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.

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.

In the talk I will present a mathematically challenging and difficult problem of the (non)existence of compact Clifford-Klein forms of homogeneous spaces G/H. These are quotients of such spaces by  discrete subgroups of G acting freely, properly and co-compactly. I will formulate the challenging Toshiyuki Kobayashi conjecture and present several partial results supporting it. The results basically are negative in the sense that I will prove the non-existence of compact Clifford-Klein forms for large families of homogeneous spaces, and the non-existence of standard compact Clifford-Klein forms for all homogeneous spaces of exceptional simple real Lie groups. The methods are purely Lie-theoretical. The approach is quite computational: after expressing the problem as some conditions on Lie subalgebras, we develop algorithms checking known obstructions to the existence of compact Clifford-Klein forms. Algorithms are implemented in the computer algebra system GAP and use classifying algorithms of semisimple Lie subalgebras developed by Willem De Graaf. We use the works of Yosuke Morita and Nicolas Tholozan. The talk is based on my joint work with Maciej Bochenski and Piotr Jastrzebski.

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.

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.

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.

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.