After recalling the formalism of perturbative Algebraic Quantum Field Theory (pAQFT), I will explain how it allows to prove finiteness of the S-matrix of the Sine Gordon model in different representations and how the net of local observables can be constructed.

I​ ​will​ ​report​ ​on​ ​some​ ​recent​ ​progress​ ​in​ ​understanding​ ​generalized​ ​symmetries​ ​in​ ​quantum​ ​field​ ​theory,​ ​in particular,​ ​rational​ ​conformal​ ​field​ ​theory​ ​in​ ​low​ ​dimensions.

I​ ​wish​ ​to​ ​present​ ​a​ ​possible​ ​solution​ ​to​ ​the​ ​old​ ​problem​ ​of​ ​self-adjointness​ ​of​ ​Wick's​ ​polynomials​ ​by enlarging​ ​the​ ​Wightman​ ​framework​ ​as​ ​to​ ​include​ ​additive​ ​non​ ​linear​ ​maps​ ​(work​ ​done​ ​in​ ​collaboration​ ​with Fredenhagen​ ​and​ ​Hollands).

The​ ​recent​ ​discovery​ ​of​ ​several​ ​gravitational​ ​wave​ ​events​ ​by​ ​the​ ​two​ ​Laser​ ​Interferometer Gravitational-Wave​ ​Observatory​ ​(LIGO)​ ​interferometers​ ​has​ ​brought​ ​the​ ​first​ ​direct​ ​evidence​ ​for​ ​the existence​ ​of​ ​black​ ​holes,​ ​and​ ​has​ ​also​ ​been​ ​the​ ​first​ ​​ ​observation​ ​of​ ​gravitational​ ​waves​ ​in​ ​the​ ​wave-zone. The​ ​talk​ ​will​ ​review​ ​the​ ​theoretical​ ​developments​ ​on​ ​the​ ​motion​ ​and​ ​gravitational​ ​radiation​ ​of​ ​binary​ ​black holes​ ​that​ ​have​ ​been​ ​crucial​ ​in​ ​interpreting​ ​the​ ​LIGO​ ​events​ ​as​ ​being​ ​emitted​ ​by​ ​the​ ​coalescence​ ​of​ ​two black​ ​holes.​ ​In​ ​particular,​ ​we​ ​shall​ ​present​ ​the​ ​Effective​ ​One-Body​ ​(EOB)​ ​formalism​ ​which​ ​led​ ​to​ ​the​ ​first prediction​ ​for​ ​the​ ​gravitational-wave​ ​signal​ ​emitted​ ​by​ ​coalescing​ ​black​ ​holes.​ ​After​ ​a​ ​suitable Numerical-Relativity​ ​completion,​ ​the​ ​(analytical)​ ​EOB​ ​formalism​ ​has​ ​allowed​ ​one​ ​to​ ​compute​ ​the​ ​bank​ ​of 200​ ​000​ ​​ ​accurate​ ​templates​ ​that​ ​has​ ​been​ ​used​ ​to​ ​search​ ​coalescence​ ​signals,​ ​and​ ​to​ ​measure​ ​the masses​ ​and​ ​spins​ ​of​ ​the​ ​coalescing​ ​black​ ​holes.

I​ ​will​ ​discuss​ ​the​ ​programme​ ​to​ ​understand​ ​conformal​ ​field​ ​theory​ ​via​ ​twisted​ ​equivariant​ ​K-theory.​ ​In particular,​ ​studying​ ​​ ​module​ ​categories​ ​and​ ​modular​ ​invariants​ ​for​ ​the​ ​twisted​ ​doubles​ ​of​ ​finite​ ​groups through​ ​correspondences​ ​as​ ​bivariant​ ​Kasparov​ ​KK​ ​elements,​ ​and​ ​realising​ ​the​ ​twisted​ ​doubles​ ​of​ ​finite groups​ ​​ ​as​ ​conformal​ ​field​ ​theories.​ ​This​ ​has​ ​also​ ​resulted​ ​in​ ​a​ ​better​ ​understanding​ ​of​ ​the​ ​double​ ​of​ ​the Haagerup​ ​subfactor,​ ​which​ ​was​ ​originally​ ​thought​ ​to​ ​be​ ​exotic​ ​and​ ​un-related​ ​to​ ​known​ ​models.

