Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

${mathbb A}^1$-localisation is a universal construction which produces « cohomology theories » for which the affine line ${mathbb A}^1$ is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I’ll discuss the analogous construction where the affine line is replaced by the projective line ${mathbb P}^1$. This is the ${mathbb P}^1$-localisation which is arguably an unnatural construction since it produces « cohomology theories » for which the projective line ${mathbb P}^1$ is contractible. Nevertheless, I’ll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)

Whereas the exponential map from a Lie algebra to a Lie group can be viewed as the monodromy  of a singular connection A dz/z on a disk, the wild character varieties are the receptacles for the monodromy data for arbitrary meromorphic connections on Riemann surfaces. This suggests one should think of the wild character varieties (or the full nonabelian Hodge triple of spaces, bringing in the meromorphic Higgs bundle moduli spaces too) as global analogues of Lie groups, and try to classify them. As a step in this direction I’ll explain some recent joint work with D. Yamakawa that defines a diagram for any algebraic connection on a vector bundle on the affine line.  This generalises the definition made by the speaker in the untwisted case in 2008 in arXiv:0806.1050 Apx. C, related to the « quiver modularity theorem », that a large class of Nakajima quiver varieties arise as moduli spaces of meromorphic connections on a trivial vector bundle the Riemann sphere, proved in the simply-laced case and conjectured in general in op.cit. (published in Pub. Math. IHES 2012), and proved in general by Hiroe-Yamakawa (Adv. Math. 2014).  In particular this construction of diagrams yields all the affine Dynkin diagrams of the Okamoto symmetries of the Painlevé equations, and recovers their special solutions upon removing one node. The case of Painlevé 3 caused the most difficulties.

I will show that Intersection Theory (for twisted de Rham cohomology) rules the algebra of Feynman integrals. In particular I will address the problem of the direct decomposition of Feynman integrals into a basis of master integrals, showing that it can by achieved by projection, using intersection numbers for differential forms. After introducing a few basic concepts of intersection theory, I will show the application of this novel method, first, to special mathematical functions, and, later, to Feynman integrals on the maximal cuts, also explaining how differential equations and dimensional recurrence relations for master Feynman integrals can be directly built by means of intersection numbers. The presented method exposes the geometric structure beneath Feynman integrals, and offers the computational advantage of bypassing the system-solving strategy characterizing the standard reduction algorithms, which are based on integration-by-parts identities. Examples of applications to multi-loop graphs contributing to multiparticle scattering, involving both massless and massive particles are presented.

I will introduce a hierarchical model for a Euclidean conformal field theory in three dimensions. This is a real valued distributional random field over Q_p^3 (instead of R^3). However, I will not assume any knowledge of p-adics. The model is a scalar phi-four theory obtained as a scaling limit of a fixed critical ferromagnetic Gibbs random field on the unit lattice. This is analogous to the scaling limit of the 2d Ising model studied recently by Dubedat, Camia, Garban, Newman, Chelkak, Hongler and Izyurov. I will review joint work with Ajay Chandra and Gianluca Guadagni which constructed not only the random field itself (the spin field) but also its pointwise square (energy field). This is based on a new rigorous renormalization group method whose main feature is the ability to handle space-dependent couplings.

The square field exhibits an anomalous scaling dimension as predicted by Wilson more than 40 years ago. This is the first rigorous construction by renormalization group methods of a bosonic field with anomalous scaling. The key to this property is a new result in dynamical system theory which is an infinite-dimensional generalization of the Poincare-Koenigs holomorphic linearization theorem.

Filtration by ramification groups of the Galois group of an extension of local fields with possibly imperfect residue fields is defined by Abbes and the speaker. The graded quotients are abelian groups and annihilated by the residue characteristic in the general case. We discuss the main ingredients of the proof and  the construction of injections of the character groups of the graded quotients.

The goal of this lecture course is to prove the universal optimality of the E8 and Leech lattices.

This theorem is the main result of a recent preprint « Universal optimality of the E8 and Leech lattices and interpolation formulas » written in collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko. We prove that E8 and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws of Gaussians).

This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.

At the last lecture, we will discuss open questions and conjectures which arose from our work.

The goal of this lecture course is to prove the universal optimality of the E8 and Leech lattices.

This theorem is the main result of a recent preprint « Universal optimality of the E8 and Leech lattices and interpolation formulas » written in collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko. We prove that E8 and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws of Gaussians).

This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.

At the last lecture, we will discuss open questions and conjectures which arose from our work.

The goal of this lecture course is to prove the universal optimality of the E8 and Leech lattices.

This theorem is the main result of a recent preprint « Universal optimality of the E8 and Leech lattices and interpolation formulas » written in collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko. We prove that E8 and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws of Gaussians).

This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.

At the last lecture, we will discuss open questions and conjectures which arose from our work.

The goal of this lecture course is to prove the universal optimality of the E8 and Leech lattices.

This theorem is the main result of a recent preprint « Universal optimality of the E8 and Leech lattices and interpolation formulas » written in collaboration with Henry Cohn, Abhinav Kumar, Stephen D. Miller and Danylo Radchenko. We prove that E8 and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws of Gaussians).

This theorem implies recently proven optimality of E8 and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.

At the last lecture, we will discuss open questions and conjectures which arose from our work.