The second one relates elliptic Schubert calculus with Felder-Tarasov-Varchenko weight functions, and Aganagic-Okounkov stable envelopes. The duality between the two recursions is an incarnation of 3d mirror symmetry symplectic duality. Joint work with A. ABSTRACT: I will discuss some combinatorial constructions and conjectures having their origin in the classification of perverse sheaves over configuration spaces.

Joint work in progress with Mikhail Kapranov. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories.

We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part. This determinant is also the discrete Hessian of the corresponding master function. Another form of the Bethe ansatz equations is given via the discrete Wronski map. I will present a formula for the norm of the Bethe vectors through the differential of the discrete Wronski map. Together with the Gaudin-Korepin formula, it provides a relation between the Hessian of the master function and the differential of the Wronski map.

About 20 years ago it was discovered that this does not exclude the chaotic behaviour of the system, which may even have positive topological entropy. I will explain that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil. Taking this into account, one may view a not necessarily commutative co module algebra as an action of a quantum group on a non-commutative space.

The above point of view was also considered by Y. Therefore, we propose a refinement of Manin's construction by studying comodule algebras up to support equivalence, which generalizes in a natural way weak equivalence of gradings. We study support equivalence and these universal Hopf algebras for group-gradings, Hopf-Galois extensions, algebraic groups and cocommutative Hopf algebras. We argue how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra co actions.

Examples include the symmetric group algebra, the Temperley-Lieb algebra and the Brauer algebra.

The motivation to study this periplectic Brauer algebra is representation theory of the periplectic Lie superalgebra, which is related to it via a Schur-Weyl type duality. Moreover, the generating functions of Demazure multiplicities will be connected to cone theta functions. For theories in more than two dimensions, the blocks have been the subject of many investigations, starting with the work of Dolan and Osborn, who determined the four-point blocks for scalar operators in an even dimensional space.

Less progress has been made in theories with supersymmetry. In this talk I will present a harmonic analysis approach to conformal partial waves, which casts them in the form of eigenfunctions of a Schroedinger problem of Calogero-Moser type.

### Invited talks

In particular, the blocks for a large class of superconformal theories will be constructed as eigenfunctions of a spin Calogero-Moser Hamiltonian perturbed by a nilpotent potential term. The talk is based on the joint work with Volker Schomerus and Evgeny Sobko. Feigin and A. Stoyanovsky, provide an interpretation of the sum side of various Rogers-Ramanujan type identities. From combinatorial bases, we obtain characters of principal subspaces and some new combinatorial identities.

We demonstrate that the reduced hierarchy belongs to the overlap of well-known trigonometric spin Sutherland and spin Ruijsenaars--Schneider type integrable many-body models, which receive a bi-Hamiltonian interpretation via our treatment. The rational pseudodifferential operator is invariant under the reproduction procedure. We expect that the coefficients of the expansion of the rational pseudodifferential operator are eigenvalues of the higher Gaudin Hamiltonians acting on the corresponding Bethe vector.

Then we study the properties of these subalgebras. Based on arXiv We proved that this isomorphism induced an isomorphism between the monoid of dominant monomials and the monoid of rectangular semi-standard Young tableaux.

### A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

Using the isomorphism and the results of Kashiwara, Kim, Oh, and Park and the results of Qin, we proved that every cluster monomial resp. These formulas are useful in studying real modules, prime modules, and compatibility of two cluster variables. We described the mutations of a Grassmannian cluster algebra using semi-standard Young tableaux and described the mutations of modules.

In particular our result implies that each common eigenspace has dimension 1. We also show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz and express the transfer matrices associated to symmetrizers and anti-symmetrizers in terms of the first transfer matrix and the center of the Yangian.

- What Makes Us Moral? On the capacities and conditions for being moral.
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- New Frontiers in Economics.

Then the approach mainly counts on J. Green's irreducible character formula of finite general linear groups and a character estimate by Liebeck et al. Time permits, I will talk about the meaning of the result in a topological setting.

One of the important aspects of the integrable systems is that these nonlinear systems possess linear problems. However, it is not easy to find a linear problem Lax equation just by looking at the nonlinear equations. Our way is to compare generic degenerations of the families of curves arising from the nonlinear problem i. We have proved that the Jacobian of generic curve of these systems has unique principal polarization, so that we can recover curves.

I will speculate about what I think this means for the spaces of Stokes data and the Hitchin integrable system. This way we confirm the conjecture on fusion rules based on the Verlinde formula. We explicitly construct the intertwining operators appearing in the formula for fusion rules.

ABSTRACT: The Grothendieck ring of the category of finite dimensional representations over a simple Lie algebra can be described via the character map, as a ring of functions invariant under the action of the Weyl group. This result was generalized to basic Lie superalgebras by A. Veselov with additional invariance conditions. In this talk, we will generalize the theorem of Sergeev and Veselov to periplectic Lie superalgebras and describe their Grothendieck rings. ABSTRACT: Recently developed approaches to scattering amplitudes in quantum field theory highlight underlying geometrical structures which allow to interpret Feynman amplitudes as periods of motives.

Techniques in algebraic geometry are applied to the motivic version of Feynman integrals to investigate their geometric properties and to give information about their numerical value.

## A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory

Mathematically, they constitute a flat connection in a vector bundle, encoded by a nonautonomus integrable quantum system. A Kaleidoscopic View of Graph Colorings. Unifying current material on graph coloring, this book describes current information on vertex and edge colorings in graph theory, including A Textbook of Graph Theory. The chapter on graph colorings has been enlarged, covering A Walk Through Combinatorics. An Introduction to Enumeration and Graph Theory. An extensive list of problems, ranging from routine exercises to research questions, is included. In each section, there are also exercises that contain.

Additive Combinatorics. This theory has seen exciting developments and dramatic changes in direction in recent years thanks to its connections with areas such as number theory, ergodic The author has written Advanced Topics in Fuzzy Graph Theory. Continuing in their tradition, it provides readers with an extensive set of tools for applying fuzzy mathematics and graph theory to social problems Advances in Commutative Algebra.

Anderson in wide-ranging areas of commutative algebra.

## Select Publications by Professor Jie Du | UNSW Research

It provides a balance of topics for experts and non-experts, with a mix of survey Advances in Variational and Hemivariational Inequalities. Theory, Numerical Analysis, and Applications. The book will be of particular interest to graduate students and. Adventures in Graph Theory.