A​ ​dilettante's​ ​view​ ​of​ ​some​ ​of​ ​the​ ​central​ ​enigmas​ ​of​ ​cosmology​ ​is​ ​presented.​ ​A​ ​proposal​ ​for​ ​the description​ ​of​ ​the​ ​state​ ​of​ ​the​ ​Early​ ​Universe​ ​is​ ​brought​ ​forward.​ ​A​ ​mechanism​ ​possibly​ ​explaining​ ​the generation​ ​of​ ​primordial​ ​magnetic​ ​fields​ ​is​ ​described​ ​and​ ​analyzed.​ ​The​ ​axion​ ​model​ ​of​ ​"Fuzzy​ ​Dark​ ​Matter" is​ ​described​ ​and​ ​criticized​ ​for​ ​its​ ​instabilities.​ ​A​ ​model​ ​of​ ​Dark​ ​Energy​ ​is​ ​introduced​ ​and​ ​discussed,​ ​and​ ​a conjectural​ ​application​ ​of​ ​this​ ​model​ ​to​ ​an​ ​explanation​ ​of​ ​the​ ​observed​ ​matter-antimatter​ ​asymmetry​ ​is sketched.

The​ ​Navier-Stokes​ ​equations​ ​are​ ​"irreversible"​ ​(are​ ​they?).​ ​I​ ​shall​ ​use​ ​them​ ​as​ ​an​ ​example​ ​of​ ​a​ ​general​ ​proposal​ ​for​ ​a theory​ ​of​ ​statistical​ ​ensembles,​ ​and​ ​their​ ​equivalence,​ ​describing​ ​the​ ​statistical​ ​properties​ ​of​ ​nonequilibrium stationary​ ​states.

Collective​ ​behavior​ ​is​ ​widespread​ ​in​ ​biological​ ​systems​ ​across​ ​many​ ​different​ ​scales​ ​and​ ​organisms.​ ​As physicists,​ ​our​ ​hope​ ​is​ ​that​ ​the​ ​(complex)​ ​details​ ​of​ ​the​ ​individuals​ ​are​ ​not​ ​important​ ​when​ ​looking​ ​at collective​ ​properties,​ ​and​ ​that​ ​large​ ​scale​ ​behavior​ ​can​ ​be​ ​characterized​ ​in​ ​terms​ ​of​ ​general​ ​laws,​ ​much​ ​as we​ ​do​ ​in​ ​condensed​ ​matter.​ ​However,​ ​this​ ​assumption​ ​cannot​ ​be​ ​given​ ​for​ ​granted​ ​and​ ​must​ ​be experimentally​ ​justified. In​ ​an​ ​attempt​ ​to​ ​improve​ ​on​ ​this​ ​situation,​ ​we​ ​present​ ​here​ ​experimental​ ​evidence​ ​of​ ​the​ ​emergence​ ​of dynamic​ ​scaling​ ​laws​ ​in​ ​natural​ ​swarms.​ ​We​ ​find​ ​that​ ​spatio-temporal​ ​correlation​ ​functions​ ​in​ ​different swarms​ ​can​ ​be​ ​rescaled​ ​by​ ​using​ ​a​ ​single​ ​characteristic​ ​time,​ ​which​ ​grows​ ​with​ ​the​ ​correlation​ ​length​ ​with a​ ​dynamical​ ​critical​ ​exponent​ ​z~1.​ ​We​ ​run​ ​simulations​ ​of​ ​a​ ​model​ ​of​ ​self-propelled​ ​particles​ ​in​ ​its swarming​ ​phase​ ​and​ ​find​ ​z~2,​ ​suggesting​ ​that​ ​natural​ ​swarms​ ​belong​ ​to​ ​a​ ​novel​ ​dynamic​ ​universality class.​ ​This​ ​conclusion​ ​is​ ​strengthened​ ​by​ ​experimental​ ​evidence​ ​of​ ​non-exponential​ ​relaxation​ ​and paramagnetic​ ​spin-wave​ ​remnants,​ ​indicating​ ​that​ ​previously​ ​overlooked​ ​inertial​ ​effects​ ​are​ ​needed​ ​to describe​ ​swarm​ ​dynamics.​ ​The​ ​absence​ ​of​ ​a​ ​purely​ ​relaxational​ ​regime​ ​suggests​ ​that​ ​natural​ ​swarms​ ​are subject​ ​to​ ​a​ ​near-critical​ ​censorship​ ​of​ ​hydrodynamics. Authors:​ ​A.​ ​Cavagna,​ ​D.​ ​Conti,​ ​C.​ ​Creato,​ ​L.​ ​Del​ ​Castello,​ ​I.​ ​Giardina,​ ​T.S.​ ​Grigera,​ ​S.​ ​Melillo,​ ​L.​ ​Parisi,​ ​M.​ ​Viale

