Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

Archive of AG&Phy talks

Archives AG&Phy September 2020-July 2021


29 Sept. 2020

Maxim Smirnov (Augsburg Universität)
Residual categories of Grassmannians [Slides]

Exceptional collections in derived categories of coherent sheaves have a long history going back to the pioneering work of A. Beilinson. After recalling the general setup, I will concentrate on some recent developments inspired by the homological mirror symmetry. Namely, I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. I will illustrate this relationship in the case of some isotropic Grassmannians. This is a joint work with Alexander Kuznetsov.


6 Oct. 2020

Noémie Combe (MPI für Mathematik in den Naturwissenschaften, Leipzig)
The NY Gravity operad and the Deligne-Mumford stack with hidden symmetry

A paracomplex Deligne-Mumford stack is considered. This structure introduces additional conical singularities, in the neighbourhood of which the curvature is negative. In particular, properties are investigated and we establish a relationship between the paracomplex Deligne-Mumford stack and Deligne-Mumford stack. Moreover, we show that the NY Gravity operad associated to this space (introduced by Manin and Combe) has a mixed Hodge structure. In this framework, the strong Hodge theorem extends and the pure Hodge structure (Kähler package) holds. Since we use tools from L2 cohomology, a parallel with the model of Atiyah-Hitchin and Donaldson is established.


13 Oct. 2020

Gaëtan Borot (HU Berlin)
Geometry of the combinatorial moduli space of curves I
Slides

The moduli space of complex curves has several descriptions, giving the same topological space but different geometric structures. The description in terms of metric ribbon graphs gives it a polytopal complex structure, and Kontsevich gave it an (almost everywhere) symplectic structure used in his proof of Witten's conjecture. I will revisit the associated geometry of this space (or rather of its universal cover, ie Teichmuller space)  making it parallel to the Weil-Petersson geometry coming from hyperbolic metrics on surfaces: we will see how to define Fenchel-Nielsen coordinates that are Darboux for Kontsevich symplectic structure. There is in fact a flow, originally studied by Bowditch-Epstein, Mondello and Do, taking hyperbolic geometry to combinatorial geometry, and I will present stronger results about the convergence of this flow. Based on joint work with Jørgen E. Andersen, Séverin Charbonnier, Alessandro Giacchetto, Danilo Lewanski, Campbell Wheeler.


20 Oct. 2020

Gaëtan Borot (HU Berlin)
Geometry of the combinatorial moduli space of curves II
Slides

Mirzakhani obtained a topological recursion for the Weil-Petersson volumes of the moduli space of bordered Riemann surfaces, based on a recursive partition of unity on the Teichmuller space that extended an identity by McShane. Her idea can be generalised to study length statistics of multicurves on surfaces. I will explain that it fits in a larger theory of "geometric  recursion", allowing to define recursively mapping class group invariant functions on the Teichmuller space, whose integration on the moduli space of curves can be computed by a topological recursion. This can be done both in the hyperbolic setting, or in the combinatorial setting using the results of the first talk. This can be applied to obtain uniform geometric proofs of Witten's conjecture, of Norbury's lattice point counting, and several proofs of topological recursion for Masur-Veech volumes of the moduli space of quadratic differentials.

Based on joint works with Jørgen E. Andersen, Séverin Charbonnier, Vincent Delecroix, Alessandro Giacchetto, Danilo Lewanski, Nicolas Orantin, Campbell Wheeler.


27 Oct. 2020

Georgios Kydonakis (MPIM Bonn)
Complex projective structures over Riemann surfaces and solutions to Hitchin's equations

R. Gunning in 1967 defined a projective coordinate system of a compact Riemann surface as one for which the transition functions are given by Möbius transformations. Such structures are alternatively described by particular flat bundles called by Gunning at the time "indigenous bundles". In modern terminology, this pertains to the structure of an SLn(C)-oper as introduced by A. Beilinson and V. Drinfeld, who, more generally, introduced G-opers for any simple simply connected complex Lie group G. We shall focus in this talk on the relationship of these structures to certain families of solutions to Hitchin's equations. Joint work with Olivia Dumitrescu, Laura Fredrickson, Rafe Mazzeo, Motohico Mulase and Andrew Neitzke. 


