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We explore the $C_2$-equivariant spectra $Tmf_1(3)$ and $TMF_1(3)$. In particular, we compute their $C_2$-equivariant Picard groups and the $C_2$-equivariant Anderson dual of $Tmf_1(3)$. This implies corresponding results for the fixed point spectra $TMF_0(3)$ and $Tmf_0(3)$. Furthermore, we prove a Real Landweber exact functor theorem., Final version to appear in AGT. 51 pages

Here we introduce reflection positive doubles, a general framework for reflection positivity, covering a wide variety of systems in statistical physics and quantum field theory. These systems may be bosonic, fermionic, or parafermionic in nature. Within the framework of reflection positive doubles, we give necessary and sufficient conditions for reflection positivity. We use a reflection-invariant cone to implement our construction. Our characterization allows for a direct interpretation in terms of coupling constants, making it easy to check in concrete situations. We illustrate our methods with numerous examples., Comment: 53 pages, 3 figures

We study the relation between convergence of partition functions (seen as general Dirichlet series) and convergence of spectra and their multiplicities. We describe applications to convergence in physical models, e.g., related to topology change and averaging in cosmology.

When the base ring is not a field, power reductivity of a group scheme is a basic notion, intimately tied with finite generation of subrings of invariants. Geometric reductivity is weaker and less pertinent in this context. We give a survey of these properties and their connections., 6 pages; Dedicated to the memory of T.A. Springer

We study the algebraic Brauer classes on open del Pezzo surfaces of degree 4. I.e., on the complements of geometrically irreducible hyperplane sections of del Pezzo surfaces of degree 4. We show that the 2-torsion part is generated by classes of two different types. Moreover, there are two types of 4-torsion classes. For each type, we discuss methods for the evaluation of such a class at a rational point over a p -adic field.

We construct Quillen equivalent semi-model structures on the categories of dg-Lie algebroids and $L_\infty$-algebroids over a commutative dg-algebra in characteristic zero. This allows one to apply the usual methods of homotopical algebra to dg-Lie algebroids: for example, every Lie algebroid can be resolved by dg-Lie algebroids that arise from dg-Lie algebras, i.e. that have a null-homotopic anchor map. As an application, we show how Lie algebroid cohomology is represented by an object in the homotopy category of dg-Lie algebroids., Comment: 35 pages

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichm\"uller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichm\"uller cusp forms on \overline{M}_g and the middle cohomology of symplectic local systems on M_g. In genus 3, we make this explicit in a large number of cases., Comment: 34 pages. In an appendix (joint work of G. Farkas, R. Pandharipande, and the second author) it is shown that Teichm\"uller modular forms extend to \overline{M}_g for g at least 3. Other minor changes

We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to anabelian geometry., Comment: 16 pages; replaces the part of arXiv:1009.0736 dealing with dynamical systems; results related to L-series have been moved to a separate preprint by the authors and Bart de Smit, of which the preliminary section (fixing notations) overlaps with that of the current paper

This paper provides an extensive study of the homotopy theory of types of algebras with units, like unital associative algebras or unital commutative algebras for instance. To this purpose, we endow the Koszul dual category of curved coalgebras, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of unital algebras. To prove such a result, we use recent methods based on presentable categories. This allows us to describe the homotopy properties of unital algebras in a simpler and richer way. Moreover, we endow the various model categories with several enrichments which induce suitable models for the mapping spaces and describe the formal deformations of morphisms of algebras., Comment: 46 pages

Consider a closed coisotropic submanifold $N$ of a symplectic manifold $(M,\omega)$ and a Hamiltonian diffeomorphism $\phi$ on $M$. The main result of this article states that $\phi$ has at least the cup-length of $N$ many leafwise fixed points w.r.t. $N$, provided that it is the time-1-map of a global Hamiltonian flow whose restriction to $N$ stays $C^0$-close to the inclusion $N\to M$. If $(\phi,N)$ is suitably nondegenerate then the number of these points is bounded below by the sum of the Betti-numbers of $N$. The nondegeneracy condition is generically satisfied. This appears to be the first leafwise fixed point result in which neither $\phi\big|_N$ is assumed to be $C^1$-close to the inclusion $N\to M$, nor $N$ to be of contact type or regular (i.e., "fibering"). It is optimal in the sense that the $C^0$-condition on $\phi$ cannot be replaced by the assumption that $\phi$ is Hofer-small., Comment: 37 pages, accepted by IMRN. I have removed the part on local coisotropic Floer homology and made this a separate article

