This page is an archive of previous talks in the Dutch Topology Intercity Seminar (TopICS), currently organized by the algebraic topologists from Radboud University Nijmegen, the University of Utrecht, and the Vrije Universiteit Amsterdam. The current program is here. If you would like to receive seminar announcements and/or participate in the seminar, please subscribe to this mailing list.
Spectral sequences have proved to be the most fruitful way to understand homotopy groups of spectra, particularly the various flavours of Adams spectral sequences. Sometimes, one can identify an Adams spectral sequence for a spectrum X with a more ‘ad hoc’ spectral sequence constructed specifically for X, one which may a priori be easier to understand. This gives one greater control over the homotopy groups of X. In this talk, I will use synthetic spectra to give much more structured versions of such identifications. Moreover, in cases where such an identification fails, this synthetic approach still goes through. I will show how in certain cases, one can define a synthetic spectrum that captures such an ‘ad hoc’ spectral sequence, and obtain from it all the benefits one would have had if such an identification had been possible. The main example is a synthetic spectrum Smf of synthetic modular forms. This is joint work with Christian Carrick and Jack Davies.
After a brief introduction to chromatic homotopy theory, I will report on recent advances in our understanding of some \(K(n)\)-local invariants using ideas from \(p\)-adic geometry. Joint work, partly in progress, with Schlank, Stapleton, and Weinstein.
Pontryagin classes were originally considered as invariants of real vector bundles, but it was realised in the 60s that they can be defined more generally for Euclidean bundles, that is, fibre bundles whose fibres are homeomorphic to Euclidean space. This led to the question whether the well-known vanishing of large-degree Pontryagin classes for small-dimensional vector bundles continues to hold in the setting of Euclidean bundles. Surprisingly, Weiss proved a few years ago that this often fails, even for bundles over spheres. I will explain a strengthening of this result resulting from joint work with A. Kupers: For every \(k \geq 0\), there exists a 6-dimensional Euclidean fibre bundle over a sphere whose kth Pontryagin class is nontrivial.
Nonlinear elliptic PDE problems can be described as zero finding problems of non-linear proper Fredholm mappings \(f:H\rightarrow H\), where \(H\) is an infinite dimensional Hilbert space. In this talk I will classify these mappings up to homotopy in terms of a non-trivial quotient of the stable homotopy groups of spheres. This is joint work with Lauran Toussaint.
The generalized Hilbert’s third problem asks about the invariants preserved under the scissors congruence operation: given a polytope \(P\) in \(\mathbb{R}^n\), one can cut \(P\) into a finite number of smaller polytopes and reassemble these to form \(Q\). Kreck, Neumann and Ossa introduced and studied an analogous notion of cut and paste relation for manifolds called the SK-equivalence (“schneiden und kleben” is German for “cut and paste”). In this talk I will explain the construction that will allow us to speak about the “K-theory of manifolds” spectrum. The zeroth homotopy group of the constructed spectrum recovers the classical groups \(\mathrm{SK}_n\). I will show how to relate the spectrum to the algebraic \(K\)-theory of integers, and how this leads to the Euler characteristic and the Kervaire semicharacteristic when restricted to the lower homotopy groups. Further I will describe the connection of our spectrum with the cobordism category.
In the 90s, Cohen- Jones, and Segal asked the question of whether various types of Floer homology theories could be upgraded to the homotopy level by constructing stable homotopy types encoding Floer data. They also sketched how one could construct these Floer homotopy types as (pro)spectra in the situation that the infinite-dimensional manifold involved is “trivially polarized”. It has since been realized that the correct home for Floer homotopy types, in the polarized situation, is twisted spectra. This is a generalization of parametrized spectra that one can roughly think of as sections of bundles of categories whose fibre is the category of spectra. The aim of this talk is to give an introduction of Floer homotopy theory and twisted spectra. I will also outline the construction of a circle equivariant twisted spectrum from Seiberg-Witten Floer data associated to a 3-manifold equipped with a complex spin structure. As there are many moving parts to this (Atiyah-Singer index theory, finite-dimensional approximation, Conley index theory etc.), I will try to keep the talk on a conceptual level that will hopefully be accessible to a large audience. This is joint work in progress with S. Behrens and T. Kragh.