3 Nov. 2020

Jörg Teschner (DESY Hamburg)
Proposal for a geometric characterisation of topological string partition functions

We propose a geometric characterisation of the topological string partition functions associated to the local Calabi-Yau (CY) manifolds used in the geometric engineering of d = 4, N = 2 supersymmetric field theories of class S. A quantisation of these CY manifolds defines differential operators called quantum curves. The partition functions are extracted from the isomonodromic tau-functions associated to the quantum curves by expansions of generalised theta series type. It turns out that the partition functions are in one-to-one correspondence with preferred coordinates on the moduli spaces of quantum curves defined using the Exact WKB method. The coordinates defined in this way jump across certain loci in the moduli space. The changes of normalisation of the tau-functions associated to these jumps define a natural line bundle.


10 Nov. 2020

Dirk Kreimer (HU Berlin)
Monodromy in physics amplitudes

The talk discusses the structure of amplitudes which emerge in high energy physics computations. Such amplitudes appear as 'blobbed' recursions of graphs.  Each of them gives rise to a multivalued function often related to (elliptic or generalized) polylogarithms. Their monodromies relate to algebraic and combinatoric properties of graphs. We will discuss this relation in some detail.


17 Nov. 2020

Gavril Farkas (HU Berlin)
Compactification of moduli of holomorphic differentials

The moduli space of holomorphic differentials (with prescribed zeros and poles) on nonsingular curves is not compact since the curve may degenerate. I will discuss various compactification of these strata in the moduli space of Deligne-Mumford stable pointed curves, which includes the space of canonical divisors as an open subset. The theory leads to geometric/combinatorial constraints on the closures of the strata of holomorphic differentials and as a consequence, one can determine the cohomology classes of the strata. This is joint work with Rahul Pandharipande.


24 Nov. 2020

Alexander Thomas (MPIM Bonn)
A geometric approach to Hitchin components via punctual Hilbert schemes

Hitchin components are generalizations of the classical Teichmüller space. I will describe a program to describe Hitchin components as the moduli space of some new geometric structure on the surface. This geometric structure generalizes the complex structure. Its construction uses the punctual Hilbert scheme of the plane. It should give a unified description of Hitchin components without fixed complex structure on the surface.


1 Dec. 2020

Alexi Morin-Duchesne (MPIM Bonn)
Boundary emptiness formation probabilities in the six-vertex model at Delta = - 1/2

The connection between statistical mechanics and the combinatorics of alternating sign matrices is known since the work of Razumov and Stroganov on the spin-1/2 XXZ chain. One important example of this combinatorial relation occurs in the study of the emptiness formation probability EFPN,m. This observable is defined as the sum of the squares of the ground state components of the Hamiltonian for the chain of length N, restricted to components where $m$ consecutive spins are aligned. At the combinatorial point Delta = -1/2, it takes the form of a simple product of integers. This was shown by Cantini in 2012.
 
In this talk, I discuss joint work with C. Hagendorf and L. Cantini where we define a new family of overlaps CN,m for the spin-1/2 XXZ chain. It is equal to the linear sum of the groundstate components that have $m$ consecutive aligned spins. For reasons that will be discussed, we refer to the ratio CN,m/CN,0 as the boundary emptiness formation probability. We compute CN,m at the combinatorial point as a simple product of integers.
 

15 Dec. 2020

Gautier Ponsinet (MPIM Bonn)
Universal norms of p-adic Galois representations and the Fargues-Fontaine curve

In 1996, Coates and Greenberg computed explicitly the module of universal norms associated with an abelian variety in a perfectoid field extension. The computation of this module is essential to Iwasawa theory, notably to prove "control theorems" for Selmer groups generalising Mazur's foundational work on the Iwasawa theory of abelian varieties over Z_p-extensions. Coates and Greenberg then raised the natural question on possible generalisations of their result to general motives. In this talk, I will present a new approach to this question relying on the classification of vector bundles over the Fargues-Fontaine curve, which enables to answer Coates and Greenberg's question affirmatively in new cases.


19 Jan. 2021

Alice Guionnet (ENS Lyon)
Topological expansions, random matrices and operator algebras theory
Slides

In this lecture, I will discuss the remarkable connection between random matrices and the enumeration of maps and some applications to operator algebras and physics. Part of my talk will be dedicated to my collaboration with Vaughan Jones on this subject.


