CONFERENCE AND WINTER SCHOOL ON:
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Lecture Series
Lecture Series I given by Ilia Itenberg (Universite Pierre et Marie Curie - Paris 6)
Title: "Tropical curves"
Abstract. The purpose of this mini-course is to give an introduction to tropical geometry
making an emphasis on applications of tropical geometry in real and complex enumerative
problems. In particular, we discuss main definitions and results concerning tropical curves,
as well as various tropical enumerative problems, including refined (or quantum) enumeration
of tropical curves and its applications.
Lecture Series II given by Erwan Brugallé (Ecole Polytechnique Palaiseaux, Paris)
Title: "Real enumerative geometry"
Abstract. Enumerative geometry is the area of mathematics which studies questions like:
how many lines pass through two points (easy)? How many conics pass through five points
(easy)? How many cubics with a crossing point pass through 8 points (less easy)?... The
aim of this course is to give a basic introduction to the enumeration of real curves in real
algebraic or real symplectic manifolds. We will particularly focus on the case of real rational
curves in real algebraic rational surfaces.
Lecture Series III given by Mattias Jonsson (University of Michigan, Ann Arbor)
Title: "An introduction to Berkovich spaces"
Abstract. Berkovich spaces are analogues of complex manifolds that appear when replacing
complex numbers by the elements of a general normed field, e.g. p-adic numbers
or formal Laurent series. They were introduced in the late 1980’s by Vladimir Berkovich
as a more honestly geometric alternative to the rigid spaces earlier conceived by Tate. In
recent years, Berkovich spaces have seen a large and growing range of applications to complex
analysis, tropical geometry, complex and arithmetic dynamics, the local Langlands
program, Arakelov geometry,... I will give a introduction to Berkovich spaces with a view
towards some of these applications.
Lecture Series IV given by Mohammed Abouzaid (Columbia University)
Title: "Mirror symmetry via non-archimedean geometry"
Abstract. I will describe a program, building on ideas of Kontsevich-Soibleman and
Fukaya, for proving Homological mirror symmetry by associating to symplectic manifolds
equipped with torus fibrations a rigid-analytic space, which is a moduli space of rank-1 local
systems on the torus fibres, and which ends up being the mirror space. The key insights
required to implements such a constructions are: (1) wall crossing maps from moduli
spaces of holomorphic discs, giving transition functions between different affinoid covers of
the mirror (2) constructions of objects of the Fukaya category associated to singular fibres
and (3) a TQFT argument for proving the non-degeneracy of the mirror functor.
Conference talks
Conference Talk 1 given by Kazushi Ueda (University of Tokyo)
Title: "Residue mirror symmetry for Grassmannians"
Abstract. Motivated by recent works on localizations in A-twisted gauged linear sigma
models, we will discuss a generalization of toric residue mirror symmetry to complete
intersections in Grassmannians. This is a joint work with Yutaka Yoshida.
Conference Talk 2 given by Cheol Hyun Cho (Seoul National University)
Title: "Non-commutative homological mirror functor"
Abstract. We explain an elementary geometric construction ( of Lagrangian Floer theory)
to construct various non-commutative mirrors of symplectic torus or punctured Riemann
surfaces. Such a mirror is given by non-commutative Landau-Ginzburg model, which
is a non-commutative algebra together with a choice of a central element. This construction
naturally provides a canonical functor from Fukaya category to the matrix factorization
category of non-commutative Landau-Ginzburg model.
Conference Talk 3 given by Oh Yong-Geun (IBS POSTECH)
Title: "Bulk deformations and nondisplaceable Lagrangian tori"
Abstract. In this talk we will explain bulk deformations of Lagrangian Floer theory
and its associated potential functions. We will then explain how one can exploit such
deformations to locate nondisplaceable Lagrangian fibers and an algorithm of locating
those as the intersections of certain collection of tropical curves selected purely in terms of
the associated moment polytope in the toric context. The talk is based on joint work with
Fukaya-Ohta and Ono, and also on the work of my two graduate students, Yoosik Kim and
Jaeho Lee.
Conference Talk 4 given by Grisha Mikhalkin (Universite de Geneve, Switzerland)
Title: "Quantum index of real plane curves and refined enumerative geometry"
Abstract. We note that under certain conditions, the area bounded by the logarithmic
image of a real plane curve is a half-integer multiple of pi square. The half-integer number
can be interpreted as the quantum index of the real curve and used to refine real enumerative
invariants. The result agrees with the Block-Göttsche numbers from the tropical world.