The language of Joyal and Lurie’s \(\infty\)-categories has now become an indispensable tool in homotopy theory. However, to encode desirable universal properties, just the theory of \(\infty\)-categories does not always suffice. Sometimes it is necessary to pass to other flavors, like equivariant, internal, or enriched \(\infty\)-categories. To illustrate, the universal property of \(G\)-spectra is best expressed in its incarnation as a \(G\)-equivariant \(\infty\)-category. In this talk, I will give an introduction to a framework that is designed to deal with different generalizations of \(\infty\)-categories via so-called \(\infty\)-equipments. These equipments give rise to category theories for their objects, incorporating concepts such as (co)limits and pointwise Kan extensions. There are suitable ambient \(\infty\)-equipments for enriched, internal, and fibered \(\infty\)-category theory (and combined flavors), and I would like to highlight some examples.
The tangle hypothesis is a variant of the cobordism hypothesis pertaining to tangles, that is, manifolds and cobordisms between them embedded in Euclidean spaces, and equipped with a suitable type of framing. This elaborate ensemble of geometric data can be assembled into a single higher categorical object, an \(E_m\)-monoidal \((\infty,n)\)-category Tang^{fr}{n,m} where \(n\) is the maximal dimension of the underlying cobordisms, and \(m\) is the fixed codimension of the Euclidean embeddings. The tangle hypothesis then asserts that Tang^{fr}{n,m} is freely generated as an \(E_m\)-monoidal \((\infty,n)\)-category with duals from a single object. A sketch of proof of the tangle hypothesis appears in Lurie’s 2009 text, and a conditional argument reducing the tangle hypothesis to another conjecture was later provided by Ayala and Francis, though a complete and formal proof has not yet appeared. In this talk I will describe work with Joost Nuiten which provides an infinitesimal version of this conjecture. More precisely, applying previous work on Quillen cohomology of higher categories we calculate the cotangent complex of Tang^{fr}{n,m} and show that, in a suitable sense, it is freely generated from a single generator. This can be considered as supporting evidence in the direction of the tangle hypothesis, but also reduces the tangle hypothesis to a statement on the level of \(E_m\)-monoidal \((n+1,n)\)-categories using a form of obstruction theory for higher categories.
The spectral Lie operad is the Koszul dual operad to the cocommutative cooperad in the category of spectra. A spectral Lie algebra is an algebra over the spectral Lie operad. M. Behrens and J. Kjaer constructed so-called Dyer-Lashof-Lie power operations acting on the mod-p homology groups of a spectral Lie algebra. However, they computed relations between these operations only for p=2. In my talk, I will explain how to compute the desired relations for any prime by using functor calculus in the category of simplicial restricted Lie algebras. The latter category might be thought of as an algebraic approximation of the category of spaces, and so, algebraic calculations may also be helpful in understanding of the topological Goodwillie spectral sequence.
Starting with the foundational work of Thomason, there has been an enormous amount of progress in chromatically localized algebraic K-theory. In this talk, I’ll survey some of this progress and explain how one can leverage these techniques to study chromatically localized TR and TC. The crucial input is the close relationship between TR and Bloch’s spectrum of curves on algebraic K-theory as first observed by Hesselholt. Part of this is joint work with Liam Keenan.
My talk concerns bordism rings of compact smooth manifolds equipped with a smooth action by a finite group. I will start by recalling classical results on the subject from the 60’s and 70’s, mostly due to Conner-Floyd, Boardman, Stong and Alexander. Afterwards I will discuss recent joint work with Stefan Schwede in which we prove an algebraic universal property for the collection of all bordism rings of manifolds with commuting involutions.
Hermitian K-theory is closely related to the classical theory of quadratic forms. We will give an overview of recent calculations of higher Hermitian K-groups of fields and rings of integers in number fields through motivic homotopy theory. The answers involve arithmetic data, while the calculational methods are rooted in homotopy theory. Joint work with Haakon Kolderup, Jonas Kylling, and Oliver Röndigs.