26 Jan. 2021

Bruno Klingler (HU Berlin)
Tame geometry and Hodge theory
Slides

Originating in Grothendieck's "Esquisse d'un programme", tame geometry has been developed by model theorists under the name "o-minimal structures". It studies structures where every definable set has a finite geometry complexity. It has for prototype real semi-algebraic geometry, but is much richer. After recalling its basic features, I will describe its recent applications to Hodge theory and period maps.


2 Feb. 2021

Tomáš Procházka (LMU München)
W algebras and integrable structures
Slides

I will review basic properties of W-algebras, in particular the WN family and the associated Winfinity algebra. These algebraic structures show up at many places in mathematical physics. Winfinity admits two different descriptions: the traditional description starts from the Virasoro algebra of 2d conformal field theory and extends it by local conserved currents of higher spin. The description discovered more recently is the Yangian description manifesting the integrable structure of the algebra. The map between these two pictures is non-trivial, but can be understood by using the Maulik-Okounkov 'instanton' R-matrix as a bridge between these two pictures.

9 Feb. 2021

Karim Adiprasito (Hebrew University of Jerusalem)
Combinatorics, Lefschetz theorems beyond positivity and transversality of primes

I will survey applications of Hodge Theory to combinatorics, and, quite suprisingly, how Hodge-Riemann relations and Lefschetz type theorems can be proven using combinatorial methods, in settings that are beyond classical algebraic geometry, at least as long as some notion of positivity is available.

I then go one step further, and ask how many triangles a PL embedded simplicial complex in R4 can have (the answer, generalizing a result of Euler and Descartes, is at most 4 times the number of edges). I discuss how to reduce this problem to a Lefschetz property beyond projective structure.

The main part of the talk is devoted to provide an indication how the proof works, explain the notion of transversal primes as well as Hall matching theorems for spaces of linear operators, and their connection in the Hall-Laman relations which replace our knowledge of the signature of the Hodge-Riemann relations in the Kähler case with nondegeneracy at monomial ideals.


16 Feb. 2021

Vivek Shende (U. Berkeley/SDU Odense)
Holomorphic curves, boundaries, skeins, and recursion
Slides

I will explain how to define all genus open Gromov-Witten invariants for Calabi-Yau 3-folds.  The key idea is to count curves by their boundary in the skein modules of Lagrangians. Then I will prove the assertion of Ooguri and Vafa that the colored HOMFLYPT polynomials of a knot are exactly the counts of holomorphic curves in the resolved conifold with boundary on a Lagrangian associated to the knot. In the process we will see the geometric origin of recursion relations for colored knot invariants. This talk presents joint work with Tobias Ekholm. 

23 Feb. 2021

Vasily Golyshev (MCCME, Moscow)
Non-abelian Abel's theorems and quaternionic rotation

I will talk on a subject which has evolved over the years in discussions with M. Kontsevich, A. Mellit, V. Roubtsov, and D. van Straten. I will explain how the kernels of non-abelian Abel's theorems can be seen as a low-technology alternative to geometric Langlands, and produce a Clausen-type lift from the Markov to a quaternionic local system on a punctured genus 1 curve


2 Mar. 2021

Jean-Louis Colliot-Thélène (CNRS & Université Paris Saclay)
Hilbert's irreducibility theorem and jumps in the rank of the Mordell-Weil group
Slides

Let k be a number field and U a smooth integral k-variety. Let X → U be an abelian scheme of relative dimension at least one. We consider the set U(k)+ ⊂ U(k) of k-rational points m ∈ U(k) such that the Mordell-Weil rank of the fibre Xm at m, which is an abelian variety over k, is strictly bigger than the Mordell-Weil rank of the generic fibre Xk(U) over the function field k(U). We prove: if the k-variety X is k-unirational, then U(k)+ is dense for the Zariski topology on U. If the k-variety X is k-rational, then U(k)+ is not a thin set in U. The second result leads us to a discussion of varieties over which Hilbert's irreducibility theorem holds.