We consider random lozenge tilings of some fixed large domain. It is well known that for relatively simple domains, the general shape of these tilings converge almost surely when the mesh of the domain goes to zero and local and global fluctuations are well known. We consider more complicated domains where holes are allowed and study their fluctuations. To do so, we remark that the distribution of horizontal tiles are given by discrete Beta-ensembles, analogues of the distributions of eigenvalues of large random Gaussian matrices for discrete variables. We analyze these distributions thanks to certain equations that Nekrasov derived, analogue to the Dyson-Schwinger equations for random matrices. Our results also include other models such as Jack deformations of the Plancherel measure. This talk is based on joint works with Borodin, Borot, Gorin, Huang.

An entanglement measure is a functional on states quantifying the amount of entanglement across two subsystems (i.e. causally disjoint regions in the context of quantum field theory). A reasonable measure should satisfy certain general properties: for example, it should assign zero entanglement to separable states, and be monotonic under separable, completely positive maps ("LOCC-operations"). The v. Neumann entropy of the "reduced state" (to one of the subsystems) is one such measure if the state for the total system is pure. But for mixed states, it is not, and one has to consider other measures. In particular, one has to consider other measures if the subsystems have a finite non-zero distance. In this talk I will present several good measures, and in particular analyze the "relative entanglement entropy", E_R, defined as the "distance" of the given state to the set of separable states, where "distance" is defined using Araki's relative entropy. I will show several features of this measure for instance: (i) charged states, where the relative entanglement entropy is related to the quantum dimension of the charge, (ii) vacuum states in 1+1 dimensional integrable models, (iii) general upper bounds for certain special regions in general CFTs in d dimensions, (iv) area law type bounds. I will also explain the relationship between E_R and other entanglement measures, such as distillable entropy. [Based on joint work with Jacobus Sanders.]

We​ ​compare​ ​two​ ​mathematical​ ​axiomatizations​ ​of​ ​chiral​ ​conformal​ ​field​ ​theory.​ ​​ ​One​ ​is​ ​a​ ​conformal​ ​net based​ ​on​ ​operator​ ​algebras​ ​and​ ​the​ ​other​ ​is​ ​a​ ​vertex​ ​operator​ ​algebra​ ​which​ ​grew​ ​out​ ​from​ ​the​ ​Moonshine conjecture.​ ​​ ​We​ ​present​ ​various​ ​results​ ​on​ ​conformal​ ​nets​ ​and​ ​compare​ ​them​ ​with​ ​those​ ​on​ ​vertex operator​ ​algebras​ ​with​ ​emphasis​ ​on​ ​comparison​ ​of​ ​the​ ​representation​ ​theories.​ ​​ ​In​ ​particular,​ ​we​ ​show that​ ​we​ ​can​ ​pass​ ​from​ ​a​ ​vertex​ ​operator​ ​algebra​ ​to​ ​a​ ​conformal​ ​net​ ​and​ ​come​ ​back​ ​under​ ​a​ ​mild​ ​technical assumption.

In​ ​1994​ ​Dorn​ ​and​ ​Otto​ ​and​ ​in​ ​1996​ ​independently​ ​Zamolodchikov​ ​and​ ​Zamolodchikov​ ​proposed​ ​a remarkable​ ​explicit​ ​expression,​ ​the​ ​so​ ​called​ ​DOZZ​ ​formula​ ​for​ ​the​ ​3​ ​point​ ​structure​ ​constants​ ​of​ ​the Liouville​ ​Conformal​ ​Field​ ​Theory​ ​(LCFT)​ ​which​ ​is​ ​expected​ ​to​ ​describe​ ​the​ ​scaling​ ​limit​ ​of​ ​large​ ​planar maps​ ​properly​ ​embedded​ ​in​ ​the​ ​sphere.​ ​I​ ​will​ ​review​ ​a​ ​rigorous​ ​construction​ ​of​ ​LCFT​ ​and​ ​sketch​ ​a​ ​recent proof​ ​of​ ​the​ ​DOZZ​ ​formula​ ​obtained​ ​together​ ​with​ ​R.​ ​Rhodes​ ​and​ ​V.​ ​Vargas.