The theory of commutative (a.k.a. \(E_\infty\)) ring spectra is a natural higher algebraic analog of commutative algebra. However, there are notions from commutative algebra that do not generalize well to the spectral world, as well as notions that admit several reasonable generalizations. One example of the latter phenomenon is the group of units of a commutative ring \(R\). The most direct generalization is the units spectrum \(gl_1(R)\) of a commutative ring spectrum. However, for various purposes (and especially the formation of quotients of \(R\)), it is natural to consider a \(\mathbb{Z}\)-linear variant \(\mathbb{G}_m(R)\) called the spectrum of “strict units.” The spectrum \(\mathbb{G}_m(R)\) admits a natural delooping, namely the strict Picard spectrum, analogous to the Picard group of a commutative ring. In the first part of my talk, I will discuss the theory of spectra and commutative ring spectra, define (strict) units and Picard spectra, and introduce some key ingredients involved in their study. In the second part, I will present the computation of the strict Picard spectrum of the sphere spectrum and its completions, and the strict units of spherical group algebras of finitely generated abelian groups. The second computation is a joint work in preparation with Thomas Nikolaus and Allen Yuan.
In recent work, Guchuan Li, Sarah Peterson, and I have constructed models for \(C_{2}\)-equivariant analogues of the integral Brown–Gitler spectra. In this talk, I will start by introducing the classical Brown–Gitler spectra, and discussing some of their applications. After that, I will sketch the construction of the \(C_{2}\)-equivariant integral Brown–Gitler spectra, and discuss the applications we are beginning to study.
Classically the Thom isomorphism relates the cohomology of the Thom space of a vector bundle to the cohomology of its base. The Thom isomorphism for equivariant vector bundles fails in \(RO(G)\)-graded cohomology, even for \(G=C_2\). However, Costenoble–Waner developed an \(RO(\Pi)\)-graded equivariant cohomology theory, extending the usual representation grading \(RO(G)\) to representations of the equivariant fundamental groupoid, and they showed the Thom isomorphism holds in this extended grading. Costenoble recently computed the \(RO(\Pi)\)-graded cohomology of \(B_{C_2}U(1)\), the classifying space for complex \(C_2\)-line bundles. In this talk I will describe these different gradings and talk about work in progress computing the \(RO(\Pi)\)-graded cohomology of \(B_{C_2}O(1)\), the classifying space for real \(C_2\)-line bundles. This is joint work with Agnès Beaudry, Chloe Lewis, Sabrina Pauli and Elizabeth Tatum.
Inspired by the work of Lurie and others, Gepner—Meier define families of equivariant cohomology theories based on oriented elliptic curves. By construction, these equivariant elliptic cohomologies are multiplicative, but only in a naïve equivariant sense—there is no obvious construction of norm maps on these theories. In this talk, I will describe how to use a moduli interpretation of the geometric fixed points of these equivariant theories due to Gepner—Meier, to construct what we call “geometric norms”. Some applications of these geometric norms will also be discussed. This is joint work-in-progress with William Balderrama and Sil Linskens.
Several structural questions have emerged at least twice in topology: once in chromatic homotopy theory and once in equivariant topology (completions and localization, fracture squares, Balmer spectra, support, telescope conjecture, sheaves, filtrations, ….). In the chromatic world they arise in hard-core form, and in equivariant topology they reach a benign algebraic manifestation in the rational case. My talk is from this gentler world. The overall project is to build an algebraic model for rational \(G\)-equivariant cohomology theories for all compact Lie groups \(G\), and when \(G\) is small or abelian this has been done. In general, the model is expected to take the form of a category of sheaves of modules over a sheaf of rings over the space of closed subgroups of \(G\). The talk will focus on structural features of the expected model for general \(G\) such as those above, and feature recent joint work with Balchin and Barthel.
In this talk, I will explain how to compute the \(C_3\)-equivariant relative Adams spectral sequence for the Borelification of \(tmf(2)\).This yields an entirely algebraic computation of the 3-local homotopy groups of \(tmf\). The final answer is well-known of course – the novelty here is that the rASS is completely determined by it \(E_1\)-page as a cochain complex of Mackey functors. Explicitly, the input consists of the Hopf algebroid structure on \(\mathbb{F}_3 \otimes_{tmf(2)}\mathbb{F}_3\) modulo transfer, which is deduced from the structure maps on the equivariant dual Steenrod algebra, as well as the knowledge of the homotopy group of the underlying \(tmf(2)\) along with the \(C_3-\)action. Then we construct a bifiltration on \(tmf(2)\) and use synthetic arguments to deduce the Adams differentials from the associated square of spectral sequences. The rASS degenerates on \(E_{12}\) for tridegree reasons and stabilizes to a periodic pattern that essentially lies within a band of slope 1/4. This is joint work with Jeremy Hahn, Andrew Senger, and Foling Zou.
