Séminaire Laurent Schwartz — EDP et applications

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.

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.

I will discuss results of recent numerical bootstrap work performed for systems with cubic symmetry. Under certain assumptions, we find an isolated region in parameter space, which given prior intuition with the numerical bootstrap, indicates the existence of a CFT in this region. We find critical exponents for the conjectured CFT which are in discrepancy with the epsilon expansion (but in good agreement with experiments for structural phase transitions). The disagreement of critical exponents for structural phase transitions calculated in the epsilon expansion with those measured in experiments is something that was noticed since the 70s. I will briefly discuss some resolutions proposed at the time.

We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large D limit, or as disordered models. Both models have a mass parameter m and the transition from the perturbative large m region to the strongly coupled « black-hole » small m region is associated with several interesting phenomena. One model, with U(n)^2 symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass m*. For 0<m<m*, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions Δ+ and Δ- for the Euclidean two-point function when t→ +∞ and t→ -∞

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.