidad de los Andes)
 +  16h00, Salle 06-13
 +  Welschinger-Witt invariants
 +
 +The "quadratically enriched" invariants defined by Kass-Levine-Solomon-Wickelgren are obtained by counting rational curves over an arbitrary ground field and take values in the associated Witt-ring of quadratic forms. Specializing these values to rank and signature reproduces the corresponding Gromov-Witten and Welschinger invariants, respectively. In my talk, I want to introduce the concept of a Witt invariant which can be computed in terms of multiquadratic and multireal field extensions. We will see that Welschinger invariants give naturally rise to such an Witt invariant which in some cases (conjecturally always) coincides with the quadratically enriched invariants.
 +
 +**2024, December 6, Friday, Université de Genève**
 +
 +  Jeffrey Hicks (Edinburgh)
 +  14h00, Salle 06-13
 +  What is the mirror to a Weinstein Neighborhood?
 +
 +A fundamental result in symplectic geometry is that every Lagrangian submanifold has a small standard neighborhood. The existence of this Weinstein Neighborhood implies that there are no "local" invariants of a Lagrangian submanifold other than the topology of the Lagrangian submanifold. Mirror symmetry is a conjectured dictionary between symplectic and algebraic geometry on a pair of mirror spaces. Under this dictionary, Lagrangian submanifolds in a symplectic manifold are related to sheaves on the mirror algebraic space.
 +
 +In this talk --- which will only use intuition, not techniques from symplectic geometry --- we give a candidate definition of the "Weinstein neighborhood" of a sheaf on an algebraic space, which extends the definition of an affine neighborhood of a point. We'll prove the existence of such a neighborhood for the structure sheaf of a hyperplane in projective space. Time permitting, we'll draw connections to tropical geometry and spaces of nonpositive curvature.
 +
 +   Francesca Carocci (Roma)
 +   16h00, Salle 06-13
 +   Realizability of tropical maps via logarithmic geometry and curves singularities
 +
 +Abstract: In this talk I will explain the interpretation of Speyer well-spacedness condition for genus 1 realizability in terms of logarithmic deformation theory and curve singularities given by Ranganathan--Santos-Parker--Wise. I will then discuss how the same ideas can be used to approach the genus 2 realizability problem and the partial results obtained so far for this case.
 +
 +
 +
 +------
 +
 +**2024, October 29, Tuesday, Université de Neuchâtel**
 +
 +  Adi Dickstein (Tel Aviv AV¶ÌÊÓƵ)
 +  14h30
 +  Symplectic versus topological quasi-states 
 + 
 +Topological quasi-states are special functionals on the algebra of continuous functions which are linear on single-generated subalgebras. They trace their origins to the von Neumann axioms of quantum mechanics. On symplectic surfaces, every topological quasi-state is symplectic, i.e., linear on Poisson-commutative subalgebras. We discuss the failure of this phenomenon in higher dimensions based on the study of symplectic embeddings of polydiscs. Furthermore, we introduce a Wasserstein-type metric on quasi-states and use it for quantitative constraints on symplectic quasistates. The talk is based on a joint work with Frol Zapolsky.
 +
 +  Vladimir Fock (Université de Strasbourg)
 +  16h30
 +  TBK-symplectic structures
 +  
 +A K-symplectic structure on an (algebraic) manifold X is a section of a certain quotient sheaf on X. The Steinberg symbol is a map of this sheaf to an Abelian group. A certain Steinberg symbol gives a symplectic structure on X. Such structures do not exist for every symplectic manifold, but once it exists (and it exists for cluster varieties) one can say much more about this manifold. In particular it gives a pre-quantum line bundle, computes hyperbolic volumes, and in addition gives certain invariants for manifolds over number fields as well as central extensions of simple and affine Lie groups.
 +
 +------
 +**2023, December 4, Monday, Université de Genève**
 +
 +  Diego MATESSI (Milano)
 +  15h00, Salle 06-13
 +  Tropical mirror symmetry and real Calabi-Yaus
 +  
 +I will present some work in progress jont with Arthur Renaudineau.  The goal is to understand the topology of real Calabi-Yaus by combining the Renaudineau-Shaw spectral sequence with mirror symmetry.  We will consider mirror pairs of Calabi-Yau hypersurfaces X and X' in toric varieties associated to dual reflexive polytopes. The first step is to prove an isomorphism between tropical homology groups of X and X', reproducing the famous mirror symmetry exchange in hodge numbers. We then expect that the boundary maps in the Renaudineau-Shaw spectral sequence, computing the homology of the real Calabi-Yaus, can be interpreted, on the mirror side, using classical operations on homology.
