School courses

Francesco Bottacin (Università di Padova)
"The Kobayashi-Hitchin correspondence" Lecture notes 1 2 3 4

Topics covered include:
- Connections on complex vector bundles
- Holomorphic structures and integrable connections
- Hermitian-Einstein vector bundles
- Stable vector bundles, Higgs bundles
- The Kobayashi-Hitchin correspondence
- The extension of these results from the projective to the quasi-projective case: parabolic Higgs bundles, pluri-harmonic metrics and tame harmonic bundles.

Damien Calaque (Universitè de Montpellier 2, France)
"From derived algebraic geometry to derived symplectic geometry"

Abstract:
These lectures are meant as an introduction to derived symplectic geometry.
We will start by reviewing the foundations of derived algebraic geometry after to Toen-Vezzosi and Lurie. We will introduce in particular derived schemes and stacks, as well as quasi-coherent sheaves on these.
In a second part we will review some recent results about symplectic and Lagrangian structures in derived geometry, and provide some examples that are relevant in ordinary symplectic geometry (with a new interpretation of Hamiltonian reduction) and mathematical physics (TFTs and the BV formalism).

Denis Charles Cisinski (Universitè de Toulouse, France)
"$\infty$-categories" Lecture notes

Abstract:
These lectures are meant as an introduction to $\infty$-categories.
After a short introduction to Quillen's theory of model categories, we will see few examples as well as some methods to produce a large class of them. We will then switch to Joyal and Lurie's theory of $\infty$-categories, by studying an appropriate model category structure on the category of simplicial sets. We will then focus on a rectification theorem: any category enriched in spaces gives rise to an $\infty$-category, and any $\infty$-category can be obtained in this way. In particular, any model category defines an $\infty$-category in a rather concrete way.
After explaining adjoints as well as limits and colimits in $\infty$-categories, we will explain the notion of stable infinity-category, and see that the Dold-Kan correspondence, together with Serre's theorem on the finiteness of homotopy groups of spheres, induce an equivalence between the homotopy theories of $\mathbb{Q}$-linear stable $\infty$-categories and of $\mathbb{Q}$-linear pre-triangulated dg-categories.
If time permits, we will say a few words about monoidal structures in this setting.

Claude Sabbah (École Polytechnique, Palaiseau, France)
"Hodge D-modules" Lecture notes Slides 1 2 3

Prerequisite:
Basic knowledge of Hodge theory and its use in algebraic geometry
(e.g., the books of Griffiths-Harris or Voisin).
Contents:
1. Overview on Mixed Hodge modules
2. Hodge modules on curves (local theory)
3. Hodge modules on curves (global theory)
4. Pure Hodge modules
References:
M. Saito's articles
C. Sabbah: Hodge theory, singularities and D-modules, available at
http://www.math.polytechnique.fr/~sabbah/livres.html

School short communications

Peter Dalakov (Bulgarian Academy of Sciences, IMI, Bulgaria)
"Langlands duality and special Kaehler metric on the Hitchin base" Lecture notes

Abstract:
I will discuss work in progress (together with Ugo Bruzzo), where we study the behaviour of the special Kaehler metric on the
Hitchin base under the Langlands duality transformation for Hitchin systems.

Carmelo Di Natale (University of Cambridge, UK)
"A Period Map for Global Derived Stacks" Lecture notes

Abstract:
In the Sixties Griffiths constructed a holomorphic map, known as the local period map, which relates the classification of smooth projective varieties to the associated Hodge structures. Fiorenza and Manetti have recently described it in terms of Schlessinger's deformation functors and, together with Martinengo, have started to look at it in the context of Derived Deformation Theory. In this talk we propose a rigorous way to lift such an extended version of Griffiths period map to a morphism of derived deformation functors and use this to construct a period morphism for global derived stacks.