I will give a leisurely overview of parametrized and higher semiadditivity. In particular I will motivate this concept by giving a variety of examples. As one such example, I will explain how it gives a conceptual interpretation of definitions in (globally) equivariant algebra and homotopy theory. I will then finish by discussing the close connection between generalized semiadditivity and the construction of transfer maps.
Tensor-triangular geometry is a lens in which one thinks of symmetric monoidal triangulated categories as categorified commutative rings. This line of thinking leads us to construct an object analogous to the Zariski spectrum, and to prove general results for these categories with respect to properties of this spectrum. One such result is that of stratification, which allows us to classify all localizing ideals of the category via arbitrary subsets of the spectrum. In the first part of this talk I will review the general theory of tensor-triangular geometry, while in the second part I will show how these results can be applied to the category of rational \(G\)-equivariant spectra for \(G\) a profinite group. In particular, we will see that it is possible to fully resolve when we have stratification based on some surprising pointset topology. This is joint work with David Barnes and Tobias Barthel.
Cut and paste or SK groups of manifolds are formed by quotienting the monoid of manifolds under disjoint union by the relation that two manifolds are equivalent if I can cut one up into pieces and glue them back together to form the other manifold. Cobordism cut and paste groups are formed by moreover quotienting by the equivalence relation of cobordism. We categorify these classical groups to spectra and lift two canonical homomorphisms of groups to maps of spectra. This is joint work with Mona Merling, Laura Murray, Carmen Rovi and Julia Semikina.
There are many different notions of "being algebraic" used in stable homotopy theory. The relationships between those turn out to be unexpectedly subtle. We will explain the different ways in which a model category of interest can be algebraic, explore the different implications between them and illustrate those with plenty of examples. (This is joint work with Jocelyne Ishak and Jordan Williamson.)
In this talk I will present a new, simpler proof of the nilpotence and periodicity theorems of Devinatz-Hopkins-Smith. The key inputs in this proof are Nishida's nilpotence theorem and an understanding of vanishing lines in synthetic spectra.
In the first part of the talk I will give an introduction to \(\infty\)-operads from the perspective of symmetric monoidal \(\infty\)-categories, using the notion of equifibered maps developed in recent joint work with Shaul Barkan. For those unfamiliar with \(\infty\)-operads this should also serve as a gentle introduction to \(\infty\)-operads, albeit from an unusual perspective. This approach to \(\infty\)-operads allows for an easy generalisation to "\(\infty\)-properads", where operation have multiple inputs and multiple outputs. In the second part of the talk I will describe the equivalence of our \(\infty\)-properads to existing models. I will also explain joint work in progress that identifies modular \(\infty\)-operads with \(\infty\)-properads with duals, and in particular proves the 1-dimensional cobordism hypothesis with singularities.
Given a map \(f\colon X \to \mathrm{Pic}(Sp)\) of \(E_n\)-spaces, the associated Thom spectrum \(M(f)\) is a comodule over the suspension spectrum of \(X\) via the Thom diagonal as well as an \(E_n\)-ring spectrum. I will discuss the compatibility between these two structures for the \(\infty\)-categorical incarnation of Thom spectra introduced by Ando–Blumberg–Gepner–Hopkins–Rezk, some of its consequences and how it fits into the bigger picture of coalgebraic structures in \(\infty\)-categories. The key step will be realizing f as an \(X\)-comodule algebra in the \(\infty\)-category of spaces over \(\mathrm{Pic}(Sp)\) in an appropriate sense, for which I will present a simple description of the \(\infty\)-category of such objects.
Classifying objects in triangulated categories up to isomorphism is generally far too hard, so we often seek to classify objects up to some operations, saying that two objects are equivalent if they can built from each other using these operations. Operations one might take here are coproducts and cones, which gives rise to the idea of thick and localizing subcategories. Instead, one can take operations from purity, which has a long history in algebra, model theory, and representation theory, leading to the notion of definable subcategories. These are closely related to many interesting questions in homotopy theory and beyond. I will explain how one can generalise and axiomatise various aspects of this algebraic theory so that it applies in triangulated categories, and then provide some applications to tensor-triangular geometry and representation theory. This is based on joint work with Isaac Bird, some of which can be found in arXiv:2202.08113.