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold $X$. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of $X$ defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of $X$ using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of $X$. After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds using a localization formula. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions., 28 pages. Published version

Consider zero-dimensional Donaldson-Thomas invariants of a toric threefold or toric Calabi-Yau fourfold. In the second case, invariants can be defined using a tautological insertion. In both cases, the generating series can be expressed in terms of the MacMahon function. In the first case, this follows from a theorem of Maulik-Nekrasov-Okounkov-Pandharipande. In the second case, this follows from a conjecture of the authors and a (more general $K$-theoretic) conjecture of Nekrasov. In this paper, we consider formal analogues of these invariants in any dimension $d \not \equiv 2 \ \mathrm{mod} \, 4$. The direct analogues of the above-mentioned conjectures fail in general when $d>4$, showing that dimensions 3 and 4 are special. Surprisingly, after appropriate specialization of the equivariant parameters, the conjectures seem to hold in all dimensions., 18 pages. Published version

Finite hypergeometric functions are functions of a finite field \({\mathbb F}_q\) to \({\mathbb C}\). They arise as Fourier expansions of certain twisted exponential sums and were introduced independently by John Greene and Nick Katz in the 1980s. They have many properties in common with their analytic counterparts, the hypergeometric functions. One restriction in the definition of finite hypergeometric functions is that the hypergeometric parameters must be rational numbers whose denominators divide q − 1. In this note we use the symmetry in the hypergeometric parameters and an extension of the exponential sums to circumvent this problem as much as possible.

We consider the localization of the $\infty$-category of spaces at the $v_n$-periodic equivalences, the case $n=0$ being rational homotopy theory. We prove that this localization is for $n\geq 1$ equivalent to algebras over a certain monad on the $\infty$-category of $T(n)$-local spectra. This monad is built from the Bousfield--Kuhn functor., 8 pages

We present measure theoretic rigidity for graphs of first Betti number b>1 in terms of measures on the boundary of a 2b-regular tree, that we make explicit in terms of the edge-adjacency and closed-walk structure of the graph. We prove that edge-reconstruction of the entire graph is equivalent to that of the "closed walk lengths"., 9 pages

Moduli spaces of stable sheaves on smooth projective surfaces are in general singular. Nonetheless, they carry a virtual class, which -- in analogy with the classical case of Hilbert schemes of points -- can be used to define intersection numbers, such as virtual Euler characteristics, Verlinde numbers, and Segre numbers. We survey a set of recent conjectures by the authors for these numbers with applications to Vafa-Witten theory, $K$-theoretic S-duality, a rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, and a virtual Segre-Verlinde correspondence. A key role is played by Mochizuki's formula for descendent Donaldson invariants., 40 pages. Published version

This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete -principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data., The authors are supported by Spanish National Research Projects MTM2016–79400–P, MTM2015-72876-EXP and SEV2015-0554. R. Casals is supported by the NSF grant DMS-1841913 and a BBVA Research Fellowship. ́A. del Pino is supported by the NWO Vici Grant no. 639.033.312 of Marius Crainic.

Recently, Nekrasov discovered a new "genus" for Hilbert schemes of points on $\mathbb{C}^4$. We conjecture a DT/PT correspondence for Nekrasov genera for toric Calabi-Yau 4-folds. We verify our conjecture in several cases using a vertex formalism. Taking a certain limit of the equivariant parameters, we recover the cohomological DT/PT correspondence for toric Calabi-Yau 4-folds recently conjectured by the first two authors. Another limit gives a dimensional reduction to the $K$-theoretic DT/PT correspondence for toric 3-folds conjectured by Nekrasov-Okounkov. As an application of our techniques, we find a conjectural formula for the generating series of $K$-theoretic stable pair invariants of the local resolved conifold. Upon dimensional reduction to the resolved conifold, we recover a formula which was recently proved by Kononov-Okounkov-Osinenko., Comment: 47 pages, published version