Hironori Sakai (University of Muenster, Germany)
"The moduli spaces of symplectic vortices on an orbifold Riemann surface" Slides

Abstract:
Given a Riemann surface and a symplectic manifold equipped with a Hamiltonian
group action, we can consider the associated PDEs over the Riemann surface,
which are called symplectic vortex equations (SVE). The theory of SVE is a
gauge theory which admits "Hitchin-Kobayashi correspondence" for some cases.
Moreover, we obtain invariants under certain conditions on the symplectic
manifold, and the invariants are equal to Gromov-Witten invariants (under
additional conditions).
In this talk we discuss SVE on an orbifold Riemann surface by making use of
differentiable stacks. We also discuss the moduli space of symplectic vortices
for a special case.

Workshop seminars (the videos are available here)

Kai Behrend (University of British Columbia, Canada)
"Derived Moduli of stable non-commutative schemes "

Abstract:
I will explain a particularly simple formalism giving rise to derived schemes: bundles of curved differential graded Lie algebras. This approach gives rise to a notion of stability for non-commutative projective schemes and a natural quasi-projective derived moduli space for stable non-commutative projective schemes. For example, certain kinds of Sklyanin (or Artin-Schelter regular) algebras are stable. This is joint work with Behrang Noohi and Junho Hwang.

Philip Boalch (ENS, France)
"Natural flat connections and wild mapping class groups"

Abstract:
As is well known in algebraic analysis a family of varieties X/S leads to a
flat connection on a vector bundle over the base S, yielding the theory of
geometric (linear) differential equations. I'll talk about the analogous
nonlinear story when one considers the first nonabelian cohomology of the
fibres, and the "wild" extension of this, where it appears natural to
extend the notion of variety. If time permits I'll also discuss the fission
operation, and recent steps to classify such nonlinear geometric
differential equations in terms of quivers.

Andrei Caldararu (University of Wisconsin, USA)
"The de Rham complex from the point of view of twisted derived intersections"

Abstract:
I shall present results of work with Arinkin and Hablicsek which allow us to understand the Frobenius push-forward of the de Rham complex as the structure sheaf of a twisted derived intersection. Similar considerations also apply for twisted de Rham complexes, yielding results which have applications in singularity theory and in the study of matrix factorizations. Using our theorems we recover and strengthen earlier results of Deligne-Illusie, Barannikov-Kontsevich, and Sabbah.

Andrea D'Agnolo (Università di Padova, Italy)
"Riemann-Hilbert correspondence for irregular holonomic D-modules" Slides

Abstract:
The classical Riemann-Hilbert correspondence establishes an equivalence between
the triangulated category of regular holonomic D-modules and that of constructible
sheaves. In a joint work with Masaki Kashiwara, we prove a Riemann-Hilbert
correspondence for holonomic D-modules which are not necessarily regular. The
construction of our target category is based on the theory of ind-sheaves by
Kashiwara-Schapira and influenced by Tamarkin's work. Among the main ingredients
of our proof is the description of the structure of flat meromorphic connections due to
Mochizuki and Kedlaya.

Alexander I. Efimov (Higher School of Economics, Russia)
"Noncommutative Hitchin system"

Abstract:
First, we construct a family of associative algebras, which are formally smooth (Qulllen-smooth), and whose representation spaces are (affine charts of) moduli spaces of semi-stable vector bundles on a smooth projective curve, with trivialization of a single fiber. Then we construct a noncommutative version of Hitchin system, which induces the usual one (for $GL_n$) by taking its "trace".
Our construction is closely related with some generalization (actually, extension) of the DG Lie algebra of Hochschild cochains of a small DG category.

Domenico Fiorenza (Università di Roma 1, Italy)
"Abel, Jacobi and the double homotopy fiber" Slides

Abstract:
Let X be a Kahler complex manifold and let Z be a complex submanifold.
Within the framework of derived deformation theory, the Abel-Jacobi map for
the pair (X,Z) has a natural interpretation as a morphism form a homotopy
fiber to a double homotopy fiber. Using this fact and the dictionary
between formal moduli problems and differential graded Lie algebras up to
homotopy, it is easy to describe a linear L-infinity morphism encoding the
Abel-Jacobi map, and in terms of this L-infinity morphism it is immediate
to reobtain the classical description of the differential of the
Abel-Jacobi map as well as the result that Bloch's semiregularity map
annihilates the obstructions to deforming Z inside X. This provides a
geometrical interpretation of recent results by Iacono-Manetti and Pridham.