Lecture Series I given by Ilia Itenberg (Universite Pierre et Marie Curie - Paris 6)
Title: "Tropical curves"
Abstract. The purpose of this mini-course is to give an introduction to tropical geometry
making an emphasis on applications of tropical geometry in real and complex enumerative
problems. In particular, we discuss main definitions and results concerning tropical curves,
as well as various tropical enumerative problems, including refined (or quantum) enumeration
of tropical curves and its applications.
Lecture Series II given by Erwan Brugallé (Ecole Polytechnique Palaiseaux, Paris)
Title: "Real enumerative geometry"
Abstract. Enumerative geometry is the area of mathematics which studies questions like:
how many lines pass through two points (easy)? How many conics pass through five points
(easy)? How many cubics with a crossing point pass through 8 points (less easy)?... The
aim of this course is to give a basic introduction to the enumeration of real curves in real
algebraic or real symplectic manifolds. We will particularly focus on the case of real rational
curves in real algebraic rational surfaces.
Lecture Series III given by Mattias Jonsson (University of Michigan, Ann Arbor)
Title: "An introduction to Berkovich spaces"
Abstract. Berkovich spaces are analogues of complex manifolds that appear when replacing
complex numbers by the elements of a general normed field, e.g. p-adic numbers
or formal Laurent series. They were introduced in the late 1980’s by Vladimir Berkovich
as a more honestly geometric alternative to the rigid spaces earlier conceived by Tate. In
recent years, Berkovich spaces have seen a large and growing range of applications to complex
analysis, tropical geometry, complex and arithmetic dynamics, the local Langlands
program, Arakelov geometry,... I will give a introduction to Berkovich spaces with a view
towards some of these applications.
Lecture Series IV given by Mohammed Abouzaid (Columbia University)
Title: "Mirror symmetry via non-archimedean geometry"
Abstract. I will describe a program, building on ideas of Kontsevich-Soibleman and
Fukaya, for proving Homological mirror symmetry by associating to symplectic manifolds
equipped with torus fibrations a rigid-analytic space, which is a moduli space of rank-1 local
systems on the torus fibres, and which ends up being the mirror space. The key insights
required to implements such a constructions are: (1) wall crossing maps from moduli
spaces of holomorphic discs, giving transition functions between different affinoid covers of
the mirror (2) constructions of objects of the Fukaya category associated to singular fibres
and (3) a TQFT argument for proving the non-degeneracy of the mirror functor.
Conference talks
Conference Talk 1 given by Kazushi Ueda (University of Tokyo)
Title: "Residue mirror symmetry for Grassmannians"
Abstract. Motivated by recent works on localizations in A-twisted gauged linear sigma
models, we will discuss a generalization of toric residue mirror symmetry to complete
intersections in Grassmannians. This is a joint work with Yutaka Yoshida.
Conference Talk 2 given by Cheol Hyun Cho (Seoul National University)
Title: "Non-commutative homological mirror functor"
Abstract. We explain an elementary geometric construction ( of Lagrangian Floer theory)
to construct various non-commutative mirrors of symplectic torus or punctured Riemann
surfaces. Such a mirror is given by non-commutative Landau-Ginzburg model, which
is a non-commutative algebra together with a choice of a central element. This construction
naturally provides a canonical functor from Fukaya category to the matrix factorization
category of non-commutative Landau-Ginzburg model.
Conference Talk 3 given by Oh Yong-Geun (IBS POSTECH)
Title: "Bulk deformations and nondisplaceable Lagrangian tori"
Abstract. In this talk we will explain bulk deformations of Lagrangian Floer theory
and its associated potential functions. We will then explain how one can exploit such
deformations to locate nondisplaceable Lagrangian fibers and an algorithm of locating
those as the intersections of certain collection of tropical curves selected purely in terms of
the associated moment polytope in the toric context. The talk is based on joint work with
Fukaya-Ohta and Ono, and also on the work of my two graduate students, Yoosik Kim and
Jaeho Lee.
Conference Talk 4 given by Grisha Mikhalkin (Universite de Geneve, Switzerland)
Title: "Quantum index of real plane curves and refined enumerative geometry"
Abstract. We note that under certain conditions, the area bounded by the logarithmic
image of a real plane curve is a half-integer multiple of pi square. The half-integer number
can be interpreted as the quantum index of the real curve and used to refine real enumerative
invariants. The result agrees with the Block-Göttsche numbers from the tropical world.