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.

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.

It is well known that any smooth projective toric surface has a full exceptional collection. I will talk about a generalization of this fact for singular surfaces. First, if the class group of Weil divisors of the surface is torsion free (for instance, this holds for all weighted projective planes), I will construct a semiorthogonal decomposition of the derived category with components equivalent to derived categories of modules over certain local finite dimensional algebras. When the class group has torsion, a similar semiorthogonal decomposition will be constructed for an appropriately twisted derived category. Many of these results extend to non-necessarily toric rational surfaces. This is a joint work with Joseph Karmazyn and Evgeny Shinder.

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.

We shall review how supergravity theories can emerge from an exceptional field theory based on the Kac-Moody group E11 (i.e. E8+++) with gauge symmetry a set of `generalised diffeomorphisms’ acting on the fundamental module while preserving E11. The construction relies on a super-algebra T  that extends E11 and provides a differential complex for the exceptional fields. A twisted self-duality equation underlying the dynamics can be shown to be invariant under generalised diffeomorphisms provided a certain algebraic identity holds for structure coefficients of the super-algebra T. The fermions of the theory belong to an unfaithful representation of the double cover of a maximal Lorentzian subgroup K(E11). We conjecture that certain tensor products of unfaithful representations are homomorphic to the quotient of specific indecomposable modules of E11. Using these conjectures, we can write a linearised Rarita-Schwinger equation and show that the E11 twisted self-duality equations are supercovariant. The conjectures are checked through computations in level decompositions with respect to maximal parabolic subgroups.

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.

Many molecules in living cells are present in such low numbers that individual probabilistic chemical events can have a great randomizing effect on the whole system. I will describe how this radically changes the dynamics of several core cellular processes, from cell fate decisions to the oscillators and DNA repair. The presentation will convey some analytical results for classes of stochatic processes, with an emphasis on unsolved problems, but also experimental results, aimed at a board audience, to illustrate what is known about these processes.

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.

I will present a new method to block-diagonalize some Hamiltonians describing quantum chains. The method is applied to study perturbations of the so called Kitaev Hamiltonian. This is a joint work with J. Froehlich.

Let G be a finitely generated group endowed with some probability measure μ and $(rho_{lambda})$ be a non-compact algebraic family of representations of G into SL(2,C). This gives rise to a random product of matrices depending on the parameter λ, so the upper Lyapunov exponent defines a function on the parameter space. Using techniques from non-Archimedean analysis and algebraic geometry, we study the asymptotics of the Lyapunov exponent when λ goes to infinity. This is joint work with Charles Favre.