In this talk I will discuss and prove a recognition principle for iterated suspensions as coalgebras over the little disks operad. This is based on joint work with Oisín Flynn-Connolly and José Moreno-Fernández.
I want to report on a calculation of the so-called stable part of the cohomology of symplectic groups over the integers, in particular at the prime 2. The approach is via the group completion theorem, which relates this stable part to sympletic K-groups of the integers. The latter has recently seen advances in the case of general number rings and I will explain how these can be brought to bear. This is joint with M.Land and T.Nikolaus.
A smooth projective variety Z is said to be Calabi-Yau if its canonical bundle is trivial. I will discuss recent joint work with Taelman, in which we use derived algebraic geometry to study how Calabi-Yau varieties in characteristic p deform. More precisely, we show that if Z has degenerating Hodge–de Rham spectral sequence and torsion-free crystalline cohomology, then its mixed chracteristic deformations are unobstructed; this is an analogue of the classical BTT theorem in characteristic zero. If Z is ordinary, we show that it moreover admits a canonical lift to characteristic zero; this extends classical Serre-Tate theory. Our work generalises results of Achinger–Zdanowicz, Bogomolov-Tian-Todorov, Deligne–Nygaard, Ekedahl–Shepherd-Barron, Schröer, Serre–Tate, and Ward.
The construction of coherent multiplicative structures on quotients in higher algebra is delicate and often impossible. It has for instance been known since the 70s that the mod p Moore spectra S/p do not admit an E_1 multiplication. Rather surprisingly, Robert Burklund showed in a recent preprint that a wide class of quotients do admit coherent multiplications. To name a few examples, he proves that S/8 admits an E_1-ring structure and S/p^{n+1} admits an E_n-ring structure for p an odd prime. The proof of these results makes use of an obstruction theory carried out in the category of synthetic spectra. I will explain this proof in my talk.
In this talk I will introduce \(G\)-global homotopy theory as a synthesis of classical \(G\)-equivariant homotopy theory on the one hand and global homotopy theory in the sense of Schwede on the other hand. I will then give an overview of several applications of the \(G\)-global theory to the study of purely equivariant or global questions, in particular regarding the corresponding notions of "coherently commutative monoids." Part of this is joint work in progress with Bastiaan Cnossen and Sil Linskens as well as with Michael Stahlhauer.
In this semester, we combined a reading seminar on Condensed Mathematics (after Clausen and Scholze) with invited talks by external speakers.
A characteristic property of compact support cohomology is the long exact sequene which connects the compact support cohomology groups of a space, an open subspace and its complement. Given an arbitrary invariant of, say, algebraic varieties, taking values in a stable infinity category C, one can wonder when it makes sense to define a "compact support" version of this invariant, such that this long exact sequence exists by construction. In this talk, I give an answer in terms of an equivalence of categories of C-valued sheaves on certain sites of algebraic varieties. I will discuss some applications of this result, and, if time permits, speculate about some related things that I haven't proven yet.
The aim of this talk is to explain a systematic formalism to construct and manipulate Thom spectra in global equivariant homotopy theory. The upshot is a colimit preserving symmetric monoidal global Thom spectrum functor from the infinity-category of global spaces over \(BOP\) to the infinity-category of global spectra. Here \(BOP\) is a particular globally-equivariant refinement of the space \(\mathbb{Z} \times BO\), which simultaneously represents equivariant \(K\)-theory for all compact Lie groups. I plan to give two applications of the formalism. Firstly, a specific and much studied morphism \(mU \to MU\) between two prominent equivariant forms of the complex bordism spectrum is a localization, in the infinity-category of commutative global ring spectra, at the ‘inverse Thom classes’. Secondly, by joint work with Gepner and Nikolaus, the infinity-category of global spectra can be describe as a pushout of parameterized symmetric monoidal infinity categories along the global Thom spectrum functor.
Global homotopy theory is the study of equivariant objects which exist uniformly and compatibly for all compact Lie groups in a certain family, and which exhibit extra functoriality. In this talk I will present new infty-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limit to formalize the idea that a global object is a collection of \(G\)-objects, one for each compact Lie group \(G\), which are compatible with the restriction-inflation functors. This is joint work with Sil Linskens and Denis Nardin.
