Algebraic and geometric structures related to integrable systems

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The beauty and attraction of mathematics is rooted in profound images of geometry and physics coming from our perception of reality and it’s analysis at different levels of sophistication. The job of mathematicians then is to express this beauty in terms of formal algebraic structures. Indeed, in forming algebraic structures we, mathematicians, can capture images of geometry and physics by creating a new, algebraic language in which to discuss them and make them accessible to our exploration, analysis and comprehension. The aim of my research program is to investigate algebraic structures arising in the theory of so-called integrable models. A simple example is the famous Korteweg–de Vries equation which is a partial differential equation for a single function u(t,x) of two variables t (time) and x (spatial coordinate). This equation has a form u_t=u_xxx+u u_x where indexes t and x stand for partial derivatives. The Korteweg–de Vries equation, in spite of its simple form, possess a rich and beautiful theory that includes interesting algebraic structures, particular solutions (the so-called solitons) and links with various fields of mathematics from algebraic geometry to functional analysis. I am proposing to study more complicated integrable models over the next few years.The first part of the proposal is devoted to quasi-linear systems of partial differential equations of the form A(u)u_t+B(u)u_x+C(u)u_y=0 where u(t,x,y) is a vector function and A, B, C are matrices depending on u. Equations of this form are useful in hydrodynamics. Such integrable systems also admit a rich mathematical theory. Many fields of mathematics (such as algebraic and differential geometry) will benefit from the development of a theory of such integrable systems.We also wish to study similar systems that are non-homogeneous and have two independent variables t and x. A typical example is a system of two equations for two unknown functions u(t,x) and v(t,x) of the form: u_t=v u_x+1/(u-v), v_t=u v_x+1/(v-u). Because this system admits many new and unusual properties, I am convinced that it's study has the potential of significantly enriching the whole theory of integrable systems.Other studies will be devoted to the so-called matrix integrable systems. A simple example of such system is the generalized Euler top which is an ordinary differential equation U_t=CU^2-U^2C where U(t) is a square matrix function of time t and C is a constant matrix.The last (but not least) part of the proposal is dedicated to algebraic structures arising in the theory of quantum integrable models: namely, the so-called elliptic algebras. To explain the idea, consider three variables x, y, z which do not commute but are subject to relations: xy-yx=z, yz-zy=x, zx-xz=y. It is well known that using these relations any monomial (say, zyxy) can be written in a unique way as a linear combination of ordered monomials such as xxyzzz. A proof of this statement is not hard and based on the observation that x, y, z actually commute up to linear terms. The theory of elliptic algebras deals with similar relations but with quadratic terms only, for example xy-3yx=5z^2, yz-3zy=5x^2, zx-3xz=5y^2. The similar statement about ordered monomials is also valid in this case but the proof is much harder.Elliptic algebras play a significant role in various branches of mathematics and mathematical physics including algebraic geometry, quantum integrable models and even homological algebra. Moreover, some structures connected with the so-called semi-classical limits of elliptic algebras are important in the theory of integrable differential equations discussed above. To summarize, the proposed research is devoted to important algebraic structures arising in modern mathematical physics.