 +
 +------
 +**2023, November 6, Monday, Université de Neuchâtel**
 +
 +  Prof. Dr. Emmanuel Opshtein (Université de Strasbourg)
 +  15:00, Université de Neuchâtel, Rue Emile-Argand 11, Room B217
 +  Liouville polarizations and their Lagrangian skeleta in dimension 4
 +  
 +In the simplest framework of a symplectic manifold with rational symplectic class, a symplectic polarization is a smooth symplectic hypersurface Poincaré-Dual to a multiple of the symplectic class. This notion was introduced by Biran, together with the isotropic skeleta associated to a polarization, and he exhibited symplectic rigidity properties of these skeleta. In later work, I generalized the notion of symplectic polarizations to any closed symplectic manifold, and showed that they are useful to construct symplectic embeddings. In the present talk, I will explain how this notion of polarization can be generalized further to the affine setting in dimension 4 and how it leads to more interesting embedding results. These refined embedding constructions provide a new way to understand the symplectic rigidities of Lagrangian skeleta noticed by Biran and get new ones. These results also lead to (seemingly new kind of) rigidities for some Legendrian submanifolds in contact geometry. I will present several examples and applications. Work in progress, in collaboration with Felix Schlenk.
 +
 +-------------
 +**2023, April 26, Wednesday, Université de Genève**
 +
 +  Lionel Lang (Gävle)
 +  Measuring holes of hypersurfaces
 +  14h00
 +
 +In 2000, Mikhalkin introduced a class of real algebraic planar curves now known as simple Harnack curves. Among their many nice properties, these curves appear as spectral curves of planar dimers. In this context, Kenyon and Okounkov showed that any simple Harnack curve is determined by the logarithmic area of some well chosen membranes bounded on the curve (plus some boundary conditions). This is a very special situation since, in general, the areas of these membranes only provide local coordinates on the space of curves under consideration. In this talk, Lionel Lang would like to discuss a generalization of this fact to arbitrary dimension, namely how logarithmic volumes of well chosen membranes provide local coordinates on linear systems of hypersurfaces. Moreover, these local coordinates have an obvious tropicalization that gives rise to global coordinates on the corresponding linear system of tropical hypersurfaces. Eventually, if time permits, he would like to discuss potential applications to deformation of real algebraic hypersurfaces.
 +
 +  Viatcheslav Kharlamov (Strasbourg)
 +  Unexpected loss of Smith-Thom maximality: the case of Hilbert squares of surfaces
 +  16h00
 +
 +Viatcheslav Kharlamov explores the maximality of the Hilbert square of maximal real surfaces, and finds that in many cases the Hilbert square is maximal if and only if the surface has connected real locus. In particular, the Hilbert square of no maximal K3-surface is maximal. Nevertheless, they exhibit maximal surfaces with disconnected real locus whose Hilbert square is maximal. This talk is based on a joint work with R. Rasdeaconu.
 +----------------
 +
 +**2023, March 21, Tuesday, Université de Neuchâtel**
 +
 +  Patricia Dietzsch (ETH Zürich)
 +  Dehn twists along real Lagrangian spheres
 +  14h00
 +  
 +A major tool in the study of the Dehn twist along a Lagrangian sphere is Seidel's long exact sequence. This sequence comes with a distinguished element $A$ in the Floer homology group of the Dehn twist. In this talk we will discuss a property of $A$ in case the Dehn twist is a monodromy in a real Lefschetz fibration. We will see that the real structure induces an automorphism on the Floer homology group of the Dehn twist and that $A$ is a fixed point.
 +
 +  Cheuk Yu Mak (AV¶ÌÊÓƵ of Southampton)
 +  Non-displaceable Lagrangian links in 4 manifolds
 +  16h00
 +
 +One of the earliest fundamental applications of Lagrangian Floer theory is detecting the non-displaceablity of a Lagrangian submanifold. Much progress and generalizations have been made since then but little is known when the Lagrangian submanifold is disconnected. In this talk, we describe a new idea to address this problem. Subsequently, we explain how to use Fukaya-Oh-Ohta-Ono and Cho-Poddar theory to show that for every $S^2×S^2$ with a non-monotone product symplectic form, there is a continuum of disconnected, non-displaceable Lagrangian submanifolds such that each connected component is displaceable. This is joint work with Ivan Smith.
 +----------------
 +
 +**2022, October 18, Université de Genève, salle 6-13**
 +
 +
 +  Ilia Itenberg (Sorbonne)
 +  Real enumerative invariants and their refinement
 +  Salle 6-13, 14h15
 +
 +**Abstract:**
 +
 +The talk is devoted to several real and tropical enumerative problems.
 +We suggest new invariants of the projective plane (and, more generally, of toric surfaces)
 +that arise as results of an appropriate enumeration of real elliptic curves.
 +These invariants admit a refinement (according to the quantum index) similar to the one introduced by Grigory Mikhalkin in the rational case.
 +We discuss the combinatorics of tropical counterparts of the elliptic invariants under consideration and establish a tropical algorithm
 +allowing one to compute them.
 +This is a joint work with Eugenii Shustin.
 ----------------     ----------------    
 **2022, September 27, Tuesday, Université de Neuchâtel** **2022, September 27, Tuesday, Université de Neuchâtel**
symplectic.1663334786.txt.gz · Dernière modification : 2022/09/16 15:26 de kalinin0
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