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”.

thousand wise men in thousand years could hardly answer”

area of multidimensional exactly integrable systems and

Tetrahedron equation.

presumably the projects for very insolent PhD students (or for

post-docs) The key criterion is: a methodology is not known.

students.

- [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.

- [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.

- [Level A+] To make all mentioned in paragraph 1 for fermionic

and mixed representations (subject of integrable field theories). - [Level A-] To classify three-dimensional boundary conditions

(in my draft calculations I’ve seen a lot of unusual scenarios) - [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) - [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.

- [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. - [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? - [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.

- High dimensional simplex equations and high dimensional

topology - 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? - 3D models are holographic image of 4-dimensional what?
- 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? - 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? - 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.