B3a) Genovese, Francesco: Effective Field theory, Renormalization group flow and Chern Simons theory, section 3.3+3.4 until and including CS example 3.22
B3b) Belmans, Pieter: Quantum master equation, deformation obstruction complex and theory on the complex plane 3.4.2 to end of 3.5
B4a) Sutherland, Tom: Classical BCOV theory and its quantization, section 4
B4b) Melani, Valerio: Quantization of BCOV on the elliptic curve I, existence and uniqueness, section 5.1 until middle of page 98 (to the end of the proof of 5.12), i.e. section 5.1.1-5.3.1
B5a) Bousseau, Pierrick: Quantization of BCOV on the elliptic curve II, proving the dilaton axiom and holomorphicity, section 5.3.2-5.4
B5b) Ruddat, Helge: Proving higher genus mirror symmetry for the elliptic curve, section 6
C
C1) Anno, Rina: "Mumford toric degenerations", section 1.1 and 1.2 of https://arxiv.org/pdf/0808.2749.pdf
C2a) Felten, Simon: "From a toric degeneration to (B,P,phi)"
C2b) Logvinenko, Timothy: "Scattering I"
C3a) Nikolaev, Nikita: "Scattering II"
C3b) Zhi, Jin: "Overview of the Gross-Hacking-Keel construction"
C4a) Kelly, Tyler: "Broken lines"
C4b) van Garrel, Michel: "Generalized theta functions"
Andrei Caldararu: "Categorical Gromov-Witten Invariants"
Caldararu I: What are categorical Gromov-Witten invariants?
I will start my lecture by reviewing the construction of classical Gromov-Witten invariants, and working through a standard example, the g=1, n=1 GW invariants of an elliptic curve. Then I will outline how we can hope to obtain these invariants from the Fukaya category by an abstract categorical construction. In the case of an arbitrary abstract category I will discuss what properties the category should have, what replaces the quantum cohomology ring, etc. I will conclude with a discussion of potential applications of the construction of categorical Gromov-Witten invariants and a general discussion of how mirror symmetry is supposed to work.
Caldararu II: Chains on moduli space of curves, ribbon graphs, and A_infinity algebras
I will begin by reviewing the main algebraic structure on the space of chains on the moduli spaces of curves, namely that they form a Batalin-Vilkovisky algebra. We will study the Quantum Master Equation in such chains; by work of Costello its solution gives rise to the string vertices of Zwiebach and Sen. I will follow with an introduction to ribbon graphs and how they give a model for the chains on M_{g,n}. I will conclude with a quick review of A_infinity algebras and a description of the original Kontsevich partition function which pairs ribbon graphs with cyclic A_infinity algebras.
Caldararu III: Algebraic structures on the periodic cyclic complex
I will introduce the Hochschild chain complex of an A_infinity algebra, and its two main differentials, the Hochschild differential b and the Connes differential B, leading to the construction of the different variants of cyclic homology (periodic, negative, positive). Using this construction I will outline Kontsevich-Soibelman's construction of the action of the PROP of chains on M_{g,n} on the cyclic complex of an A_infinity algebra. Following Costello, part of this action allows us to regard the periodic cyclic complex as a symplectic vector space with a Lagrangian inside it. This leads to the construction of a Weyl algebra and Fock module associated with this structure. Costello constructs a deformation of this Fock module induced using a choice of string vertices. This leads to the construction of an abstract Gromov-Witten potential (a line in a Fock space).
Caldararu IV: Hodge-de Rham degeneration and applications
The periodic cyclic homology carries a natural filtration, the Hodge filtration. I will outline its construction, and explain how a choice of a splitting of this filtration allows us to regard the abstract Gromov-Witten potential previously constructed as an actual categorical Gromov-Witten potential. Everything will be put together into the calculation of the categorical B-model potential of the family of elliptic curves (joint work with Junwu Tu). I will conclude with a list of potential directions in which the whole construction can be extended, and expected applications and conjectures.
Tim Perutz: "From homological to Hodge-theoretic mirror symmetry"
(Based on work of S. Ganatra, N. Sheridan and myself, and various subsets of this trio.)
Perutz I: Fukaya categories, open-closed maps, and pairings.
I will review some of the mathematics of Fukaya categories of symplectic manifolds. These categories come with important structure: the closed-open map CO, which is a map of algebras from quantum cohomology to Hochschild cohomology of the category, and the open-closed map OC, from Hochschild homology to quantum cohomology, which is an isometry with respect to a “Mukai pairing” on Hochschild homology and a cup-product pairing on quantum cohomology. (The B-model analog of this structure was studied by Andrei Caldararu several years ago, and an abstract of OC is key to Costello’s construction, as discussed in Andrei’s lectures.) Abouzaid has shown that a subcategory generates the Fukaya category if OC is surjective on that subcategory.
Perutz II: Smoothness, automatic generation, and homological mirror symmetry.
Symplectic topologists have recently been learning a principle which Kontsevich has advocated for some time: the importance of categorical smoothness. The prototypical example of a smooth DG category is the derived category of a smooth algebraic variety. In the setting of homological mirror symmetry (HMS), the Fukaya category of the mirror symplectic manifold will then also be smooth. From smoothness many things flow: sharp versions of generation for the Fukaya category; that OC and CO are isomorphisms; that OC coincides with Costello’s abstract version, hence that his formal constructions are in fact geometric; and that HMS implies closed-string mirror symmetry.
Perutz III: The cyclic open-closed map, the categorical Gauss—Manin connection, and the mirror map.
The cyclic homology of an A-infinity category, defined over (say) a formal punctured disc, carries a Gauss—Manin connection, as constructed by Getzler. In the case of the derived category of a smooth algebraic variety over the punctured disc, this is expected (but not actually proven) to coincide with the classical GM connection in algebraic de Rham cohomology. I will explain that, in the case of a smooth Fukaya category, it coincides with the quantum differential operator, a connection in quantum cohomology. Under the assumption of HMS, the consequences are as follows: (i) that with an undetermined mirror map, the mirror map can be characterized Hodge-theoretically; and (ii) that the quantum differential equation matches the (derived) algebro-geometric GM connection. Statement (ii) is (genus 0) Hodge-theoretic mirror symmetry. It encompasses the famous rational curve-counts on the quintic 3-fold.