Ezra Getzler (Northwestern University, USA)
"Chern-Weil Theory for Perfect Complexes"

Abstract:
Using the Connes-Karoubi theory of non-commutative differential forms, I construct
an extension of the usual Chern-Weil theory for vector bundles to perfect complexes.

Florian Ivorra (Université de Rennes 1, France)
"Intersection theory and $A_\infty$-algebra"

Abstract:
In his paper Chow Group with Coefficients, M. Rost has developed a generalization of the classical Chow groups based on Milnor K-theory and an axiomatic generalization of it called cycles modules. For a cycle module M with a ring structure and a smooth algebraic variety X over a field, in this talk I will explain that Rost's cycle complex on X with coefficients in M has a structure of an $A_\infty$-algebra. In the case of Milnor K-theory it provides an homotopy model for the classical intersection theory of cycles and provides some new higher order operations on Chow groups similar to Massey products in singular cohomology.

Dominic Joyce (Oxford University, UK)
Mini-course (3 hours): "Shifted symplectic derived algebraic geometry, and extensions of Donaldson-Thomas theory" Slides 1 2 3

Abstract:
"Donaldson-Thomas invariants" are numbers which "count" semistable coherent sheaves on Calabi-Yau $3$-folds, which are important in String Theory as "numbers of BPS states", and have nice properties (deformation-invariance, wall-crossing formulae under change of stability condition). Kontsevich and Soibelman outlined wide-ranging generalizations of D-T theory in the directions of "motivic invariants" (replacing numbers by elements of a larger ring of motives) and "categorification" (finding canonical vector spaces whose dimensions are D-T invariants, and putting algebraic structures on these vector spaces -- Cohomological Hall Algebras), but these are still rather incomplete, especially the categorification picture. It would also be very interesting to know if any parts of this picture extend to Calabi-Yau $m$-folds for $m>3$, in particular $m=4$.
In a beautiful paper, Pantev, Toen, Vezzosi and Vaquie (arXiv:1111.3209) introduced the notion of "$k$-shifted symplectic structure" on a derived scheme or derived stack, for all integers $k$, where $0$-shifted symplectic structures on derived schemes are just classical algebraic symplectic structures on classical smooth schemes. They proved that derived moduli stacks of (complexes of) coherent sheaves on a Calabi-Yau m-fold have a $(2-m)$-shifted symplectic structure. Thus, PTVV give us a new geometric structure on Calabi-Yau moduli stacks, which we can try to use to make rigorous the proposed extensions of Donaldson-Thomas theory.
This mini-course will report on a programme to do this, in a series of papers arXiv:1304.4508, arXiv:1305.6302, 1211.3259, arXiv:1304.6428, arXiv:1312.0090 and work in progress, joint work with O. Ben-Bassat, D. Borisov, C. Brav, V. Bussi, D. Dupont, S. Meinhardt, and B. Szendroi.
The first lecture will explain PTVV's shifted symplectic geometry, and a "Darboux theorem" which gives a local canonical form (Zariski local for schemes, smooth-local for Artin stacks) for $k$-shifted symplectic derived schemes and stacks. When $k = - 1$, this implies that classical Calabi-Yau $3$-fold moduli schemes and stacks are Zariski locally / smooth-locally equivalent to the critical locus of a regular function on a smooth scheme.
The second lecture will explain a project to categorify Donaldson-Thomas theory of Calabi-Yau $3$-folds using perverse sheaves. Given a PTVV $-1$-shifted symplectic derived scheme or stack $X$ with an "orientation", we construct a natural perverse sheaf $P_X$ on $X$ by gluing together perverse sheaves of vanishing cycles on the critical loci from the "Darboux theorem", such that in the scheme case, the graded dimension of the hypercohomology $H(P_X)$ of $P_X$ is the Donaldson-Thomas invariant of $X$. It turns out that the construction factors through a new classical geometric structure on the classical CY3 moduli scheme or stack, called a "$d$-critical locus".
The third lecture will hopefully discuss several topics:
(a) how to define a "multiplication" on the perverse sheaves $P_X$, and so obtain Cohomological Hall Algebras (in progress);
(b) motivic invariants of $-1$-shifted symplectic derived scheme and stacks, and motivic D-T theory;
(c) defining new, deformation-invariant Donaldson-Thomas-style invariants "counting" coherent sheaves on Calabi-Yau $4$-folds, starting with the "Darboux Theorem" for $-2$-shifted symplectic derived schemes (in progress); and
(d) defining "Fukaya categories" of complex or algebraic Lagrangians in complex or algebraic symplectic manifolds, using the perverse sheaves $P_X$, noting that derived Lagrangian intersections have $-1$-shifted symplectic structures (in progress).