The Eichler-Shimura isomorphism relates holomorphic modular cusps forms of positive even integral weight to cohomology classes. The Haberland formula uses the cup product to give a cohomological formulation of the Petersson scalar product. In this paper we extend Haberland's formula to modular cusp forms of positive real weight. This relation is based on the cup product of an Eichler cocycle and a Knopp cocycle. We may also consider the cup product of two Eichler cocycles. In the classical situation this cup product is almost always zero. However we show evidence that for real weights this cup product may very well be non-trivial. We approach the question whether the cup product is a non-trivial coinvariant by duality with a space of entire modular forms. The cup product yields a bilinear map over C from pairs of holomorphic modular forms (not necessarily of the same weight, one of them may have large growth at the cusps) to coinvariants in infinite-dimensional modules. To investigate whether this bilinear map is non-trivial we test the result against entire modular forms of a suitable weight. Under some conditions on the weights, this leads to an explicit triple integral, which can be investigated numerically, thus providing evidence that the cup product is non-trivial at least in some situations.

For more than three decades, fierce debates raged both in private letters and across public spaces over a formula expressed in 1864 by the French geometer Michel Chasles. Proofs and refutations thereof abounded, to no avail: the formula was too useful to be abandoned by its defenders, too elusive to be made rigorous for its detractors. The disputes over Chasles’s formula would not be solved by a definitive proof or rebuttal; rather, the core epistemic issues at stake shifted from generality to rigor and from truth to geometrical significance. This essay tracks the main lines of circulation of Chasles’s formula and shows how the disputes to which it gave rise embody conflicting mathematical selves—that is to say, different normative accounts of what being a mathematician entails. This perspective allows for a renewed understanding of what historians have described as the conflicted rise of modernism in mathematics and a firmer rooting of it within broader late nineteenth-century cultural trends.

I reply to recent arguments by Peter Barker & Tofigh Heidarzadeh, Arun Bala, and F. Jamil Ragep claiming that certain aspects Copernicus’s astronomical models where influenced by late Islamic authors connected with the Maragha school. In particular, I argue that: the deleted passage in De revolutionibus that allegedly references unspecified previous authors on the Tusi couple actually refers to a simple harmonic motion, and not the Tusi couple; the arguments based on lettering and other conventions used in Copernicus’s figure for the Tusi couple have no evidentiary merit whatever; alleged indications that Nicole Oresme was aware of the Tusi couple are much more naturally explained on other grounds; plausibility considerations regarding the status of Arabic astronomy and norms regarding novelty claims weight against the influence thesis, not for it.

We show that some natural problems that are XNLP-hard (hence $$\textrm{W}[t]$$ -hard for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds, Minimum Maximum Outdegree, and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set. Our hardness claims beat existing results. The FPT algorithms use a new parameter “treebreadth”, associated to a weighted tree partition, as well as DP and ILP.

Mathematicians used to be highly invested in the study of the history of their own field, but their voice in historiographical discussions has diminished in influence in the past century. One prominent narrative paints this as a justified fall from grace: mathematicians wedded to present mathematical values looked at the past with prejudiced eyes, whereas a new generation of historians were better able to appreciate the past proper, in its own terms. But the best internalist mathematical historiography of old needed no such external corrective. It was already committed to avoiding presentism and anachronism, for reasons that were not in opposition to mathematical values but rather derived directly from a positive vision of the role that history could play in the mathematical community. In this vision, a historical understanding of how a field developed is a proxy for first-hand research experience in that field. It follows that it is essential for historical accounts to thoroughly convey the scope and limitations of alternative conceptions and approaches, including dead-end developments, since this is precisely what sets the critical knowledge gained by first-hand research experience apart from the doctrinal knowledge gained merely from a textbook. Hence, from this point of view, presentist historiography is not the natural outlook of the mathematician, but rather a direct antithesis of the mathematician’s most fundamental reason for studying history. To study past mathematics precisely as it appeared to active researchers at the time is not foreign to the mathematician, but a direct corollary of the mathematician’s core conviction that only those with first-hand research experience in a field of mathematics truly understand it. A sophisticated internalist historiography derived from these ideals was articulated to a greater extent in the past than is commonly recognized today. By going back to its roots, the mathematician’s historiography could revive some of the virtues that have been neglected in recent years.

