Unbounded capillary surfaces in domains with a sharp corner or a cusp are studied. It is shown how numerical study using a proposed computational methodology leads to two new conjectures for open problems on the asymptotic behavior of capillary surfaces in domains with a cusp. The numerical methodology contains two simple but important ingredients, a change of variable and a change of coordinates, which are inspired by known asymptotic approximations for unbounded capillary surfaces. These ingredients are combined with the finite volume element or Galerkin finite element methods. Extensive numerical tests show that the proposed computational methodology leads to a global approximation method for singular solutions of the Laplace–Young equation that recovers the proper asymptotic behavior at the singular point, is more accurate and has better convergence properties than numerical methods considered for singular capillary surfaces before. Using this computational methodology, two open problems on the asymptotic behavior of capillary surfaces in domains with a cusp are studied numerically, leading to two conjectures that may guide future analytical work on these open problems.

We study the contact equivalence problem for toric contact structures on S-3-bundles over S-2. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To show that two toric contact structures with the same first Chern class are contact inequivalent, we use Morse-Bott contact homology. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contacto-morphism group. We treat a subclass of contact structures which includes the Sasaki-Einstein contact structures Y-p,Y-q studied by physicists with the anti-de Sitter/conformal field theory conjecture. In this case we give a complete solution to the contact equivalence problem by showing that Y-p,Y-q and Y-p',Y-q' are inequivalent as contact structures if and only if p not equal p'.

This is a follow-up to our 2013 paper "Categorifications of the extended affine Hecke algebra and the affine quantum Schur algebra (S) over cap (n, r) for 3 <= r < n" in which we categorified the affine q -Schur algebra <(S)over cap>(n, r) for 2 < r < n using a quotient of the categorification of U-q ((Sl) over cap (n)) of Khovanov and Lauda (2009, 2010, 2011). In this paper we categorify (S) over cap (n, n) for n >= 3 using an extension of the aforementioned quotient.

In the first part of this paper the projective dimension of the structural modules in the BGG category O is studied. This dimension is computed for simple, standard and costandard modules. For tilting and injective modules an explicit conjecture relating the result to Lusztig's a-function is formulated ( and proved for type A). The second part deals with the extension algebra of Verma modules. It is shown that this algebra is in a natural way Z(2)-graded and that it has two Z-graded Koszul subalgebras. The dimension of the space Ext(1) into the projective Verma module is determined. In the last part several new classes of Koszul modules and modules, represented by linear complexes of tilting modules, are constructed.

5.

Mazorchuk, Volodymyr

et al.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Geometry and Logic.

In this paper we show that every block of the category of cuspidal generalized weight modules with finite dimensional generalized weight spaces over the Lie algebra sp(2n)(C) is equivalent to the category of finite dimensional C[[t(1,) t(2,) ... , t(n)]]-modules.

6.

Nilsson, Jonathan

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.

A new family of simple gl_{2n}(C)-modules2016In: Pacific Journal of Mathematics, ISSN 0030-8730, E-ISSN 1945-5844, Vol. 283, no 1, p. 1-19Article in journal (Refereed)

Abstract [en]

We construct a new family of simple gl_{2n}-modules which depends on n^{2} generic parameters. Each module in the family is isomorphic to the regular U(gl_{n} )-module when restricted the gl_{n} -subalgebra naturally embedded into the top-left corner.