Student projects

Dear prospective student,

Some day this page will contain a good collection of my PhD
projects. However, currently this page is under construction. In
the present form this is a collection of few raw ideas. The
principle of this page in its current form can be formulated as
“One fool man in one minute can ask as many questions that
thousand wise men in thousand years could hardly answer”
.

So, below there are some thoughts about further development in the
area of multidimensional exactly integrable systems and
Tetrahedron equation.

Classification of levels:

A+: Extremely challenging problems — my own priorities and
presumably the projects for very insolent PhD students (or for
post-docs) The key criterion is: a methodology is not known.

A: Challenging problems: for me, post-docs and for high level PhD
students.

A-: Very good projects. A good start for PhD study.

Mainstream problems.

  1. [Level A+] Extraordinary challenging problems. This is a
    long-standing problem to find a proper form of algebraic equations
    defining spectra of integrable field theories in three-dimensional
    space-time suitable in the truly three-dimensional thermodynamic
    limit. This is the open problem for the auxiliary
    transfer-matrices, layer-to-layer transfer matrices and for the
    evolution operators as well. This subject covers

    • The cyclic representation for the Weyl algebra models (see the next paragraph for some progress and more recent development)
    • The Fock space representation of q-oscillator model (see also the next paragraph)
    • The modular representation — the most complicated one, even in 2D only a few exact results are known.


  2. [Level A] Challenging problems. This paragraph contains
    some known development of the previous one.

    There are known spectral equations describing a two-dimensional
    lattice Bose gas. The latter model can be seen as a model
    describing a non-relativistic Hamiltonian of a real Bose gas in
    tow-plus-one dimension of space-time. The spectral equations are
    much more complicated than e.g. nested Bethe Ansatz equations. The
    real problems are:


    • First, to study them numerically and

    • To obtain a limiting form of them (i.e. an integral form of equations
      in the limit when the number of degrees of freedom goes to
      infinity).

    Spin lattices. We know the spectral equations for what
    is called the spin lattice (a multidimensional extension of spin
    chains) and, moreover, we know the limiting form of these
    equations. However, only a structure of the ground state is known.
    As well, the model is formulated on the level of auxiliary
    transfer matrix (a quantum curve). Some recent results concern a
    factorization quantum transfer matrix (much more complicated
    object than the quantum curve) and evolution operator. The further
    development includes:


    • To find a good numerical and analytical
      method to solve the spectral equations, and

    • To find a theta-functions analogue of polynomial factorization in the
      thermodynamic limit, and

    • To find finally a proper form of spectral equations for evolution operator and find a structure of elementary field-theoretical excitations.

  3. [Level A+] To make all mentioned in paragraph 1 for fermionic
    and mixed representations (subject of integrable field theories).
  4. [Level A-] To classify three-dimensional boundary conditions
    (in my draft calculations I’ve seen a lot of unusual scenarios)
  5. [Level A+] To solve some q-oscillator statistical mechanics for
    non-unitary representations (completely positively defined
    Boltzmann weights but non-linear boundary conditions — standard
    machinery does not work)
  6. [Level A-] To understand a fixed boundary conditions and
    presumably B, C and D series of quantum groups entirely in the
    algebraic terms. To prove, at least, that the fixed boundary
    conditions indeed give the representation theory for B, C and D.

Curious problems



  1. [Level A]. There exist BKP (B-type Kadomcev-Petviashvili
    hierarchy) solitons. I believe, the existence of solitons
    guarantees the existence of quantization. Discrete BKP hierarchy
    is very well known, but its quantization scheme is a total
    mystery.
  2. [Level A-] We know a couple of solutions to quantum 4-simplex
    equation. These solutions for a simple square four-dimensional
    lattice do not produce an interesting series of three-dimensional
    models (the result is a disjoint set of independent Hirota-type
    three-dimensional models). A good question for topological
    students is: what will happen if we will topologically modify the
    simple square lattice?
  3. [Level A-] Solutions to 4-simplex equation mentioned before
    are related to a class of three-dimensional models with Weyl
    algebra of observables. There must be their analogue for
    q-oscillator algebra of observables. Find them.


Unclassified problems [A+/A-]



  1. High dimensional simplex equations and high dimensional
    topology
  2. Applications to knot and link invariants. In general, the
    tetrahedral scheme provides a definition of Hamiltonian for every
    knot and link — does it make any sense?
  3. 3D models are holographic image of 4-dimensional what?
  4. Solution of the Tetrahedron equation in R and W forms can be
    used to produce an infinite seria of solutions to the Yang-Baxter
    equation in R and W forms. However, a solution of the Tetrahedron
    equation in S form produce a series of solutions to YBE in a mixed
    R and W form which are not yet classified in terms of quantum
    groups. Are there non-trivial examples?
  5. In the framework of quantum curve for q-oscillator model a
    very unusual object appeared: a non-local transfer matrix. It has
    no interpretation in terms of quantum groups (as well as the
    lattice Bose gas mentioned above). What is this?
  6. Search for solutions of the Tetrahedron equation with
    non-local algebra of observables. Such theories a priori will have
    no quantum group interpretation. Presumably, a solution of p.1 of “curiuos problems” lays
    here.