The BV formalism for L$_\infty$-algebras

The notions of a BV$_\infty$-morphism and a category of BV$_\infty$-algebras
are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms
is characterized as a certain subcategory of the category of
BV$_\infty$-algebras. This provides a Fourier-dual, BV alternative to the
standard characterization of the category of L$_\infty$-algebras as a
subcategory of the category of dg cocommutative coalgebras or formal pointed dg
manifolds. The functor assigning to a BV$_\infty$-algebra the
L$_\infty$-algebra given by higher derived brackets is also shown to have a
left adjoint.

Graviton-Photon Scattering

We use that the gravitational Compton scattering factorizes on the Abelian QED amplitudes to evaluate various gravitational Comp- ton processes. We examine both the QED and gravitational Compton scattering from a massive spin-1 system by the use of helicity am- plitudes. In the case of gravitational Compton scattering we show how the massless limit can be used to evaluate the cross-section for graviton-photon scattering and discuss the difference between photon interactions and the zero mass spin-1 limit. We show that the forward scattering cross-section for graviton photo-production has a very pecu- liar behaviour, differing from the standard Thomson and Rutherford cross-sections for a Coulomb-like potential.

Three dimensional Sklyanin algebras and Groebner bases

We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field K with 3 generators x,y,z given by 3 relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin, Odesskii, and Artin, Tate, Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three cubes of p, q and r are distinct, then S
is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 variables.
It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van den Bergh.

Geometry of Morphogenesis

On Koszulity for operads of Conformal Field Theory

We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Gr ̈bner bases theory for operads and an operadic analogue of the Priddy criterion. An example of deformation of an operad coming from the Hom structures is considered. In particular we study possible deformations of the Associative operad from the point of view of the confluence property. Only one deformation, the operad which governs the identity (α(ab))c = a(α(bc)) turns out to be confluent. We introduce a new Hom structure, namely Hom–Gelfand-Dorfman algebras and study their basic properties.

The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring R, let J1 and J2 be the following noncommutative birational partly defined involutions on the set M3 (R) of 3 × 3 matrices over R: J1 (M ) = M −1 (the usual matrix inverse) and J2 (M )jk = (Mkj )−1 (the transpose of the Hadamard inverse).
We prove the following surprising conjecture by Kontsevich (1996) saying that (J2 ◦ J1 )3 −1 is the identity map modulo the DiagL × DiagR action (D1 , D2 )(M ) = D1 M D2 of pairs of invertible diagonal matrices.
That is, we show that for each M in the domain where (J2 ◦J1 )3 is defined, there are invertible −1 diagonal 3 × 3 matrices D1 = D1 (M ) and D2 = D2 (M ) such that (J2 ◦ J1 )3 (M ) = D1 M D2.

Topos-theoretic background

This text, which will form the first chapter of my book in preparation "Lattices of theories", is a self-contained introduction to topos theory, geometric logic and the 'bridge' technique.

Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective

We establish, generalizing Di Nola-Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.

Quasi-exact-solvability of the $A_2$ elliptic model: Algebraic form, $sl(3)$ hidden algebra, polynomial eigenfunctions

Dimensional exactness of self-measures for random countable iterated function systems with overlaps.

We study projection measures for random countable (finite or infinite) conformal
iterated function systems with arbitrary overlaps. In this setting we extend Feng's and Hu's result from [6] about deterministic finite alphabet iterated function systems. We
prove, under a mild assumption offinite entropy, the dimensional exactness of the projections
of invariant measures from the shift space, and we give a formula for their dimension,
in the context of random infinite conformal iterated function systems with overlaps. There
exist numerous differences between our case and the finite deterministic case. We give then
applications and concrete estimates for pointwise dimensions of measures, with respect to
various classes of random countable IFS with overlaps. Namely, we study several types of
randomized systems related to Kahane-Salem sets; also, a
random system related to a statistical problem of Sinai; and randomized infinite IFS in the
plane for which the number of overlaps is uniformly bounded from above.

Principe de fonctorialité et transformations de Fourier non linéaires : proposition de définitions et esquisse d'une possible (?) démonstration

Ce texte rassemble les notes écrites d'une série de quatre exposés donnés à l'IHES les 19 juin, 26 juin, 3 juillet et 8 juillet 2014.
Il introduit une nouvelle approche pour une éventuelle démonstration - encore à vérifier - du transfert automorphe de Langlands sur les corps globaux.
En attendant donc de vérifier soigneusement si cela marche ou bien non, d'abord dans le cas de GL(2) et des représentations de puissances symétriques de son dual. Le point le plus essentiel, sur lequel tout est fondé, est la propriété de stabilité du tore maximal par convolution (définie comme la transformée de Fourier de la multiplication point par point des fonctions) apparue dans la dernière partie de la partie III.

Extensions of flat functors and theories of presheaf type

We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos.

Cyclic theories

We describe a geometric theory classified by Connes-Consani's epicylic topos and two related theories respectively classified by the cyclic topos and by the topos $[{\mathbb
N}^{\ast}, \Set]$.

A Feynman integral via higher normal functions

We study the Feynman integral for the three-banana graph defined as the scalar two-point
self-energy at three-loop order. The Feynman integral is evaluated for all
identical internal masses in two space-time dimensions. Two calculations are given
for the Feynman integral; one based on an interpretation of the
integral as an inhomogeneous solution of a classical Picard-Fuchs
differential equation, and the other using arithmetic algebraic
geometry, motivic cohomology, and Eisenstein series. Both methods use
the rather special fact that the Feynman integral is a family of
regulator periods associated to a family of $K3$ surfaces.
We show that the integral is given by a sum of elliptic
trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the
regulator of a class in the motivic cohomology of the $K3$
family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman
integral is given by a critical value of the Hasse-Weil $L$-function
of the $K3$ surface. This result is shown to be a particular case of Deligne's
conjectures relating values of $L$-functions inside the critical strip
to periods.