9 Mar. 2021

Thomas Willwacher (ETH Zürich)
Embeddings of manifolds in euclidean space and Feynman diagrams
Slides

The spaces of embeddings of manifolds M in Rn are classical objects in topology. I will discuss the computation of their rational homotopy type, for fairly general M, based on recent joint work with B. Fresse and V. Turchin. The answer can be formulated in terms of combinatorial (Feynman) diagrams.


16 Mar. 2021

Pedro Tamaroff (Trinity College Dublin)
Poincaré-Birkhoff-Witt theorems: homotopical and effective computational methods for universal envelopes
Slides

In joint work with V. Dotsenko, we developed a categorical framework for Poincaré-Birkhoff-Witt type theorems about universal enveloping algebras of various algebraic structures, and used methods of term rewriting for operads to obtain new PBW theorems, in particular answering an open question of J.-L. Loday. Later, in joint work with A. Khoroshkin, we developed a formalism to study Poincaré–Birkhoff–Witt type theorems for universal envelopes of algebras over differential graded operads, motivated by the problem of computing the universal enveloping algebra functor on dg Lie algebras in the homotopy category. Our formalism allows us, among other things, to obtain a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras, and extend Quillen's quasi-isomorphism C(g) → BU(g) to homotopy Lie algebras. I will survey and explain the role homological algebra, homotopical algebra, and effective computational methods play in the main results obtained with both V. Dotsenko (1804.06485) and A. Khoroshikin (2003.06055) and, if time allows, explain a new direction in which these methods can be used to study certain operads as universal envelopes of pre-Lie algebras.


23 Mar. 2021

Gereon Quick (NTNU Oslo)
From rational points to étale homotopy fixed points
Slides

To decide whether integral or rational solutions to polynomial equations exist is a classical problem in mathematics. Such solutions correspond to what we now call rational points on algebraic varieties. To detect such points is still a notoriously difficult task. In this talk we will build a bridge from rational points to homotopy theory and discuss how étale homotopy fixed points under the Galois action can be used to define obstructions for the existence of rational points. Along the way we will review ideas from étale homotopy theory and review the difference between fixed and homotopy fixed points under group actions.

30. Mar. 2021

Mark Gross (Cambridge Univ.)
Intrinsic mirror symmetry
Slides

I will talk about the program developed jointly with Bernd Siebert to understand mirror symmetry. In particular, I will focus on the general construction of mirrors. Associated to any log Calabi-Yau pair (X,D) with maximally degenerate boundary D or to any maximally unipotent degeneration of Calabi-Yau manifolds X→S, we associate a mirror object as either the Spec or Proj of a ring. This ring, analogous to the degree 0 part of symplectic homology of X\D, is constructed using a flavor of Gromov-Witten invariants called punctured invariants, developed jointly with Abramovich, Chen and Siebert.


6 Apr. 2021

François Loeser (Institut de Mathématiques de Jussieu)
Limits of complex integrals and non-archimedean geometry
Slides

Chambert-Loir and Ducros recently developed a theory of real valued (p,q)-forms and currents on Berkovich spaces which is an analogue of the theory of differential forms on complex spaces. This theory allows to integrate compactly supported forms of maximal bidegree, the output being a real number. In this talk I will explain how these integrals occur as limits of one-parameter families of complex integrals. This is joint work with A. Ducros and E. Hrushovski.
 

13 Apr. 2021

Hülya Argüz (Université Versailles St-Quentin)
Donaldson-Thomas invariants of quivers with potentials from the flow tree flormula
Slides

A categorical notion of stability for objects in a triangulated category was introduced by Bridgeland. Donaldson-Thomas (DT) invariants are then defined as virtual counts of semistable objects. We will focus attention on a natural class of triangulated categories defined via the representation theory of quivers with potentials, and explain how to compute DT invariants in this case from a smaller subset of "attractor invariants'' which are known in many cases. For this we investigate wall-crossing in the space of stability conditions, and prove a flow tree formula conjectured by Alexandrov-Pioline in this setup. This is joint work with Pierrick Bousseau.


20 Apr. 2021

Anna Wienhard (Heidelberg Universität)
Positivity, higher Teichmüller spaces and (non-commutative) cluster algebras
Slides

Higher Teichmüller theory emerged as a new field in mathematics about twenty years ago. It generalizes Thurston’s view on Teichmüller space, but also draws on new connections to representation theory, Higgs bundles, theoretical physics and cluster algebras. In this talk I will provide an overview of the different approaches to higher Teichmüller spaces, and explain how a generalization of total positivity provides a (conjectural) unifying framework. It also leads to new predictions and connections to Lie groups over non-commutative rings.