Spring School for Master & PhD students and Postdocs
March 25 - 31, 2018 in Italy http://www.montegufoni.it/
Senior Speakers: Andrei Caldararu Tim Perutz Organizers:
Helge Ruddat (ruddat)
Tom Sutherland (sutherland)
Estelle Bonmann (ebonmann) (respectively @uni-mainz.de)
Accepted participants will be provided shared rooms in the castle. We might be able to modestly subsidize travel costs by a lump sum per person.We will organize and prepare meals ourselves.Each participant will be asked to give a talk presenting material that we assign together beforehand. This will consist of works of Barannikov-Kontsevich, Costello-Li, Gross-Siebert and others.Sorry, the event has reached its participant capacity limit.Literature:
Das Seminar findet an den wie in der folgenden Tabelle angegebenen Terminen statt. Montags von 12-14 Uhr sind wir im Raum 04-432 und für die zusätzlichen Sitzungen um 10-12 Uhr, sind wir im Raum 04-230 für diese Uhrzeit.
Die Tabelle zeigt die Aufteilung der Seminarthemen und Seminartermine sowie Übungsaufgaben pro Student(in). Sie bezieht sich auf das unten angegebene Buch.
Jeder Vortrag soll nicht viel länger als eine Stunde gehen und danach noch um das Vortragen der Übungsaufgabe zu dem Kapitel (von dem/der jeweils Vortragenden, ca. 15min) ergänzt werden. Außerdem sollte Zeit für Fragen eingeplant werden. (Gutes Zeitmanagement gehört zu einem guten Vortrag)
Jede(r) Vortragende teilt zu Beginn ihres/seines Vortrags einen Handzettel aus, auf dem die wesentliche Begriffe, Definitionen und Resultate des Vortrags zusammengefasst sind. Bitte zur Erstellung davon Latex verwenden. Übungsaufgaben sind bis eine Woche nach dem letzten Vortrag zum jeweiligen Thema einzureichen (auch per Latex-pdf) per Email an elementare.alg.geo_ät_web.de.
Achtet bitte darauf, dass ihr auch die Aufgabenstellung mit aufschreibt und die Aufgaben in ein einziges pdf Dokument kompiliert (jeweils um jede neue Aufgabe ergänzen, bzw gegebenenfalls auch Korrekturen der alten Aufgaben machen, jede neue Aufgabe beginnt auf einer neuen Seite).
Bei mindestens einer Aufgabe ist ein Computerprogramm zu schreiben, schreibt dafür einfach die Beschreibung des Algorithmus auf. In die Seminarnote fließen der eigene Vortrag, die allgemeine Beteiligung im Seminar und die Bearbeitung der Aufgaben ein.
Bitte lest auch das (recht kurze) Kapitel Null im Buch, weil wir darüber keinen Vortrag haben werden, es aber als Allgemeinbildung und Hintergrundwissen hilfreich ist. Auch zum Appendix von Kapitel eins haben wir keinen Vortrag, bitte ebenso bei Zeiten eigenständig anschauen.
Je früher ihr mit der Materialsichtung anfangt, desto besser!
Topics around Calabi-Yau and Fano manifolds, deformation theory and mirror symmetry
This colloquium invites mostly external speakers. It is loosely associated with the Emmy Noether
grant "Degenerations of Calabi-Yau Manifolds and Related Geometries".
Organizers are: Simon Felten, Matej Filip, Andrea Petracci, Helge Ruddat
Regular times: Tuesday 5-6 pm & Wednesday 9-10 am (German time zone)
23.06.2021 09:00 Wei Hong BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties
07.07.2021 09:00 Lawrence Barrott - cancelled due to illness - postphoned until october Deforming Calabi-Yau subvarieties
20.07.2021 17:00 Benjamin Gammage Homological mirror symmetry over the SYZ base
Abstracts
12.05. 2021 09:00 Ziming Ma - SYZ Mirror Symmetry and Maurer-Cartan Equation
The Strominger-Yau-Zaslow conjecture for understanding Mirror Symmetry geometrically, leads to the Fukaya's conjectural reconstruction of mirror manifolds which solves Maurer-Cartan equation near large limits using quantum corrections. In this talk, we will discuss progesses of the Fukaya's conjecture and the formulation of the Maurer-Cartan equation near large structure limits by constructing a dgBV algebra PV *(X), a generalized version of the Kodaira-Spencer dgLa, associated to possibly degenerate Calabi-Yau variety X equipped with local thickening data. This talk is based on joint works with Kwokwai Chan, Conan Leung and Yat-Hin Suen.
19.05.2021 08:30 Taro Sano - Construction of non-Kähler Calabi-Yau manifolds by log deformations
Calabi-Yau manifolds (in the strict sense) form an important class in the classification of algebraic varieties. One can also consider its generalisation by removing the projectivity assumption. Clemens and Friedman constructed infinitely many topological types of non-Kähler Calabi-Yau 3-folds whose 2nd Betti numbers are zero. In this talk, I will present examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers. The construction is by smoothing normal crossing varieties. The key tools of the construction are some isomorphisms between general rational elliptic surfaces which induce isomorphisms between Calabi-Yau manifolds of Schoen type.
26.05.2021 09:00 Yuto Yamamoto - Tropical contractions to integral affine manifolds with singularities
We construct contraction maps from tropical Calabi -Yau varieties to the integral affine maifolds with singularities that arise as the dual intersection complexes of toric degenerations of Calabi-Yau varieties in the Gross-Siebert program. We show that the contractions preserve tropical cohomology groups, and send the eigenwaves to the radiance obstructions. As an application, we also prove the Poincaré-Verdier duality for integral affine manifolds with singularities.
09.06.2021 09:00 Taro Fujisawa - Geometric polarized log Hodge structures on the standard log point
I will talk about the following fact: a projective vertical exact log smooth morphism over the standard log point yields polarized log Hodge structures on the base. In the proof of this fact, the case of a strict log deformation is essential. So, I will mainly talk about this case, and explain how to relate my previous results on the mixed Hodge structures to log Hodge structures for a projective strict log deformation. If the time remaines, I will discuss a generalization to the case of a general base point. This talk is based on a joint work with C. Nakayama.
15.06.2021 17:00 Pieter Belmans - Hochschild cohomology of Fano 3-folds(notes/script)
The Hochschild-Kostant-Rosenberg decomposition gives a description of the Hochschild cohomology of a smooth projective variety in terms of the sheaf cohomology of exterior powers of the tangent bundle. In all but a few cases it is a non-trivial task to compute this decomposition, and understand the extra algebraic structure which exists on Hochschild cohomology. I will give a general introduction to Hochschild cohomology and this decomposition, and explain what it looks like for Fano 3-folds (joint work with Enrico Fatighenti and Fabio Tanturri), and time permitting also for partial flag varieties (joint work with Maxim Smirnov).