Marco Manetti (Università di Roma 1, Italy)
"A formality criterion for differential graded Lie algebras" Slides

Abstract:
In the first part of the talk, addressed to a general audience, we review the notion of differential graded Lie algebras (DGLA) , the notion of formality and their use in deformation theory.
In the second part we shall explain how the formality of a DGLA $L$ is completely controlled by the natural spectral sequence computing the Chevalley-Eilenberg cohomology groups $H^*(L,L)$.

Ryszard Nest (University of Copenhagen, Denmark)
"Index and determinant of n-tuples of commuting operators"

Abstract:
Suppose that $A$ is an n-tuple of commuting operators on a Hilbert space and $f$ is an n-tuple of functions holomorphic in a neighborhood of the (Taylor) spectrum of $A$. The n-tuple of operators $f(A)=( f_1(A_1),..., A_n),..., f_n(A_1,..., A_n) )$ give rise to a complex $K(f(A),H)$, its so called Koszul complex, which is Fredholm whenever $f^{-1}(0)$ does not intersect the essential spectrum of $A$.
Given that $f$ satisfies the above condition, we will give a local formulae for the index and determinant of $K(f(A),H)$. The index formula is a generalization of the fact that the winding number of a continuous nowhere zero function $f$ on the unit circle can be computed locally in the interior of the unit disc.
The explicit local formula for the determinant of $K(f(A),H)$ can be seen as an extension of the Tate tame symbol to, in general, singular complex curves.

Behrang Noohi (University of London, UK)
"Introduction to topological stacks"

Abstract:
In this short course I will introduce topological stacks and after reviewing the basic properties of the 2-category of topological stacks and some examples I will explain how one can extend various familiar tools from algebraic topology to the setting of topological stacks. The key role here is played by the 'classifying space' of a topological stack. Time permitting, I will also discuss ongoing work (with Thomas Coyne) on the construction of singular chains on topological stacks.

Tony Pantev (University of Pennsylvania, USA)
"Constructions in shifted symplectic geometry"

Abstract:
I will describe various geometric constructions that produce Lagrangian substacks in shifted symplectic stacks. I will discuss examples of shifted Lagrangian fibrations and foliations and their relevance for the global existence of potentials. This is a joint work with Katzarkov, Toen, Vaquie, and Vezzosi.

Thomas Reichelt (Max Planck Institute for Mathematics, Germany)
"Hodge theory of GKZ systems and mirror symmetry"

Abstract:
We will explain a close relationship between GKZ hypergeometric systems after Gelfand, Kapranov and Zelevinsky, Gauss-Manin systems of families of Laurent polynomials and the intersection cohomology of hyperplane sections in a toric variety.
As an application, we will show how to construct non-affine Landau-Ginzburg models which are mirror partners for complete intersections in toric varieties with a numerical effective anti-canonical divisor.