The​ ​Yang-Baxter​ ​equation​ ​(YBE)​ ​lies​ ​at​ ​the​ ​heart​ ​of​ ​many​ ​subjects,​ ​including​ ​quantum​ ​statistical mechanics,​ ​QFT,​ ​knot​ ​theory,​ ​braid​ ​groups,​ ​subfactors,​ ​quantum​ ​groups,​ ​quantum​ ​information​ ​...​ ​.​ ​In​ ​this talk,​ ​I​ ​will​ ​consider​ ​mainly​ ​involutive​ ​solutions​ ​of​ ​the​ ​YBE​ ​("R-matrices").​ ​Any​ ​such​ ​R-matrix​ ​defines​ ​a representation​ ​and​ ​an​ ​extremal​ ​character​ ​of​ ​the​ ​infinite​ ​symmetric​ ​group​ ​as​ ​well​ ​as​ ​a​ ​corresponding​ ​tower of​ ​subfactors.​ ​Using​ ​these​ ​structures,​ ​I​ ​will​ ​describe​ ​how​ ​to​ ​find​ ​all​ ​R-matrices​ ​up​ ​to​ ​a​ ​natural​ ​notion​ ​of equivalence​ ​inherited​ ​from​ ​applications​ ​in​ ​QFT​ ​(given​ ​by​ ​the​ ​character​ ​and​ ​the​ ​dimension),​ ​how​ ​to completely​ ​parameterize​ ​the​ ​set​ ​of​ ​solutions,​ ​and​ ​how​ ​to​ ​decide​ ​efficiently​ ​whether​ ​two​ ​given​ ​R-matrices are​ ​equivalent.​ ​Examples​ ​include​ ​diagonal​ ​R-matrices​ ​as​ ​they​ ​appear​ ​in​ ​DHR​ ​theory,​ ​or​ ​Temperley-Lieb type​ ​R-matrices​ ​at​ ​parameter​ ​q=2.

I​ ​will​ ​review​ ​some​ ​results​ ​on​ ​energy​ ​transport​ ​in​ ​weakly​ ​coupled​ ​classical​ ​Hamiltonian​ ​systems​ ​and discuss​ ​ongoing​ ​attempts​ ​to​ ​go​ ​beyond​ ​the​ ​weak​ ​coupling​ ​limit.

The​ ​talk​ ​will​ ​be​ ​based​ ​on​ ​a​ ​joint​ ​paper​ ​https://arxiv.org/abs/1704.08746​ ​with​ ​Mina​ ​Aganagic.​ ​In​ ​this paper,​ ​we​ ​essentially​ ​complete​ ​the​ ​program​ ​of​ ​Nekrasov​ ​and​ ​Shatashvili​ ​who​ ​explained​ ​the​ ​meaning​ ​of Bethe​ ​roots,​ ​Bethe​ ​equations,​ ​etc.​ ​of​ ​quantum​ ​integrable​ ​systems​ ​via​ ​their​ ​correspondence​ ​with supersymmetric​ ​gauge​ ​theories.​ ​We​ ​explain​ ​the​ ​meaning​ ​of​ ​off-shell​ ​Bethe​ ​eigenfunctions​ ​(which​ ​also give​ ​solutions​ ​of​ ​the​ ​quantum​ ​Knizhnik-Zamolodchikov​ ​equations​ ​and​ ​related​ ​difference​ ​equations).​ ​Our formulas​ ​may​ ​be​ ​seen​ ​from​ ​a​ ​geometric,​ ​representation–theoretic,​ ​combinatorial,​ ​and​ ​other​ ​angles

The non linear Schroedinger equation. NLS for short, in its simplest resonant form is   iu_t+Δ u = κ |u|^2qu,   q ≤ 1 ∈ℕ.   Δ is the Laplace operator.
We study this on an n—dimensional torus. It is well known that the NLS can be treated as infinite dimensional Hamiltonian system, perturbation of the linear one.
We show the existence of a strong reducible normal form and by applying a suitable KAM algorithm, the existence of large families of quasi—periodic solutions with various properties of stability. Joint work with Michela Procesi.

While​ ​path​ ​integral​ ​approach​ ​to​ ​quantum​ ​field​ ​theories​ ​has​ ​come​ ​to​ ​dominate​ ​the​ ​field,​ ​the​ ​Hamiltonian methods​ ​have​ ​been​ ​unjustly​ ​neglected.​ ​I​ ​will​ ​discuss​ ​some​ ​work,​ ​inspired​ ​by​ ​the​ ​Rayleigh-Ritz​ ​method​ ​in quantum​ ​mechanics,​ ​which​ ​uses​ ​the​ ​Hamiltonian​ ​approach​ ​to​ ​do​ ​approximate​ ​but​ ​precise​ ​non-perturbative computations​ ​in​ ​strongly​ ​coupled​ ​quantum​ ​field​ ​theory​ ​in​ ​1+1​ ​dimensions.