Analyses in Greek geometry are traditionally seen as heuristic devices. However, many occurrences of analysis in formal treatises are difficult to justify in such terms. I show that Greek analysies of geometrics can also serve formal mathematical purposes, which are arguably incomplete without which their associated syntheses are arguably incomplete. Firstly, when the solution of a problem is preceded by an analysis, the analysis latter proves rigorously that there are no other solutions to the problem than those offered in the synthesis. Secondly, whenever some construction assumption beyond ruler and compass is made, the problem is not only solvable by that assumption but is in fact equivalent to that assumption in a rigorous sense. Dans la géométrie grecque, l’analyse est traditionnellement vue comme une technique heuristique. Cependant, de nombreuses analyses dans les traités mathématiques anciens sont difficiles à expliquer en ces termes. Cet article vise à montrer que l’analyse géométrique grecque peut aussi servir des enjeux mathématiques déterminés, qui, sans son apport, rendraient la synthèse incomplète. D’abord, lorsque la solution d’un problème est précédée par une analyse, celle-ci démontre rigoureusement qu’il n’y a pas d’autres solutions au problème que celles offertes dans la synthèse. Ensuite, chaque fois qu’on est en présence d’une hypothèse de construction qui dépasse l’utilisation de la règle et du compas, l’analyse démontre que le problème n’est pas seulement résolvable à partir de cette assomption, mais qu’il est en fait rigoureusement équivalent à elle.

In this survey we explain a stratification of a Hilbert modular variety $\mathscr{M}_{E}$ in characteristic $p \gt 0$ attached to a totally real number field $E$. This stratification refines the stratification of $\mathscr{M}_{E}$ by Lie type, and has the property that many strata are central leaves in $\mathscr{M}_{E}$, called distinguished central leaves. ¶ In the case when the totally real field $E$ is unramified above $p$, this stratification reduces to the stratification of $\mathscr{M}_{E}$ by $\alpha$-type first introduced by Goren and Oort and studied by Yu, and coincides with the EO stratification on $\mathscr{M}_{E}$. Moreover it is known that every non-supersingular $\alpha$-stratum of $\mathscr{M}_{E}$ is irreducible. To treat the general case where $E$ may be ramified above $p$, a key ingredient is the notion of congruity, a $p$-adic numerical invariant for abelian varieties with real multiplication by $\mathcal{O}_E$ in characteristic $p$. For every Lie stratum $\mathcal{N}_{\underline{e}}$ on $\mathscr{M}_{E}$, this new invariant defines a finite number of locally closed subsets $\mathcal{Q}_{\underline{c}}(\mathcal{N}_{\underline{e}})$, and $\mathcal{N}_{\underline{e}}$ is the disjoint union of these Lie-congruity strata $\mathcal{Q}_{\underline{c}}(\mathcal{N}_{\underline{e}})$ in $\mathcal{N}_{\underline{e}}$. ¶ The incidence relation between the Lie-congruity strata enables one to show that the prime-to-$p$ Hecke correspondences operate transitively on the set of all irreducible components of any distinguished central leaf in $\mathscr{M}_{E}$, see Theorems 7.1, 8.1 and 9.1. The Hecke transitivity implies, according to the method of prime-to-$p$ monodromy of Hecke invariant subvarieties, that every non-supersingular distinguished central leaf in a Hilbert modular variety $\mathscr{M}_{E}$ is irreducible. The last irreducibility result is a key ingredient of the proof the Hecke orbit conjecture for Siegel modular varieties.

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this $v_n$-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T(n)-local spectra. We also compare it to the homotopy theory of commutative coalgebras in T(n)-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity., Final version to appear in Annals of Mathematics. Added a short section on the Whitehead bracket

We consider local-global principles for rational points on varieties, in particular torsors, over one-variable function fields over complete discretely valued fields. There are several notions of such principles, arising either from the valuation theory of the function field, or from the geometry of a regular model of the function field. Our results compare the corresponding obstructions, proving in particular that a local-global principle with respect to valuations implies a local-global principle with respect to a sufficiently fine regular model.

Galileo claimed the phases of Venus as his own discovery. His priority claim hinges on the authenticity of his December 1610 record of observations ostensibly dating back to October. Scholars have argued that Galileo’s report must be truthful, since it is too accurate and detailed to have been forged after the fact. However, I show that Galileo could easily have forged these data by means of a basic simulation with a physical sphere rather than actual observations. This calls into question the received view that Galileo must have known about the phases of Venus already before it was brought to his attention by others.