Claude Sabbah (École Polytechnique, Palaiseau, France)
"Aspects of the irregular Hodge filtration" Slides

Abstract:
Given a regular function $f$ on a smooth quasi-projective variety $U$, the de Rham complex of $U$ relative to the twisted differential $d+ df$ can be equipped canonically with a filtration (the irregular Hodge filtration) for which the associated hypercohomology spectral sequence degenerates at $E_1$. A logarithmic version of this de Rham complex (relative to a suitable compactification of $U$) has been introduced by M. Kontsevich, who showed the independence of the dimension of the corresponding cohomologies with respect to the differential $ud + vdf$, for $u,v$ arbitrary complex numbers. This leads to bundles on the projective line of the $(u:v)$ variable, on which we construct a natural connection for which the Harder-Narasimhan filtration satisfies the Griffiths transversality property and standard limiting properties at $v=0$. This is a joint work with Hélène Esnault (Berlin) and Jeng-Daw Yu (Taipei).

Dmytro Shklyarov (Freiburg University, Germany)
"A categorical approach to the higher residue pairings"

Abstract:
The higher residue pairings are one of the main ingredients in Kyoji Saito's
construction of Frobenius manifolds associated with germs of analytic functions. The
original definition of the higher residues relies on complex geometry. We will describe an
alternative (still conjectural) approach to the pairings based on the categories of matrix
factorizations.

Ping Xu (Penn State University, USA)
"Holomorphic Poisson manifolds and symplectic realizations"

Abstract:
For a complex manifold $X$, it is well known that $T^*X$ is a holomorphic symplectic manifold.
The complex manifold $X$ can be seen as a (rather trivial) holomorophic Poisson manifold when endowed with the zero Poisson bracket.
The canonical projection from $T^*X$ to $X$ is a Poisson map and $X$ is embedded as a Lagragian submanifold of $T^*X$ (by the zero section).
In this talk, I will show that a similar phenomenon holds when the Poisson bracket on $X$ is not trivial.

Amnon Yekutieli (Ben Gurion University, Israel)
"Residues and Duality for Schemes and Stacks" Slides

Abstract:
Let K be a regular noetherian commutative ring. I will begin by explaining the theory of rigid residue complexes over
essentially finite type K-algebras, that was developed by J. Zhang and myself several years ago. Then I will talk about the
geometrization of this theory: rigid residue complexes over finite type K-schemes. An important feature is that the rigid residue complex over a scheme X is a quasi-coherent sheaf in the etale topology of X. For any map f : X → Y between K-schemes there is the ind-rigid trace homomorphism (that usually does not commute with the differentials). When the map f is proper, the ind-rigid trace does commute with the differentials (this is the Residue Theorem), and it induces Grothendieck Duality.
Then I will move to finite type Deligne-Mumford K-stacks. Any such stack X has a rigid residue complex on it, and for any map f : X → Y between stacks there is the ind-rigid trace homomorphism. These facts are rather easy consequences of the corresponding facts for schemes, together with etale descent. I will finish with the Residue Theorem, which holds when the map f : X → Y is proper and coarsely schematic; and the Duality Theorem, which also requires the map f to be tame. If there is time I will outline the proofs.

Workshop short communications (the videos are available here)

Oren Ben-Bassat (University of Oxford, UK)
"Analytic Geometry As Relative Algebraic Geometry"

Abstract:
I will review symmetric monoidal categories and explain how one can work
with "algebras and modules" in such a category. Toen, Vaquie, and Vezzosi
promoted the study of algebraic geometry relative to a closed symmetric
monoidal category. By considering the closed symmetric monoidal category of
Banach spaces, we recover various aspects of Berkovich analytic geometry.
The category opposite to the category of commutative algebra objects in a
closed symmetric monoidal category carries several interesting topologies
introduced by Toen, Vaquie and Vezzosi. In the case that the symmetric
monoidal category is the category of groups some of these topologies
specialize to the ordinary Zariski topology on the category of affine
schemes. We consider the case where the symmetric monoidal category is the
category of Banach spaces over a non-Archimedean complete valuation field.
We show in this case that one of the abstractly defined topologies
specializes to the G-topology of Berkovich analytic geometry. In our
context, the quasi-abelian categories of Banach spaces or modules as
developed by Schneiders and Prosmans are very helpful. This is joint work
with Kobi Kremnizer (Oxford). No background in derived algebraic geometry
or non-Archimedean geometry will be assumed. This work is available at
http://arxiv.org/abs/1312.0338.