27 Apr. 2021

Yuri B. Suris (TU Berlin)
Bilinear discretization of quadratic vector fields: integrability and geometry
Slides

We discuss dynamics of birational maps which appear as bilinear discretizations of quadratic vector fields. The corresponding dynamical systems turn out to be integrable much more often than could be expected. Various aspects of integrability of birational dynamical systems will be discussed, along with remarkable geometric structures behind some of the particular examples.


4 May 2021

Yuri Manin (MPIM Bonn)
Non-associative Moufang loop of point classes on a cubic surface
Slides

In this talk I will explain, how the problem stated in my book Cubic Surfaces, was recently solved by Dimitri Kanevsky, after about fifty years since its publication.


11 May 2021

Omid Amini (École Polytechnique)
Hybrid geometry of curves and their moduli spaces

It is now well-understood that the Deligne-Mumford compactification of the moduli spaces of curves is not large enough for capturing several limit phenomena concerning geometry of curves and their families.

The aim of this talk is to present a hybrid refinement of the Deligne-Mumford compactification, that uses the combinatorics and geometry of graphs and their minors (in the sense of Robertson-Seymour graph minor theory), and which allows to address some of these limit questions arising in the study of Riemann surfaces and their asymptotic geometry.

Among the resulting applications, I will provide a complete solution to the problem of limits of Arakelov-Bergman measures and present a refined analysis of the degenerations of Arakelov Green functions, close to the boundary of the moduli spaces.

The talk is based on joint works with Noema Nicolussi.


18 May 2021

Inna Zakharevich (Cornell)
Point counting to detect non-permutative elements of K1(Var)
Slides

The Grothendieck ring of varieties is defined to be the free abelian group generated by varieties (over some base field k) modulo the relation that for a closed immersion Y → X there is a relation that [X] = [Y] + [X \ Y]. This structure can be extended to produce a space whose connected components give the Grothendieck ring of varieties and whose higher homotopy groups represent other geometric invariants of varieties. This structure is compatible with many of the structures on varieties. In particular, if the base field k is finite for a variety X we can consider the "almost-finite" set X(kbar), which represents the local zeta function of X. In this talk we will discuss how to detect interesting elements in K1(Var) (which is represented by piecewise automorphisms of varieties) using this zeta function and precise point counts on X.


25 May 2021

Will Sawin (Columbia University)
Is freeness enough for counting rational points ?
Slides

Manin's conjecture predicts the number of rational points of bounded height on a Fano variety, but for the predictions to hold, we must first remove a "thin set" consisting of rational points lying on certain special subvarieties and lifting to certain special covering spaces. It might be better if we could instead identify the bad rational points to remove by their intrinsic geometry. It may seem that rational points do not have any intrinsic geometry, but recently Peyre has given two proposals to do this, one measuring the freeness of a point and the other using all the heights. I will explain why the freeness proposal is not, alone, sufficient.


1 June 2021

Nathalie Wahl (University of Copenhagen)
String topology operations

String topology for a manifold can be defined as a certain set of operations on the homology of its free loop space, that is the space of all maps from a circle into the manifold. I’ll give an overview of some of the many non-trivial string topology operations we know, both from an algebraic model (Hochschild homology) perspective, and from a more geometric perspective, as directly defined on the homology of the loop space. 


8 June 2021

Jacopo Stoppa (SISSA)
Log Calabi-Yau surfaces and Jeffrey-Kirwan residues

I will discuss joint work in progress with Riccardo Ontani (SISSA). We use Gross-Hacking-Keel mirror symmetry for log Calabi-Yau surfaces in order to provide a geometric interpretation for certain remarkable formulae appearing in the physical literature, in the context of supersymmetric gauge theories, which involve Jeffrey-Kirwan residues of meromorphic forms.