In joint work with Nima Rasekh and Martina Rovelli, we are developing a new approach to limits in an \((\infty,2)\)-category, by defining them as terminal objects in the corresponding double \((\infty,1)\)-category of cones. To verify that this gives the correct universal property, we need to compare our definition to the established definition of limits in an \((\infty,2)\)-category seen as a category enriched in \((\infty,1)\)-categories. The difficulty of this comparison arises in the fact that there is no direct Quillen equivalence between the \((\infty,2)\)-categorical models of categories enriched in complete Segal spaces and 2-fold complete Segal spaces. As a first step towards the comparison, we construct a direct Quillen equivalence between the above mentioned models. This construction is not specific to the case n=2 and we in fact obtain a direct Quillen equivalence between categories enriched in complete Segal Theta-n-spaces and complete Segal objects in Theta-n-spaces, which both model \((\infty,n+1)\)-categories. In particular, this construction generalizes the homotopy coherent nerve from Kan-enriched categories to quasi-categories.
The classical Hochschild-Kostant-Rosenberg theorem identifies Hochschild homology of a commutative ring which is smooth over the base field with its de Rham complex. In this talk, we provide a generalisation to topological Hochschild homology of commutative ring spectra, replacing the de Rham complex by an "\(\eta\)-deformed de Rham complex" which incorporates the \(E_{\infty}\)-structure. (Joint with Thomas Nikolaus).
Joyal proved that symmetric sequences in sets (or “species”) can be identified with certain endofunctors of Set, namely the “analytic" functors. Under this identification, the composition product on symmetric sequences corresponds to composition of endofunctors, and this allows us to identify operads in Set with certain “analytic” monads. Moreover, the monad corresponding to an operad O is precisely the monad for free O-algebras in Set. In this talk I will explain how to obtain an analogous identification for infinity-operads: assigning to an infinity-operad O (in Lurie’s sense) the monad for free O-algebras in spaces identifies infinity-operads with analytic monads. This builds on previous work with Gepner and Kock where we developed the theory of analytic monads in the infinity-categorical setting.
Log geometry is a variant of algebraic geometry in which mildly singular varieties can be treated as if they were smooth. Rognes has extended the definition of Hochschild homology to allow for log rings - the affine schemes of log geometry - as input. As the Hochschild-Kostant-Rosenberg theorem identifies the Hochschild homology of smooth rings with its de Rham complex, it is natural to ask whether Hochschild homology of log smooth log rings are related to the log de Rham complex. I will give a reformulation of Rognes' definition that will allow us to tackle this problem in much the same way as for ordinary Hochschild homology. Parts of the talk will be based on joint work with Binda-Park-Østvær.
Lurie’s straightening theorem is one of the cornerstones of \(\infty\)-category theory. It provides an equivalence between functors from an \(\infty\)-category \(C\) valued in \(\mathrm{Cat}_{\infty}\), the \(\infty\)-category of small \(\infty\)-categories, and particular kinds of \(\infty\)-categories fibered over C called cocartesian fibrations. This equivalence gives an efficient way of writing down \(\mathrm{Cat}_{\infty}\)-valued functors via these fibered \(\infty\)-categories, which are otherwise hard to write down directly because of the coherence issues one then has to deal with. In this talk we will see a handful of applications of this straightening construction, and we will give an outline of a new proof of the straightening theorem (this is joint work with Fabian Hebestreit and Gijs Heuts).
In this semester, we had reading seminars on synthetic spectra and buildings.
In this semester, we primarily had talks by group members about results that are useful for a broader audience.
In this semester, we primarily had talks by group members about their own work and about "Basic Notions" relevant for Algebraic Topology and related subjects.
In this semester, we primarily had talks by group members about their own work and about "Basic Notions" relevant for Algebraic Topology and related subjects.
In this semester, we primarily had talks by group members about their own work and about "Basic Notions" relevant for Algebraic Topology and related subjects.
I will describe a model for configuration spaces (or, more generally, for the configuration category) of points on a manifold which is given in terms of a triangulation of the manifold. This gives rise to combinatorial descriptions for embedding calculus and associated spectral sequences, extending Sinha’s description when the manifold in question is the interval. Joint work with Pascal Lambrechts and Paul-Arnaud Songhafouou.
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