Sheng-Fu Chiu (Northwestern University, USA)
"Contact non-squeezability" Slides

Abstract:
Let $U_R = B_R × S^1$ be a contact ball. It is proved by Eliashberg, Kim, and
Polterovich that if there exists an positive interger $m$ in the interval $[\pi r^2;\pi R^2]$,
then $U_R$ cannot be squeezed into $U_r$. The case $m <\pi r^2 <\pi R^2 < m + 1$ remains
unsolved in their paper. Inspired by the work of Guillermou et al., we apply the
microlocal sheaf theory to give an algebraic approach to this problem. This is a
joint work with Dmitri Tamarkin.

Oleksandr Kravets (Moscow Higher School of Economics, Russia)
"Exceptional collections in categories of singularities of3-dimensional Landau-Ginzburg models"

Abstract:
Derived categories of singularities measure how far is an algebraic variety
from being smooth. They appear on the algebraic side of the Homological
Mirror Symmetry (HMS) for the Landau-Ginzburg models. The talk will be
dedicated to the case when the superpotential is given by the so called
invertible polynomial. In the case of small dimensions, I will give
the description of mentioned categories in terms of full strongly exceptional collections
of good kind. I will also recall several conjectures concerning HMS for the
current case and will show how the constructed collections help to assert
these conjectures.

Alice Rizzardo (SISSA, Italy)
"Representability of cohomological functors over extension fields"

Abstract:
We will present a new approach to studying whether an exact
functor between the bounded derived categories of two smooth projective
varieties is isomorphic to a Fourier-Mukai transform. This involves
studying the representability of a cohomological functor from the derived
category of a projective variety to the category of vector spaces over an
extension field. This will give us, in low dimensions, a "generic
Fourier-Mukai behavior".

Jean-Baptiste Teyssier (Free University of Berlin, Germany)
"On a characterization of regular holonomic $\mathcal D$-modules"

Abstract:
Let $X$ be a smooth complex manifold. Let us denote by $D^b_c(X;\mathbb C)$ the derived category of complexes of sheaves in $\mathbb C$-vector spaces on $X$ with bounded and constructible cohomology. Let $Sol$ be the functor of solutions for $\mathcal D$-modules on $X$.
Traditionally, the fully-faithfulness of Riemann-Hilbert correspondance is proved by showing that if $\mathcal M_1$ and $\mathcal M_2$ are regular holonomic $\mathcal D_X$-modules, then the canonical morphism
$$RH_{\mathcal M_1;\mathcal M_2} : R\mathcal Hom(\mathcal M_1;\mathcal M_2) \longrightarrow R\mathcal Hom(Sol(\mathcal M_2); Sol(\mathcal M_1))$$
is an isomorphism in $D^b_c(X;\mathbb C)$.
This talk will deal with the converse statement. We will namely give a sketch of proof of the following Theorem. If $\mathcal M$ is an holonomic $\mathcal D_X$-module for which $RH_{\mathcal M;\mathcal M}$ is an isomorphism, then $\mathcal M$ is regular.

Jesse Wolfson (Northwestern University, USA / IPMU, Japan)
"The Index Map in Algebraic K-Theory"

Abstract:
In the 1960s, Atiyah and Janich constructed an "index map" from
the space of Fredholm operators to the classifying space of topological
K-theory and showed this map is an equivalence. We explain how a
dictionary due to Sato, Sato, Segal and Wilson leads to a natural analogue
of the index map in algebraic K-theory. Recent work of Sho Saito gives
rise to an analogue of Atiyah and Janich's equivalence. This is joint work
with Oliver Braunling and Michael Groechenig.