23.06.2021 09:00 Wei Hong - BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties
The vector space of holomorphic polyvector fields on any complex manifold has a natural Gerstenhaber algebra structure. In this paper, we study BV operators of the Gersten- haber algebras of holomorphic polyvector fields on smooth compact toric varieties. We give a necessary and sufficient condition for the existence of BV operators of the Gerstenhaber algebra of holomorphic polyvector fields on any smooth compact toric variety
postphoned until October 2021 09:00 Lawrence Barrott - Deforming Calabi-Yau subvarieties
The Doran-Harder-Thompson conjecture is a duality on geometric structures under mirror symmetry. On one side we have smooth degenerations of Calabi-Yau's (CY's) to unions of normal crossings components, and on the other we have fibrations of the mirror CY's by CY subvarieties. In the simplest case it proposes that CY's with a Tyurin degeneration should be mirror to CY's fibred over P1. I will explain how some of the recent machinery of deformation theory for singular CY's of Chan-Leung-Ma and Felton-Filip-Ruddat, together with the mirror construction of the Gross-Siebert program leads to a proof of one direction of this conjecture in classes of examples. If time permits I will sketch how this relates to the period calculations appearing in other papers (Doran-Kostiuk-You) via more recent techniques in the Gross-Siebert program. This is based on joint work with Chuck Doran.
20.07.2021 17:00 Benjamin Gammage - Homological mirror symmetry over the SYZ base
The Gross-Siebert program suggests that mirror symmetry is mediated by the combinatorial data of a dual pair of integral affine manifolds with singularities and polyhedral decomposition. Much is now understood about the passage from the combinatorial data to complex spaces "near the large complex structure limit" - a toric degeneration and its smoothing. In this talk, we discuss the mirror procedure for moving from the combinatorial data to symplectic spaces "near the large volume limit" - a Weinstein symplectic manifold and its compactification -- and we will explain a proof of homological mirror symmetry between the complex and symplectic manifold associated to local pieces of the combinatorial data. This is part of a program with Vivek Shende to prove homological mirror symmetry globally over the SYZ base.
The Strominger-Yau-Zaslow conjecture for understanding Mirror Symmetry geometrically, leads to the Fukaya's conjectural reconstruction of mirror manifolds which solves Maurer-Cartan equation near large limits using quantum corrections. In this talk, we will discuss progesses of the Fukaya's conjecture and the formulation of the Maurer-Cartan equation near large structure limits by constructing a dgBV algebra PV *(X), a generalized version of the Kodaira-Spencer dgLa, associated to possibly degenerate Calabi-Yau variety X equipped with local thickening data. This talk is based on joint works with Kwokwai Chan, Conan Leung and Yat-Hin Suen.
19.05.2021 Taro Sano
Construction of non-Kähler Calabi-Yau manifolds by log deformations
Calabi-Yau manifolds (in the strict sense) form an important class in the classification of algebraic varieties. One can also consider its generalisation by removing the projectivity assumption. Clemens and Friedman constructed infinitely many topological types of non-Kähler Calabi-Yau 3-folds whose 2nd Betti numbers are zero. In this talk, I will present examples of non-Kähler Calabi-Yau manifolds with arbitrarily large 2nd Betti numbers. The construction is by smoothing normal crossing varieties. The key tools of the construction are some isomorphisms between general rational elliptic surfaces which induce isomorphisms between Calabi-Yau manifolds of Schoen type.
26.05.2021 Yuto Yamamoto
Tropical contractions to integral affine manifolds with singularities
We construct contraction maps from tropical Calabi -Yau varieties to the integral affine maifolds with singularities that arise as the dual intersection complexes of toric degenerations of Calabi-Yau varieties in the Gross-Siebert program. We show that the contractions preserve tropical cohomology groups, and send the eigenwaves to the radiance obstructions. As an application, we also prove the Poincaré-Verdier duality for integral affine manifolds with singularities.
09.06.2021 Taro Fujisawa
15.06.2021 Pieter Belmans
23.06.2021 Wei Hong
BV operators of the Gerstenhaber algebras of holomorphic polyvector fields on toric varieties
21.10.2020 Katharina Hübner
Logarithmic differentials on adic spaces
The object of interest in this talk is a certain subsheaf Omega_X of the sheaf of differentials Omega_X of a discretely ringed adic space X over a field k. The first part will be dedicated to an introduction to discretely ringed adic spaces. We will then define $\Omega^+_X$ using K\"ahler seminorms and establish a relation with logarithmic differentials. Finally we study the case where $X = Spa(U,Y)$ for a scheme $Y$ over $k$ and a subscheme $U$ such that the corresponding log structure on $Y$ is log smooth. It turns out that $\Omega^+_X(X)$ equals $\Omega^{log}_{(U,Y)}(U,Y)$.
22.01.2020 Thomas Prince
Calabi-Yau toric hypersurfaces using the Gross-Siebert algorithm
We explain how to form a novel dataset of simply connected Calabi-Yau threefolds via the Gross-Siebert algorithm. These are related to Calabi-Yau toric hypersurfaces with certain Gorenstein (not necessarily isolated) singularities by a family over a possibly reducible base. Specifically, we explain how to smooth the boundary of a class of 4-dimensional reflexive polytopes to obtain a polarised tropical manifolds. We also describe how to compute the Betti numbers of these Calabi-Yau threefolds. We also explain an extension of this project generalising work of Batyrev-Kreuzer on conifold transitions.
21.06.2018 Michel van Garrel
Number of rational curves in log versus local geometries
Let X be a smooth projective variety and let D be a nef divisor. It is well known that D corresponds to a line bundle O(-D), which leads one to consider two geometries associated to D. On one hand, there is the logarithmic geometry of the pair (X,D). On the other hand, there is the local geometry of the total space of
O(-D). In this collaboration with Tom Graber and Helge Ruddat, we show that in an appropriate sense (in terms of log and local Gromov-Witten invariants), the number of log rational curves of (X,D) equals (up to a factor) the number of rational curves of O(-D).
08.06.2018 Ilia Zharkov
Topological model for affine hypersurfaces
Given an affine complex hypersurface I will define a phase tropical hypersurface and show that is homeomorphic to the complex one. I will also describe some immersed spheres which suppose to represent Lagrangian objects generating the Fukaya category of the hypersurface.