15 June 2021

Andras Szenes (Université de Genève)
The Verlinde formula and parabolic Hecke correspondences

The Verlinde formula for the Hilbert function of the moduli space of vector bundles on a Riemann surface is one of the most fascinating results in enumerative geometry. I will review several approaches to this theorem, and then present a brand new proof (joint work with Olga Trapeznikova) based on a new look at the Drinfeld-Hecke correspondences on curves.
Chaired by Alexander Thomas.


22 June 2021

Nicolò Sibilla (SISSA)
Fukaya category of surfaces and pants decompositions

In this talk I will explain some results joint with James Pascaleff on the Fukaya category of Riemann surfaces. I will explain a local-to-global principle which allows us to reduce the calculation of the Fukaya category of surfaces of genus g greater than one to the case of the pair-of-pants, and which holds both in the punctured and in the compact case. The starting point are the sheaf-theoretic methods which are available in the exact setting, and which I will review at the beginning of the talk. This result has several interesting consequences for HMS and geometrization of objects in the Fukaya category. The talk is based on 1604.06448 and 2103.03366.


29 June 2021

Ashkan Nikeghbali (Zürich Universität)
Convergence of random holomorphic functions with real zeros, random matrices and the distribution of the zeros of the Riemann zeta function

The GUE conjecture states that the ordinates of the zeros of the Riemann zeta function on the critical line should behave statistically like eigenvalues of large random matrices: more precisely they should be asymptotically distributed like a sine kernel determinantal point process. In the past two decades, a model has emerged to understand and predict the distribution of values of the Riemann zeta function on the critical line: the characteristic polynomial of random unitary matrices. It has been thought that there should exist a random holomorphic emerging as some scaling limit of the characteristic polynomial. We give a construction of this function and describe its relation to the GUE conjecture. We then show how it naturally appears in ratios in random matrix theory. We then discuss several generalisations of this construction by other authors as well as by J. Najnudel and myself.


6 July 2021

Marta Mazzocco (Birmingham University)
Isomonodromic deformations: confluence, reduction and quantization
Slides

In this talk we study the theory of isomonodromic deformations for systems of differential equations with poles of any order on the Riemann sphere. Our initial motivation was to generalise a theorem by Reshetikhin that the quasiclassical solution of the standard KZ equations (i.e. with simple poles) is expressed via the isomonodromic τ-function arising in the case of Fuchsian systems. Along the way of pursuing this project, we have found a number of interesting results, some of which were already known as folklore (i.e. either done is very specific examples or not really proved formally), others completely original.


13 July 2021

Piotr Sulkowski (Warsaw University)
Permutohedra for knots and quivers
Slides

The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be assigned to a given knot and encode the same information. I will explain that this phenomenon is generic rather than exceptional. First, I will present conditions that characterize equivalent quivers. Then I will show that equivalent quivers arise in families that have the structure of permutohedra, and the set of all equivalent quivers for a given knot is parameterized by vertices of a graph made of several permutohedra glued together. These graphs can be also interpreted as webs of dual 3d N=2 theories. All these results are intimately related to properties of homological diagrams for knots, as well as to multi-cover skein relations that arise in counting of holomorphic curves with boundaries on Lagrangian branes in Calabi-Yau three-folds.


20 July 2021

Margaret Bilu (IST Austria)
Zeta statistics
Slides

Many questions in number theory have a natural analogue, of more geometric nature, formulated in the Grothendieck ring of varieties. For example, Poonen's finite field Bertini theorem has a motivic counterpart due to Vakil and Wood; however, despite the clear similarities between these two results, none of the two can be deduced from the other. The aim of this talk is to describe and motivate a conjectural way of comparing such statements in arithmetic and motivic statistics, by reformulating them in terms of the convergence of zeta functions in different topologies. We will finish by mentioning some concrete settings where our conjectures are satisfied. This is joint work with Ronno Das and Sean Howe.


27 July 2021

Samuele Giraudo (Université Gustave Eiffel)
Operads of musical phrases and generation
Slides

Operads can be used as a framework to represent musical phrases. A musical phrase is seen in this way as an operation that can be composed with some others to construct bigger phrases. We recall the T construction, a functorial construction from the category of monoids to the category of operads which is used to build the music box operad encoding musical phrases. We then describe how to use it to randomly generate some elements from small musical phrases. This is based on specific generation algorithms using colored operads and bud generating systems. The latter are generalizations of context-free grammars.