Цель семинара — знакомство участников с различными идеями и конструкциями, связанными с деформационным квантованием, теорией квантовых групп и (более общо) некоммутативной геометрии. Темы, которые мы постараемся обсудить даны внизу, но порядок их обсуждения зависит от вкусов и запросов участников; при желании некоторые из темы могут быть отложены на потом, а вместо них будут рассмотрены новые темы, предложенные участниками семинара. Студентам, которые примут участие в работе семинара, будет предложено сделать один или несколько докладов на заинтересовавшую их тему (что будет учтено при простановке оценок). Участие в работе семинара в первом семестре не является необходимым для того, чтобы присоединиться к нам весной!
26 сентября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Yuancheng Xie from Peking University
Тема: Space curves and solitons of the KP hierarchy
Abstract:
It is well known that algebro-geometric solutions of the KdV hierarchy are constructed from the Riemann theta (or Klein sigma) functions associated with hyperelliptic curves, and soliton solutions can be obtained by rational limits of the corresponding curves.
In this talk, I will associate a family of singular space curves indexed by the numerical semigroups〈l, lm+1, . . . , lm+k〉where m ≥ 1 and 1 ≤ k ≤ l−1 with a class of generalized KP solitons. Some of these curves can be deformed into smooth "space curves", and they provide canonical models for the l-th generalized KdV hierarchies (KdV hierarchy corresponds to the case l = 2). If time permits, we will also see how to construct the space curves from a commutative ring of differential operators in the sense of the well-known Burchnall-Chaundy theory.
This talk is based on a joint work with Professor Yuji Kodama.
19 сентября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Grigoriy Papayanov from University of Haifa
Тема: Formal geometry and deformation quantizations
Abstract:
Every smooth manifold, of any kind --- C-infinity, algebraic, complex analytic etc. is endowed with a natural algebra bundle together with a flat connection: the jet bundle. Every geometric structure on a manifold, understood in a broad sense, for example, as a differential operator between some tensor bundles, defines the corresponding structure on the jet bundle. It follows that if the geometric structure has a unique local form (like Darboux form for the symplectic structure), it is uniquely defined by the reduction of the structure group of the jet bundle, together with the structure Lie algebra of the flat connection. This observation is known as Gelfand-Kazhdan formal geometry. I want to explain in what sense the (symplectic) deformation quantization is a structure to which formal geometry mechanism is applicable, how to use it to give a conceptual explanation of the results of Fedosov-Nest-Tsygan-Bezrukavnikov-Kaledin on the classification of quantizations, and to which calculations this method might lead.
12 сентября 2023 (вторник), 17:00, дистанционно в Zoom (
Идентификатор конференции: 899 4326 1657
Код доступа: ium )
Докладчик: Andrey Smilga from the University of Nantes
Тема: Dynamical systems with higher time derivatives
Abstrac:
Mechanical and field theory systems with higher time derivatives in the Lagrangian are "ghost-ridden". This means that the spectra of the corresponding Hamiltonians have no bottom and include states with arbitrarily low energies. In most such nonlinear systems, these ghosts are "malignant" — the classical equations of motion have blow-up solutions (which run into a singularity at finite time), which lead to breaking of unitarity in quantum theory.
There are, however, some special nonlinear higher-derivative systems (in particular, exactly soluble systems) where the ghosts are "benign" and, in spite of the absence of the ground state in the spectrum, there is no blow-up and unitarity is preserved. We discuss in detail one of such systems.
We consider also the simplest nontrivial exactly soluble higher-derivative field systems with the equation of motion
u_{ttt} + 6uu_t + u_x = 0,
and
u_{ttt} + 6u^2u_t + u_x = 0.
These are the "rotated" KdV and the modified KdV system where the spatial coordinate and time change their roles. Higher time derivatives bring about the ghosts.
In the rotated KdV, the ghosts are "malignant" and there are blow-up solutions. But in the modified rotated KdV, such solutions seem to be absent: we present arguments that the ghosts are benign in this case, i.e. the classical dynamics of this system does not involve a blow-up. In addition, we solved the Cauchy problem for the rotated modified KdV numerically and did not see a trace of blow-up. Unfortunately, at some point, the solutions become unstable and there is a horizon, beyond which we could not go with the numerical methods that we used.