18.10.2017 Robin Guilbot
On embedded Mirror Symmetry
A wide majority of the known instances of Mirror Symmetry between families of Calabi-Yau varieties are realized as complete intersections in toric varieties.
In these examples the features of Mirror Symmetry are more or less direct consequences of convex-combinatorial dualities. But the elegance of these constructions
is somehow balanced by their peculiarity : toric complete intersections are expected to form a small minority of all the Calabi-Yau varieties.
I will Review the most famous toric mirror constructions, describe a generalization of the hypersurface case introduced in joint work with M. Artebani and P. Comparin,
and sketch the foundations of a new construction for non-complete intersections based on embedded toric degenerations, following J. Böhm’s PhD thesis.
18.10.2017 Valentin Tonita
K-theoretic mirror formulae
Permutation equivariant K-theoretic Gromov Witten invariants, introduced by Givental, are
certain Euler characteristics on the moduli spaces of stable maps to a (smooth, projective)
variety X. I will define the invariants and show how to write K-theoretic I-functions for large classes of varieties
(e.g. toric, certain complete intersections), i.e. certain q-hypergeometric series which are
generating series of these invariants in genus zero . Time permitting, I will discuss the ideas behind
the proofs of these results.
16.10.2017 Matej Filip
Hochschild cohomology and Deformation quantization of affine toric varieties
For an affine toric variety we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Using this description we prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization. Restricting to the commutative case, recent developments in constructing the versal deformation of an affine toric variety will be explained.
27.06.2017 Martin Ulirsch (University of Michigan, USA)
A moduli stack of tropical curves
In this talk I am going to give an introduction to these fascinating moduli spaces and discuss recent work with Renzo Cavalieri, Melody Chan, and Jonathan Wise (arXiv 1704.03806), where we propose the notion of a moduli stack of tropical curves as a geometric stack over the category of rational polyhedral cones. Using this 2-categorical framework one can give a natural interpretation of the forgetful morphism as a universal curve. Moreover, I will propose a way of describing the process of tropicalization via logarithmic geometry in the sense of Kato-Illusie using the theory of Artin fans. Finally, given time, I will also report on an ongoing follow-up project (joint with Margarida Melo, Filippo Viviani, and Jonathan Wise) that uses these techniques to construct the universal Picard variety in logarithmic and tropical geometry.
02.05.2017 Michel van Garrel (Universität Hamburg)
Rational curves in log K3 surfaces
In this talk, we address the basic question of how to count rational curves in log K3 surfaces. We will present partial results in that direction and give a full conjectural description. This is baded on two joints works, one with T. Graber and H. Ruddat, and the other one with J. Choi, N. Takahashi ad S. Katz.
03.11.2016 Tom Sutherland (Università degli Studi di Pavia)
Stability conditions from periods of elliptic curves
I will describe spaces of stability conditions on some Calabi-Yau-3 categories with a simple combinatorial presentation through the study of the period map of meromorphic differentials on associated families of elliptic curves.
19.07.2016 Tim Kirschner (Universität Duisburg-Essen)
Finite quotients of three-dimensional complex tori
I will report on a current project with Patrick Graf (Bayreuth). Using Graf's recent results about the algebraic approximation of Kähler threefolds of Kodaira dimension zero, we show that a three-dimensional compact, connected Kähler space X with isolated canonical singularities is the finite quotient of a complex torus if and only if the first and second Chern classes of X vanish. This brings together an old theorem of Yau (where X is smooth) and a theorem of Shepherd-Barron and Wilson (where X is projective).
14.07.2016 James Pascaleff (University of Illinois)
Symplectic Cohomology and Wall Crossing
In this talk I will describe a way that certain wall-crossing formulae can be seen in terms of symplectic cohomology, which is a Floer theoretic invariant of non-compact symplectic manifolds. In the case of log Calabi-Yau manifolds, this invariant is supposed to be mirror-dual to the poly-vector fields. I will draw connections to the theory of cluster varieties as studied by Gross-Hacking-Keel-Kontsevich. This is partially based on discussions with Dmitry Tonkonog (Cambridge).
26.04.2016 Mohammad Akhtar (Imperial College London)
Mirror symmetry and the classification of Fano varieties
The classification of Fano varieties is an important long-standing problem in algebraic geometry. A new approach to this problem via mirror symmetry was recently proposed by Coates-Corti-Galkin-Golyshev-Kasprzyk. Their philosophy was that Fano varieties can be classified by studying their Laurent polynomial mirrors. This talk will survey the results of a collaborative effort to apply this philosophy to the classification of Fano orbifold surfaces. We will describe a conjectural picture which suggests that classifying suitable deformation classes of certain Fano orbifold surfaces is equivalent to classifying Fano lattice polygons up to an appropriate notion of equivalence. Central to this framework is the notion of mirror duality (between a Fano orbifold surface and a Laurent polynomial) and the closely related operations of algebraic and combinatorial mutations. We will also discuss how combinatorial mutations allow us to find mirror dual Laurent polynomials in practice and will give experimental evidence supporting our conjectures.
07.01.2016 Andreas Gross (Universität Kaiserslautern)
Intersection Theory on Tropicalizations of Torodial Embeddings
A central goal of tropical geometry is to give combinatoric descriptions of algebro-geometric objects. In enumerative geometry, these description ideally give rise to so-called correspondence theorems, which state that some given algebraic enumerative problem can be translated into a tropical enumerative problem with the same solution. The tropical intersection theory of Allermann and Rau has become a useful tool in tropical enumerative geometry, its connection to algebraic geometry being based on the description of the intersection ring of complete toric varieties by Fulton and Sturmfels. Unfortunately, moduli spaces are rarely toric, yet in many cases they are toroidal.
In my talk I will outline how to extend the scope of tropical intersection theory to be able to describe certain intersections on toroidal varieties.
17.12.2015 Sara Filippini (Universität Zürich)
Refined curve counting and the tropical vertex group
The tropical vertex group of Kontsevich and Soibelman is generated by formal symplectomorphisms of the 2-dimensional algebraic torus. It plays a role in many problems in algebraic geometry and mathematical physics. Based on the tropical vertex group, Gross, Pandharipande and Siebert introduced an interesting Gromov-Witten theory on weighted projective planes which admits a very special expansion in terms of tropical counts.