There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree of a finite harmonic morphism from a refinement of G to a tree, as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and sgon(G) are NP-hard by a reduction from the maximum independent set problem and the vertex cover problem, respectively. Both constructions show that computing gonality is moreover APX-hard., The previous version only dealt with hardness of the divisorial gonality. The current version also shows hardness of stable gonality and discusses the relation between the two graph parameters

A partial combinatory algebra (PCA) is a set equipped with a partial binary operation that models a notion of computability. This paper studies a generalization of PCAs, introduced by W. Stekelenburg, where a PCA is not a set but an object in a given regular category. The corresponding class of categories of assemblies is closed both under taking small products and under slicing, which is to be contrasted with the situation for ordinary PCAs. We describe these two constructions explicitly at the level of PCAs, allowing us to compute a number of examples of products and slices of PCAs. Moreover, we show how PCAs can be transported along regular functors, enabling us to compare PCAs constructed over different base categories. Via a Grothendieck construction, this leads to a (2-)category whose objects are PCAs and whose arrows are generalized applicative morphisms. This category has small products, which correspond to the small products of categories of assemblies, and it has finite coproducts in a weak sense. Finally, we give a criterion when a functor between categories of assemblies that is induced by an applicative morphism has a right adjoint, by generalizing the notion of computational density.

We consider two preorder-enriched categories of ordered PCAs: OPCA, where the arrows are functional morphisms, and PCA, where the arrows are applicative morphisms. We show that OPCA has small products and finite biproducts, and that PCA has finite coproducts, all in a suitable 2-categorical sense. On the other hand, PCA lacks all nontrivial binary products. We deduce from this that the pushout, over Set, of two nontrivial realizability toposes is never a realizability topos.

The moduli space of stable pairs on a local surface $X=K_S$ is in general non-compact. The action of $\mathbb{C}^*$ on the fibres of $X$ induces an action on the moduli space and the stable pair invariants of $X$ are defined by the virtual localization formula. We study the contribution to these invariants of stable pairs (scheme theoretically) supported in the zero section $S \subset X$. Sometimes there are no other contributions, e.g. when the curve class $\beta$ is irreducible. We relate these surface stable pair invariants to the Poincar\'e invariants of D\"urr-Kabanov-Okonek. The latter are equal to the Seiberg-Witten invariants of $S$ by work of D\"urr-Kabanov-Okonek and Chang-Kiem. We give two applications of our result. (1) For irreducible curve classes the GW/PT correspondence for $X = K_S$ implies Taubes' GW/SW correspondence for $S$. (2) When $p_g(S) = 0$, the difference of surface stable pair invariants in class $\beta$ and $K_S - \beta$ is a universal topological expression., Comment: 25 pages. Published version. Content the same. Exposition completely changed following referee's suggestions

A classical theorem of Honda and Tate asserts that for every Weil q-number , there exists an abelian variety over the nite eld Fq, unique up to Fq-isogeny, whose q- standard proof (of the existence part in the Honda-Weil theorem) uses the the fact that for a given CM eld L and a given CM type for L, there exists a CM abelian variety with CM type (L; ) over a eld of characteristic 0. The usual proof of the last statement uses complex uniformization of (the set of C-points of) abelian varieties over C. In this short note we provide an algebraic proof of the existence of a CM abelian variety over an integral domain of characteristic 0 with a given CM type, resulting in an algebraic proof of the existence part of the Honda-Tate theorem which does not use complex uniformization. Dedicated to the memory of Taira Honda.

We conjecture a formula for the generating function of virtual $\chi_y$-genera of moduli spaces of rank 2 sheaves on arbitrary surfaces with holomorphic 2-form. Specializing the conjecture to minimal surfaces of general type and to virtual Euler characteristics, we recover (part of) a formula of C. Vafa and E. Witten. These virtual $\chi_y$-genera can be written in terms of descendent Donaldson invariants. Using T. Mochizuki's formula, the latter can be expressed in terms of Seiberg-Witten invariants and certain explicit integrals over Hilbert schemes of points. These integrals are governed by seven universal functions, which are determined by their values on $\mathbb{P}^2$ and $\mathbb{P}^1 \times \mathbb{P}^1$. Using localization we calculate these functions up to some order, which allows us to check our conjecture in many cases. In an appendix by H. Nakajima and the first named author, the virtual Euler characteristic specialization of our conjecture is extended to include $\mu$-classes, thereby interpolating between Vafa-Witten's formula and Witten's conjecture for Donaldson invariants., Comment: 44 pages. Published version. Appendix C by first named author and H. Nakajima