L4

We provide lattice-based protocols allowing to prove relations among committed integers. While the most general zero-knowledge proof techniques can handle arithmetic circuits in the lattice setting, adapting them to prove statements over the integers is non-trivial, at least if we want to handle exponentially large integers while working with a polynomial-size modulus qq. For a polynomial L, we provide zero-knowledge arguments allowing a prover to convince a verifier that committed L-bit bitstrings x, y and z are the binary representations of integers X, Y and Z satisfying Z=X+Y over the integers. The complexity of our arguments is only linear in L. Using them, we construct arguments allowing to prove inequalities X <Z among committed integers, as well as arguments showing that a committed X belongs to a public interval [α,β], where α and β can be arbitrarily large. Our range arguments have logarithmic cost (i.e., linear in L) in the maximal range magnitude. Using these tools, we obtain zero-knowledge arguments showing that a committed element X does not belong to a public set S using soft-O(n⋅log|S|) bits of communication, where n is the security parameter. We finally give a protocol allowing to argue that committed L-bit integers X, Y and Z satisfy multiplicative relations Z=XY over the integers, with communication cost subquadratic in L. To this end, we use our protocol for integer addition to prove the correct recursive execution of Karatsuba's multiplication algorithm. The security of our protocols relies on standard lattice assumptions with polynomial modulus and polynomial approximation factor.

TBA

The celebrated localisation theorem of Beilinson-Bernstein asserts that there is an equivalence between representations of a Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. In this talk we discuss certain analogues of this result in various contexts. Namely, there is a localisation theorem for quantum groups due to Backelin and Kremnizer and, more recently, Ardakov and Wadsley also proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. We then explain how to combine the ideas involved in these results to construct
a p-adic analytic quantum flag variety and a category of D-modules on it, and we show that the global section functor on these D-modules yields an equivalence of categories.

One of the main open problems in loop quantum gravity is to reconcile the fundamental quantum discreteness of space with general relativity in the continuum. In this talk, I present recent progress regarding this issue: I will explain, in particular, how the discrete spectra of geometric observables that we find in loop gravity can be understood from a conventional Fock quantisation of gravitational edge modes on a null surface boundary. On a technical level, these boundary modes are found by considering a quasi-local Hamiltonian analysis, where general relativity is treated as a Hamiltonian system in domains with inner null boundaries. The presence of such null boundaries requires then additional boundary terms in the action. Using Ashtekar’s original SL(2,C) self-dual variables, I will explain that the natural such boundary term is nothing but a kinetic term for a spinor (defining the null flag of the boundary) and a spinor-valued two-form, which are both intrinsic to the boundary. The simplest observable on the boundary phase space is the cross sectional area two-form, which generates dilatations of the boundary spinors. In quantum theory, the corresponding area operator turns into the difference of two number operators. The area spectrum is discrete without ever introducing spin networks or triangulations of space. I will also comment on a similar construction in three euclidean spacetime dimensions, where the discreteness of length follows from the quantisation of gravitational edge modes on a one-dimensional cross section of the boundary.
The talk is based on my recent papers: arXiv:1804.08643 and arXiv:1706.00479.
 

The (total) Steenrod square is a ring homomorphism from the cohomology of a topological space to the Z/2-equivariant cohomology of this space, with the trivial Z/2-action. Given a closed monotone symplectic manifold, one can define a deformed notion of the Steenrod square for quantum cohomology, which will not in general be a ring homomorphism, and prove some properties of this operation that are analogous to properties of the classical Steenrod square. We will then link this, in a more general setting, to a definition by Seidel of a similar operation on Floer cohomology.
 

The Hitchin fibration is a natural tool through which one can understand the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing the type of questions and methods considered in the area, we shall dedicate this talk to the study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations. Through Cayley and Langlands type correspondences, we shall provide a geometric description of these objects, and consider the implications of our methods in the context of representation theory, Langlands duality, and within a more generic study of symmetries on moduli spaces.

We will talk about the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations. When viscosity coefficients are given as a constant multiple of density's power, based on some analysis  of  the nonlinear structure of this system, by introducing some new variables and the initial layer compatibility conditions, we identify the class of initial data admitting a local regular solution with far field vacuum and  finite energy  in some inhomogeneous Sobolev spaces, which solves an open problem of degenerate viscous flow partially mentioned by Bresh-Desjardins-Metivier (2006, Anal. Simi. Fluid Dynam.),  Jiu-Wang-Xin (2014, JMFM) and so on. Moreover, in contrast to the classical well-posedness theory in the case of  the constant viscosity,   we show   that one can not obtain any global classical solution whose $L^\infty$  norm of $u$ decays to zero as time $t$ goes to infinity under the assumptions on the conservation laws of total mass and momentum.