I will describe a refinement or "q-deformation" of this expansion, motivated by wall-crossing ideas, using Block-Goettsche invariants. This leads naturally to the definition of a class of putative q-deformed curve counts. We prove that this coincides with another natural q-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined. This is joint work with Jacopo Stoppa.
29.10.2015 Aleksey Zinger (MPIM Bonn)
Normal Crossings Divisors for Symplectic Topology
I will describe purely symplectic notions of normal crossings divisor and configuration. They are compatible with the existence of the desired auxiliary almost Kahler structures, provided ``existence" is suitably interpreted. These notions lead to a multifold version of Gompf's symplectic sum construction. They also imply that Brett Parker's work on exploded manifolds concerns a multifold version of the usual symplectic sum (or degeneration) formula for Gromov-Witten invariants. We hope our approach can be extended to more general singularities and provide purely symplectic analogues of the singularities and their deformations appearing in the Gross-Siebert program. This is joint work with Mark McLean and Mohammad Tehrani.
25.06.2015 Peter Overholser (Leuven / Imperial College London)
Descendent tropical mirror symmetry for P2
The Gross-Siebert program can be seen as an attempt to understand mirror symmetry from a tropical perspective. Gross has realized this goal in a particular example, giving a tropical description of mirror symmetry for P2. I will show how his construction can be modified to to yield a novel mirror symmetric relationship.
Der Eulerkreis ist eine Gruppe historisch interessierter Mathematiker und Mathematikerinnen, die sich mit den Schriften und dem Werk Leonhard Eulers (1707-1783) beschäftigen.
The Euler Circle Mainz is a Group of historically interested mathematicians who are interested in Leonhard Euler's (1707-1783) work and writings.
Die nachfolgenden Übersetzungen Eulerscher Werke ins Deutsche oder Englische wurden von A. Aycock angefertigt. Wir danken dem Institut für Mathematik für großzügige Unterstützung, welche diese Übersetzungen ermöglichte.
The following translations of Euler's work into German or English were prepared by A. Aycock. We wish to thank the Institute of Mathematics for generous support, that made These translations possible.
Contact:
D. van Straten: straten@Mathematik.uni-mainz.de
T. Sauer: tsauer@uni-mainz.de
Eine Methode sich der Eigenschaft des Maximums oder Minimums erfreuender Kurven zu finden, oder die Lösung des im weitesten Sinn aufgefassten isoperimetrischen Problems
Originaltitel: Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattissimo sensu accepti, erstmals publiziert 1744, Nachdruck in "Opera Omnia", Eneström-Nummer E065
Grundlagen des Differentialkalküls, der vollständige Erklärung dieses Kalküls enthält, Teil 1
Originaltitel: "Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, Partis Prioris, erstmals publiziert im Jahre 1755", Nachdruck in "Opera Omnia", Eneström-Nummer E212
E010 - Eine neue Methode unzählige Differentialgleichungen zweiten Grades auf Differentialgleichungen ersten Grades zurückzuführen
E011 - Konstruktion gewisser Differentialgleichungen, die die Trennung der Unbestimmten nicht zulassen
E019 - Über transzendente Progressionen oder deren allgemeine Terme algebraisch nicht gegeben werden können E019 - On transcendental Progressions or those whose general terms can not be given algebraically
E020 - Über die Summation von unzähligen Progressionen
E031 - Konstruktion der Differentialgleichung \( ax^n dx = dy + y^2 dx\)
E041 - Über die Summen reziproker Reihen E041 - On the sums of series of reciprocals
E043 - Beobachtungen über harmonische Progressionen E043 - Observations on harmonic Progressions
E047 - Das Finden einer jeden Summe einer Reihe aus dem gegebenen allgemeinen Term
E051 - Über die Konstruktion von Gleichungen mit Hilfe von Schleppbewegung und anderen sich auf die inverse Methode der Tangenten beziehenden Dinge
E052 - Lösung der Probleme, die die Rektifikation der Ellipse erfordern
E053 - Lösung eines sich auf die "Geometria situs" beziehenden Problems
E055 - Die allgemeine Methode Reihen zu summieren - weiterentwickelt E055 - The universal Method to sum series further promoted
E059 - Theorems on the reduction of integral formulas to the quadrature of the cirlce
E060 - Über das Finden von Integralen, wenn nach der Integration der variablen Größe ein bestimmter Wert zugeteilt wird
E061 - Eine andere Dissertation über die Summen der aus den Potenzen der natürlichen Zahlen entspringenden Reihen, in welcher dieselben Summationen aus einer gänzlich anderen Quelle deriviert werden E061 - Another Dissertation on the sums of the series of reciprocals arising from the powers of the natural
numbers, in which the same summations are derived from a completely difference source
E062 - Über die Integration von Differentialgleichungen höherer Grade E062 - On the integration of differential equations of higher orders
E164 - Theoreme über die in dieser Form paa ± qbb enthaltenen Teiler von Zahlen
E188 - Die Methode Differentialgleichungen höherer Grade zu integrieren weiter entwickelt
E189 - Über die Bestimmung von Reihen oder eine neue Methode, die allgemeinen Terme von Reihen
zu finden E189 - On the Determination of Series or a new Method to find the General Terms of Series
E190 - Consideration of certain series having singular properties
E191 - Über die Partition von Zahlen E191 - On the partition of numbers
