Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In these pedagogical lectures, I will explore causality constraints on conformal field theory. First, I will show how causality is encoded in crossing symmetry and reflection positivity of Euclidean correlators, and derive constraints on the interactions of low-lying operators directly from the conformal bootstrap. Then, I will explain the connection between these causality constraints and the averaged null energy condition. Finally, I will use causality to show that the averaged null energy is positive in interacting quantum field theory in flat spacetime. Based on arXiv:1509.00014, arXiv:1601.07904, arXiv:1610.05308.

The Teichmüller space parametrizes Riemann surfaces of
fixed topological type and is fundamental in various contexts of mathematics
and physics.  It can be defined as a component of the moduli space of flat G=PSL(2,R)
connections on the surface.  Higher Teichmüller space extends these notions
to appropriate higher rank classical Lie groups G, and N=1 super Teichmüller
space likewise studies the extension to the super Lie group G=OSp(1|2).
In this talk, which reports on joint work with Yi Huang, Ivan Ip and Robert Penner,
I will discuss our solution to the long-standing problem of giving Penner-type
coordinates on super-Teichmüller space and its higher analogues and will
also talk about several applications of this theory including our recent
generalization of the McShane identity to the super case.

This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.

Let G be a real reductive Lie group and H a closed subgroup of G. According to the Benoist-Kobayashi criterion, the properness of an action on G/H is controlled by the Cartan projection of H. In this talk, we give some examples of closed subgroups that are not reductive in G but whose Cartan projection is computable (one example is an abelian horospherical subgroup of G). Applying these nonreductive subgroups, we show that some homogeneous spaces of reductive type do not have compact Clifford-Klein forms.

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.

Séminaire Laurent Schwartz — EDP et applications

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.

Séminaire Laurent Schwartz — EDP et applications

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.

${mathbb A}^1$-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.)