MIT Infinite Dimensional Algebra Seminar (Spring 2015)

I will tell a few interrelated stories that link natural probabilistic systems with different families of symmetric functions.

We will talk about several combinatorial structures related to the positive Grassmannian. They include a class of convex polytopes, certain 3-valent graphs, certain 2-dimensional surfaces, etc. We'll discuss equalities between minors of matrices. We'll mention an application of combinatorics of the positive Grassmannian to physics of scattering amplitudes. The talk is based on recent joint works with Thomas Lam and with Miriam Farber, as well as with Nima Arkani-Hamed at al.

The aim of this talk is to study the Tate cohomology, specifically for any finite dimensional Hopf algebra A over a field k. In the first half of the talk, we describe the construction and structure of the Tate cohomology of A. In particular, we look at the algebra relation between its Tate and Tate-Hochschild cohomology, followed by some applications to the Sweedler algebra and finite group algebras.
While the ordinary cohomology rings of some finite dimensional Hopf algebras are known to be finitely generated, the same may not be true when we extend them to negative cohomology. In the second half of the talk, we investigate this finite generation property when A is symmetric. This problem is motivated from a construction of the universal ghost map in the stable module category of A. It turns out that if a module in a connected component of the stable Auslander-Reiten quiver associated to A has finitely generated Tate cohomology, then so does every module in that component. If time permits, we will apply some of these results to an algebra defined by Radford and the restricted universal enveloping algebra of sl_2.

A remarkable conjecture of Feigin-Finkelberg-Kuzntesov-Mirkovic says that semi-infinite cohomology of the quantum group with coefficients in a tilting module can be expressed as a certain multiplicity space for geometric Eisenstein series for the curve P^1. In the fall I indicated the proof of this conjecture that goes through localization of Kac-Moody representations and the Kazhdan-Lusztig equivalence. In the present talk I'll present a different derivation of this conjecture: this time from the geometric Langlands equivalence for P^1.

Let A be an associative algebra, a Lie algebra or, more generally, an algebra over an arbitrary (k-linear) operad defined over a field k of characteristic zero. The set of all representations of A in a finite-dimensional vector space V can be given the structure of an affine k-scheme, called the represesentation scheme Rep_V(A). In this talk, I will discuss the properties of the derived representation scheme DRep_V(A), a higher homological extension of Rep_V(A) obtained by deriving the representation functor Rep_V in the sense of non-abelian homological algebra. As a main example, I will consider the derived schemes associated to the classical commuting schemes of complex reductive Lie algebras. I will present a general conjecture about the structure of these derived commuting schemes and discuss its implications.

(The talk is based on joint work with G. Felder, A. Patotski, A. Ramadoss and T. Willwacher)

BPS quivers are a very special (yet humongous) class of quivers with potentials. The study of the representation theory of these algebras leads to very pleasant surprises. In this talk we are going to discuss relations with cluster algebras and formulate a conjecture relating the number theoretical properties of the algebras underlying the BPS quivers with the properties of the corresponding cluster algebras. Based on joint works with Sergio Cecotti.

We will define to DAHA-Jones polynomials (also called refined due to an extra parameter "t") of iterated torus knots, which class includes all algebraic knots. The DAHA superpolynomials will be the main theme; they presumably coincide with the Poincare polynomials of the HOMFLY-PT homology for algebraic knots (equivalently, stable Khovanov-Rozansky polynomials). They are defined in type "A" or for the classical series; the latest paper of the speaker and Ross Elliot indicates that they may exist even for the series "E".

We will begin with the DAHA-Jones polynomials for A_1 and then explain how uncolred DAHA-superpolynomials for the torus knots T(2n+1,2) can be calculated. They will be compared with those due to Evgeny Gorsky and others in terms of the rational DAHA. Then we will switch to general theory for arbitrary root systems, including superpolynomials. A surprising application is the justification of the formula for colored superpolynomials for T(2n+1,2) suggested by physicists, using the theory of DAHA of type C-check-C_1, directly related to the covers of CP^1 with 4 punctures. The latter theory has an interesting NT twist when PSL(2,Z) is replaced by the absolute Galois group over Q.

If time permits we will discuss the iterated knots and plane curve singularities. Conjecturally, DAHA provides the formulas for Betti numbers of the compactified Jacobian factors of the latter, which are very difficult to calculate in algebraic geometry. This is for very special a=0,q=1. However it is a great test of the maturity of the new theory, closely related to the Oblomkov-Rasmussen-Shende conjecture, which extends the OS-conjecture, generalized and proved by Davesh Maulik. I will state the ORS conjecture (modulo the definition of the weight filtration).

A quasi-Coxeter algebra is a bialgebra which carries actions of a given generalised braid group and of Artin’s braid groups on the tensor products of its representations. A basic example is the quantum group U_h(g) endowed with Lusztig’s quantum Weyl group elements and its universal R-matrix. I will explain how to construct such a structure on the enveloping algebra Ug of a semisimple Lie algebra g, which accounts for the monodromy of the rational Casimir and KZ connections of g. Given that such structures are rigid, this implies in particular that the monodromy of the rational Casimir connection is described by the quantum Weyl group operators of U_h(g).

We will discuss upper and lower bounds for the effective Nullstellensatz for systems of algebraic PDEs. These are uniform bounds for the number of differentiations to be applied to a system of PDEs in order to discover algebraically whether it is consistent (i.e., has a solution in a differential field extension). The bounds are functions of the degrees and orders of the equations of the system and the numbers of dependent and independent variables in them. Seidenberg was the first to address this problem in 1956. The first explicit bounds appeared in 2009, with the upper bound expressed in terms of the Ackermann function. In the case of one derivation, the first bound is due to Grigoriev (1989). In 2014, another bound was obtained if restricted to the case of one derivation and constant coefficients. Our new result does not have these restrictions.

In 2004, Enriquez-Etingof-Marshall suggested a new approach to the Ginzburg-Weinstein linearization theorem. This approach is based on solving a system of PDEs for a gauge transformation between the standard classical r-matrix and the Alekseev-Meinrenken dynamical r-matrix. In the talk, we explain that this gauge transformation can be constructed as a monodromy (connection matrix) for a certain irregular Riemann-Hilbert problem. Geometrically, this leads to a symplectic neighborhood version of the Ginzburg-Weinstein linearization theorem. Our construction is based on earlier works by Boalch. As an application, we give a new description of the Lu-Weinstein symplectic double.

To each Coxeter group there is an associated Artin-Tits braid group. While Coxeter groups are relatively well-understood objects, the associated braid groups are less well-understood, in part because these groups seem hard to study using finite-dimensional representations on vector spaces. On the other hand, actions of such groups on finite linear categories seem well-suited to work as a substitute. The goal of this talk will be to explain in some detail the example of the free group, which is the Artin-Tits group associated to the universal Coxeter system.