E212 Kapitel 1 - On the transformation of series Kapitel 2 - On the investigation of summable series Kapitel 3 - On Finding finite differences Kapitel 4 - On the Conversion of Functions into Series Kapitel 5 - Investigation of the sum of series from the general Term Kapitel 6 - On the Summation of Progressions by means of infinite Series Kapitel 7 - A further Generalization of the summation method treated in chapter V Kapitel 8 - On the Use of Differential Calculus in the Formation of Series Kapitel 9 - On the use of Differential Calculus in the resolution of Equations Kapitel 10 - On Maxima and Minima Kapitel 11 - On Maxima and Minima of multiform functions and such containing several variables Kapitel 12 - On the Use of Differentials in the Investigation of the real Roots of Equations Kapitel 13 - On Criteria for imaginary roots Kapitel 14 - On Differentials of Functions in only certain cases Kapitel 15 - On the values of functions which in certain cases seem to be undetermined Kapitel 16 - On the Differentiation of inexplicable Functions Kapitel 17 - On the Interpolation of series Kapitel 18 - On the use of Differential Calculus in the Resolution of Fractions
E228 - Über Zahlen, die Aggregate zweier Quadrate sind
E261 - Ein anderes Beispiel der neuen Methode transzendente Größen miteinander zu vergleichen - Über den Vergleich von Ellipsenbogen E261 - Another Specimen of the new Method to compare transcendental Quantities to each other - On the Comparison of the Arcs of an Ellipse
E265 - Über Differentialgleichungen zweiten Grades
E269 - On the Integration of Differential Equations
E271 - Zahlentheoretische Theoreme, mit einer neuen Methode bewiesen
E273 - Betrachtung von Formeln, deren Integration mithilfe von Kegelschnitten durchgeführt werden kann
E274 - Die Konstruktion der Differenzen-Differentialgleichung Aydu2 + (B + Cu)dudy + (D + Eu + Fuu)ddy = 0 für konstant angenommenes Element du
E275 - Bemerkungen zu einem gewissen Auszug des Descartes, der sich auf die Quadratur des Kreises bezieht
E280 - Über Progressionen von Kreisbogen, deren Tangenten nach einem gewissen Gesetz fortschreiten
E281 - Ein Beispiel für einen einzigartigen Algorithmus
E285 - Investigation of Functions from a given condition of the differentials
E295 - Über die Rückführung von Integralformeln auf die Rektifikation der Ellipse und der Hyperbel
E296 - Elemente des Variationskalküls E296 - Elements of the Calculus of Variations
E301 - On the motion of bodies attracted to two fixed centres of force
E321 - Beobachtungen über die Integrale der Formeln \( \int x^{p-1} dx (1 - x^n)^{\frac{q}{n}-1} \) nachdem nach der Integration \( x = 1 \) gesetzt worden ist E321 - Observations on the integral of the formulas \( \int x^{p-1} dx (1 - x^n)^{\frac{q}{n}-1} \) having put x = 1 after the integration
E322 - Über die Nützlichkeit von unstetigen Funktionen in der Analysis
E323 - Über den Gebrauch des neuen Algorithmus' beim Lösen des Pell'schen Problems
E324 - Eigenschaften von Dreiecken, deren Winkel in einem bestimmten Verhältnis zueinander stehen
E325 - Leichte Lösung gewisser sehr schwieriger geometrischer Probleme
E345 - Integration der Gleichung \( \frac{dx}{\sqrt{A+Bx+Cx^2+Dx^3+Ex^4}} = \frac{dy}{\sqrt{A+By+Cy^2+Dy^3+Ey^4}} \) E345 - Integration of the Equation dx√ A+Bx+Cx2+Dx3+Ex4 = dy √A+By+Cy2+Dy3+Ey4
E347 - More general Discussion of the Formulas serving for the comparison of curves
E352 - Bemerkung über die wunderbare Relation zwischen der Reihe der direkten und reziproken Potenzen
E366 - Über die Konstruktion von Differenzen-Differentialgleichungen mit Quadraturen der Kurven
E368 - On the hypergeometric Curve expressed by the equation
E385 Kapitel 1 - Über das Variationskalkül im Allgemeinen Kapitel 2 - Über die Variation zwei Variablen involvierender Differentialformeln Kapitel 4 - Über die Variation zwei Variablen involvierender zusammengesetzter Integralformeln Kapitel 5 - Über die Variation drei Variablen involvierender und zwei Relationen verwickelnder Integralformeln Kapitel 6 - Über die Variation drei Variablen involvierender Differentialformeln, deren Relation in einer einzigen Gleichung enthalten ist Kapitel 7 - Über die Variation drei Variablen involvierender Integralformeln, von denen eine wie eine Funktion der zwei übrigen angesehen wird
E390 - Betrachtungen über orthogonale Trajektorien
E392 - Entwicklung eines außerordentlichen Paradoxons über die Gleichheit von Flächen
E393 - Über Summen, deren Reihen die Bernoulli-Zahlen involvieren
E394 - Über die Partition von Zahlen in so von der Anzahl wie von der Gattung her gegebene Teile
E396 - Zweiter Abschnitt über die Grundsätze der Bewegung von Fluiden
E406 - Beobachtungen zu den Wurzeln von Gleichungen
E408 - Über rektifizierbare Kurven auf einer Kugeloberfläche
E419 - Über Körper, deren Oberfläche sich in die Ebene ausbreiten lassen E419 - On Solids whose surface can be unfolded onto a plane
E420 - Eine neue und leichte Methode die Variationsrechnung zu behandeln
E421 - Entwicklung der Integralformel \( \int x^{f-1} DX (lnx)^{\frac{m}{n}} \) nach Erstreckung der Integration vom Wert \( x = 0 \) bis zu \( x = 1 \) E421 - Expansion of the integral formula \( \int x^{f-1} DX (lnx)^{\frac{m}{n}} \) having extended the integration
from the value \( x = 0 \) to \( x = 1 \)
E422 - Entwicklung eines völlig einzigartigen geometrischen Problems
E445 - Neue Beweise über die Auflösung von Zahlen in Quadrate
E447 - Summation der Progressionen ... E447 - Summation of the progressions … (englischer Text)
E448 - Eine neue unendliche Reihe, die sehr stark konvergiert und die Perimetrie einer Ellipse ausdrückt
E450 - A new way to express irrational quantities approximately
E454 - Über die Auflösung von Irrationalitäten durch Kettenbrüche, wo zugleich eine gewisse neue und einzigartige Gattung des Minimums dargestellt wird
E463 - Über den Wert der Integralformel zlwzl+w 1z2l dz z (ln z)m in dem Fall, in dem nach Integration z = 1 gesetzt wird
E464 - Eine neue Methode Integralgrößen zu bestimmen
E465 - Beweis des Newton'schen Lehrsatzes über die Entwicklung der Potenzen des Binoms für die Fälle, in denen die Exponenten keine ganzen Zahlen sind
E 465 - Proof of the Newtonian Theorem on the Expansion of the Powers of the Binomial for the cases in which the Exponents are not integer numbers