Beltrami-Courant Differentials and $G_{\infty}$-algebras

Using the symmetry properties of two-dimensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it.
We conjecture that the generalized Maurer-Cartan equation for the corresponding $L_{\infty}$ subalgebra gives solutions to the Einstein equations.

2-CY-tilted algebras that are not Jacobian

Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials.
In another direction, we compute the fractionally Calabi-Yau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2.

Algebras of quasi-quaternion type

We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi quaternion type.
We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by Labardini to triangulations of closed surfaces with punctures on the other hand, hence it serves as a bridge between modular representation theory of finite groups and cluster algebras.

SL(2,Z)-invariance and D-instanton contributions to the $D^6 R^4$ interaction

The modular invariant coefficient of the $D^6R^4$ interaction in the low energy expansion of type~IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine its exact $SL(2,\Z)$-invariant solution $f(condition as $y\to \infty$ (the weak coupling limit). The solution is presented as a Fourier series with modes $\widehat{f}_n(y) e^{2\pi i n x}$, where the mode coefficients, $\widehat{f}_n(y)$ are bilinear in $K$-Bessel functions. Invariance under $SL(2,\Z)$ requires these modes to satisfy the nontrivial boundary condition $ \widehat{f}_n(y) =O(y^{-2})$ for small $y$, which uniquely determines the solution. The large-$y$ expansion of $f(ower-behaved) terms, together with precisely-determined exponentially decreasing contributions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs.

Algebraic rational cells, equivariant intersection theory, and Poincaré duality

We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of Q-filtrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any Q-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more general spherical varieties. In view of the localization theorem for equivariant operational Chow rings, we get some conditions for Poincaré duality in this setting.

Motivic Cohomology Spectral Sequence and Steenrod Algebra

For an odd prime number $p$, it is shown that differentials $d_n$ in the motivic cohomology spectral sequence with $p$-local coefficients vanish unless $p-1$ divides $n-1$.
We obtain an explicit formula for the first non-trivial differential $d_p$, expressing it in terms of motivic Steenrod $p$-power operations and Bockstein homomorphisms.
Finally, we construct examples of varieties, having non-trivial differentials $d_p$ in their motivic spectral sequences.

Boundedness of non-homogeneous square functions and $L^q$ type testing conditions with $q \in (1,2)$

Calculabilité de la cohomologie étale modulo l

Scattering Equations and String Theory Amplitudes

Scattering equations for tree-level amplitudes are viewed in the context of string theory.
As a result of the comparison we are led to define a new dual model
which coincides with string theory in both the small and large $\alpha'$ limit, and
whose solution is found algebraically on the surface of solutions to the scattering equations.
Because it has support only on the scattering equations, it can be solved exactly,
yielding a simple resummed model for $\alpha'$-corrections to all orders.
We use the same idea to generalize scattering equations to amplitudes with fermions
and any mixture of scalars, gluons and fermions. In all cases checked we find
exact agreement with known results.

Localization in equivariant operational K-theory and the Chang-Skjelbred property

We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are
deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifications of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings.

The physics of quantum gravity

Quantum gravity is still very mysterious and far from being well under- stood. In this text we review the motivations for the quantification of gravity, and some expected physical consequences. We discuss the remarkable rela- tions between scattering processes in quantum gravity and in Yang-Mills theory, and the role of string theory as an unifying theory.

Polylogarithms and multizeta values in massless Feynman amplitudes

The last two decades have seen a remarkable development of analytic methods in the study of Feynman amplitudes in perturbative quantum field theory. The present lecture offers a physicists' oriented survey of Francis Brown's work on singlevalued multiple polylogarithms, the associated multizeta periods and their application to Schnetz's graphical functions and to $x$-space renormalization. To keep the discussion concrete we restrict attention to explicit examples of primitively divergent graphs in a massless scalar QFT.

Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability

The physics and the mixed Hodge structure of Feynman integrals

This expository text is an invitation to the relation between quantum
field theory Feynman integrals and periods.
We first describe the relation between the Feynman parametrization of
loop amplitudes and world-line methods, by explaining
that the first Symanzik polynomial is the determinant of
the period matrix of the graph, and the second Symanzik polynomial
is expressed in terms of world-line Green s functions.
We then review the relation between Feynman graphs and variations
of mixed Hodge structures.
Finally, we provide an algorithm for generating the Picard-Fuchs
equation satisfied by the all equal mass banana graphs in a two-dimensional space-time to all loop orders.

Du transfert automorphe de Langlands aux formules de Poisson non linéaires

Non-Abelian Lie algebroids over jet spaces

We associate Hamiltonian homological evolutionary vector fields
-- which are the non-Abelian variational Lie algebroids' differentials --
with Lie algebra-valued zero-curvature representations for partial differential equations.

Une loi de réciprocité explicite pour le polylogarithme elliptique

On démontre une compatibilité entre la réalisation p-adique et la réalisation de de Rham des sections de torsion du profaisceau polylogarithme elliptique. La preuve utilise une variante pour H1 de la loi de réciprocité explicite de Kato pour le H2 des courbes modulaires.

Le système d'Euler de Kato (II)

Ce texte est le deuxième article d’une série de trois articles sur une généralisation de système d’Euler de Kato. Il est consacré e d’Euler de Kato raffiné associé systèmes d’Euler de Kato.