E490 - Über die Repräsentation einer sphärischen Oberfläche auf einer Ebene
E491 - Über die geographische Projektion einer sphärischen Oberfläche
E492 - Über die von De Lisle gebrauchte geographische Projektion bei der gesamten Karte des russischen Reichs
E499 - On the Integration of the Formula ∫ dx log x / √1-xx extended from x = 0 to x = 1
E500 - Über den Wert der von der Grenze \(x=0 \) bis hin zu \(x=1\) erstreckten Integralformal \( \int \frac{x^{a-1}dx}{ln \,x} \frac{(1-x^b)(1-x^c)}{1-x^n} \)
E555 - Über den außerordentlichen Nutzen der Interpolationsmethode in der Lehre der Reihen E555 - On the extraordinary use of the method of Interpolation in the doctrine of Series
E562 - How Sines and Cosines of multiple angles can be expressed by Products
E565 - Über die vielen transzendenten Größen, die sich in keiner Weise mit Integralformeln ausdrücken lassen
E575 - On the remarkable Properties of the Coefficients which occur in the Expansion
of the binomial raised to an arbitrary power
E581 - Eine umfassendere Darstellung des Vergleiches der in der Integralformel \( \int \frac{Zdz}{\sqrt{1+mzz+nz^4}} \) enthaltenen Größen, während \( Z \) irgendeine rationale Funktion \( zz \) bezeichnet E581 - More Complete Explanation of the Comparison of Quantities contained in the Integral formula
R p Zdz 1+mzz+n4 while Z denotes any rational function of zz
E583 - Über die bemerkenswerte Zahl, die bei der Summation der natürlichen harmonischen Progression auftaucht E583 - On the memorable number occurring in the summation of the natural harmonic Progression
E587 - Bemerkungen zu einigen Lehrsätzen des höchst illustren Lagrange
E588 - Investigation of the integral Formula∫((x^(m-1) dx)/(1+x^(k)n) ) in the case in which one set x = ¥ after the integration
E589 - Investigation of the Value of the Integral ∫xm-1dx\1-2xk cosq+x2k extended from x = 0 to x = ¥
E592 - Über die Auflösung von transzendenten Brüchen in unendlich viele einfache Brüche
E593 - Über die Transformation von Reihen in Kettenbrüche, wo zugleich die Theorie nicht unwesentlich erweitert wird
E594 - Eine Methode Integralformen zu finden, die in gewissen Fällen in einem gegebenen Verhältnis zueinander stehen, wo zugleich eine Methode angegeben wird, Kettenbrüche zu summieren E594 - A Method of Finding Integral Formulas which in certain Cases have a given ratio, where at the same time a method of summing continued fractions is presented
E595 - Summation des Kettenbruches, dessen Indizes eine arithmetische Progression festlegen, während alle Zähler Einheiten sind, wo zugleich die Auflösung der Riccati-Gleichung durch Brüche
von dieser Art gelehrt wird
E598 - Über einen gewaltigen Fortschritt in der Zahlentheorie
E602 - Eine leichte Methode alle Eigenschaften von Raumkurven zu finden
E604 - Über rechtwinklige sowie schiefwinklige reziproke Trajektorien
E605 - Über die wundersamen Eigenschaften der Curvae elasticae, die in der Gleichung \( y = \int \frac{xx\mathrm{d}x}{\sqrt{1-x^4}} \) enthalten ist
E606 - Spekulation über die Integralformel \( \int \frac{x^n \mathrm{d}x}{\sqrt{aa-2bx+cxx}} \), wo zugleich herausragende Beobachtungen über Kettenbrüche entstehen E606 - Speculations on the integral formulaR xndx √aa−2bx−cxx where at the same time extraordinary observations on continued fractions occur
E609 - Betrachtungen über orthogonale wie schiefwinklige Trajektorien
E610 - Neue Beweise über die Teiler von Zahlen der Form xx + nyy
E613 - Erläuternde Darstellungen zu den letzten Kapiteln meines Buches "Institutiones Calculi Differentialis" über unerklärbare Funktionen E613 - Further Explanations to the last Chapter of my book Calculi Differentialis on inexplicable functions
E616 - Über die Transformation der divergenten Reihe \( 1 - mx + m (m+n)x^2-m(m+n)(m+2n)x^3 + \) etc. in einen Kettenbruch E616 - On the transformation of the divergent series 1−mx +m(m+n )x2−m(m+n)(m+2n)x3 +etc. into a continued fraction
E617 - Über die Summation von Reihen, in denen die Vorzeichen der Terme alternieren
E620 - Eine leichte Methode, das Integral ∂x x · xn+p−2xn cos ζ+xn−p x2n−2xn cos θ+1 im Fall zu finden, in welchem nach der Integration entweder x = 1 oder x = ∞ gesetzt wird
E621 - Über den immensen Nutzen des Kalküls imaginärer Größen in der Analysis
E629 - Entwicklung der von der Grenze \( x = 0 \) bis hin zu \( x = 1 \) erstreckten Integralformel \( \int \partial x \left( \frac{1}{1-x} + \frac{1}{\log x} \right) \)
E631 - Leichte und klare Analysis, die zu höchst abstrusen Reihen führt, mit welchen nicht nur die Wurzeln aller algebraischen Gleichungen sondern auch jedwede Potenzen derer ausgedrückt werden können
E637 - Ein neuer Beweis, dass die Newtonsche Entwicklung der Potenzen des Binoms auch für gebrochene Exponenten gilt
E652 - Über den allgemeinen Term der hypergeometrischen Reihen E652 - On the general Term of hypergeometric Series
E655 - Beobachtung über Reihen, deren Terme nach den Sinus oder Kosinus vielfacher Winkel fortschreiten
E656 - Über höchst bemerkenswerte aus dem Kalkül der imaginären Größen herstammende Integrationen
E658 - Über das Finden der Kraftmomente bezüglich beliebiger Achsen, wo viele außerordentliche Eigenschaften über zwei Geraden, die nicht in derselben Ebene liegen, erklärt werden
E659 - Eine leichte Methode die Momente aller Kräfte bezüglich einer beliebigen Achse zu bestimmen
E661 - Verschiedene Betrachtungen über hypergeometrische Reihen E661 - Various Consideration on hypergeometric Series
E662 - On the true Value of the integral formula ∂ x log 1 xn extended from the limit x = 0 to x = 1
E663 - Eine umfassendere Darstellung jener merkwürdigen Reihen, die aus den Koeffizienten der Potenzen des Binoms gebildet werden E663 - A more complete Explanation of those memorable series which are formed from the binomial coefficients
E684 - Über die Wurzeln der unendlichen Gleichung 0=1 - x^2 / n(n+1) + x^4 / n(n+1)(n+2)(n+3) -
x^6 / n ... (n+5) + etc.
E671 - Über höchst irrationale von Winkeln abhängige Differentialformeln, welche sich dennoch mit Logarithmen und Kreisbogen integrieren lassen
E672 - Ein höchst bemerkenswertes Theorem über die Integralformel
E673 - Eine auf Vermutungen beruhende Untersuchung über die Integralformel
E674 - Beweis des über eine Vermutung gefundenen außerordentlichen Theorems über die Integration der
Formel
E675 - Über die Werte der von der Grenze der Variable x = 0 bis hin zu x = ∞ erstreckten Integrale
E676 - Eine kürzere Methode die Vergleiche der in der Form ... enthaltenen transzendenten Größen zu finden
E678 - A new Method to investigate the cases, in which it is possible to solve the difference-differential equation (1 - )- - ^2= 0
E681 - Ein Beispiel von Differentialgleichungen unbestimmten Grades und deren Interpolation
E694 - Eine weitere Untersuchung über imaginäre Integralformeln
E698 - Verschiedene Betrachtungen über die Fläche von Kugeldreiecken
E700 - On differential formulas of second degree that admit an integration
E704 - Weitere Untersuchungen über die Reihen, die nach den Vielfachen eines Winkels fortschreiten
E707 - Über den außerordentlichen Nutzen des Kalküls mit imaginären Größen in der Integralrechnung
E710 - Beweis einer einzigartigen Transformation von Reihen E710 - A specimen of a singular transformation of series
E711 - Eine neue und leichte Methode, nicht nur die Wurzeln selbst sondern auch beliebige Potenzen derer aller algebraischen Gleichungen mittels gefälliger Reihen auszudrücken
E722 - Analytische Untersuchungen über die Entwicklung der Trinomialpotenz \( (1 + x + xx) \) E722 - Analytical Investigations on the Expansion of the trinomial Power (1+ x + xx)n
E726 - Beweis des ausgezeichneten numerischen Theorems über die Koeffizienten der Binomialpotenzen
E734 - Integration dieser Differentialgleichung \(dy + yydx = \frac{Adx}{(a+bx+cxx)^2}\)
E735 - Über ein riesiges Paradoxon, welches in der Analysis der Maxima und Minima auftritt
E736 - Über die Summation der Reihen, die in dieser Form enthalten sind \( \frac{a}{1}+\frac{a^2}{4}+\frac{a^3}{9}+\frac{a^4}{16}+\frac{a^5}{25}+\frac{a^6}{36}+etc\) E736 - On the Summation of the series contained in this form \( \frac{a}{1}+\frac{a^2}{4}+\frac{a^3}{9}+\frac{a^4}{16}+\frac{a^5}{25}+\frac{a^6}{36}+etc\)
E740 - Über nicht in derselben Ebene gelegene gekrümmte Linien, die mit der Eigenschaft des Maximums oder Minimums versehen sind
E742 - Bemerkungen über die in dieser Form enthaltenen Kettenbrüche
E743 - Über eine höchst bemerkenswerte Reihe, mit welcher jede Binomialpotenz ausgedrückt werden kann
E744 - Über die Teiler der in der Form mxx + nyy enthaltenen Zahlen
E745 - Über die Kettenbrüche von Wallis E745 - On Wallis’ continued fractions
E746 - Eine kurze Methode, Summen unendlicher Reihen durch Differentialformeln zu untersuchen
E750 - Ein Kommentar zum Kettenbruch, mit welchem der bedeutende Lagrange die Binomialpotenzen ausgedrückt hat
E751 - Leichte Analysis, die Riccati-Gleichung durch einen Kettenbruch aufzulösen
E752 - Über gewisse sehr schwer zu findende Integrale
E759 - Eine genauere Untersuchung über Brachystochronen
E760 - Über die wahre Brachystochrone oder die Linie des schnellsten Herabsinkens in einem widerstehenden Medium
E761 - Über die Brachystochrone in einem widerstehenden Medium, während
der Körper auf irgendeine Weise zu einem festen Kraftzentrum hingezogen wird
E768 - On the Binomial Coefficients and their Interpolation
E814 - Abschnitt III der Grundlagen des Differentialkalküls
Rationalizing roots: an algorithmic approach Marco Besier, Duco van Straten, Stefan Weinzierl
Communications in Number Theory and Physics Vol. 13 No. 2, 253–297 (2019) hep-th; hep-ph; math-ph; MITP/18-091 arXive: 1809.10983
On a theorem of Greuel and Steenbrink
in: Singularities and computer algebra
(Festschrift for Gert-Martin Greuel on the Occasion of his 70th Birthday)
Eds.: W. Decker, G. Pfister, M. Schulze
353–364, Springer (2017)
Twisted cubics on cubic fourfolds Christian Lehn, Manfred Lehn, Christoph Sorger, Duco van Straten
Crelle‘s J. reine u. angew. Math., Vol. 731, 87-128 (2017)
From Briancon-Skoda to Scherk-Varchenko
Duco van Straten
in: Commutative Algebra and Noncommutative Algebraic Geometry I
Eds: D. Eisenbud, S. B. Iyengar, A. K. Singh, J. T. Stafford, M. Van den Bergh
MSRI Publications, Vol. 67, 347-370 (2015)
Calabi-Yau conifold expansions Slawomir Cynk, Duco van Straten
in: Arithmetic Geometry of K3 Surfaces and Calabi-Yau Threefolds
Eds.: R. Laza, R. Schütt, N. Yui
Fields Institute Communications, Vol. 67, 499-515 (2013)
On symplectic hypersurfaces
Manfred Lehn, Yoshinori Namikawa, Christoph Sorger, Duco van Straten
in: Minimal models and extremal rays (Kyoto 2011)
Eds.: J. Kollár, O. Fujino, S. Mukai, N. Nakayama
Advanced Studies in Pure Mathematics, Vol. 99, 1-24 (2013)
Apéry Limits of Differential Equations of Order 4 and 5 Gert Almkvist, Duco van Straten, Wadim Zudilin
in: Modular Forms and String Duality
Eds.: N. Yui, C. Doran, H. Verrill
Fields Institute Communications, Vol. 54, 105-123 (2008)
Some Problems on Lagrangian Singularities Duco van Straten
in: Singularities and computer algebra
Eds.: C. Lossen, G. Pfister
London Math. Soc. Lecture Notes, Vol 324, 333-349, Cambridge Univ. Press (2006)
Disentanglements Theo de Jong, Duco van Straten
in: Singularity Theory and its Applications, Warwick 1989, vol 1
Eds.: D. Mond, J. Montaldi
Lecture Notes Math., Vol. 1462, 199-211, Springer (1991)
