Dualities, affine vertex operator algebras, and geometry of complex polynomials

Sammanfattning: This thesis consists of two parts which deal with different subjects. In the first part we study certain aspects of the representation theory of affine Kac-Moody Lie algebras and related structures. We notice that for any positive integer k, the set of (1,2)-specialized characters of the level k standard A₁¹-modules is the same as the set of rescaled graded dimensions of the subspaces of the level 2k+1 standard A₂²-modules that are vacuum spaces for the action of the principal Heisenberg subalgebra of A₂². We conjecture the existence of a semisimple category induced by the ``equal level'' representations of some algebraic structure which would naturally explain this duality-like property, and we study potential such structures in the context of (generalized) affine vertex operator algebras. We also propose a combinatorial approach for explaining these coincidences. To this end, we determine certain sets of annihilating fields of standard modules for an arbitrary affine Kac-Moody Lie algebra. This yields in particular a characterization of standard modules in terms of irreducible loop modules,a useful tool for combinatorial constructions of bases for standard modules. The second part of the thesis contains three papers which focus on Sendov's conjecture about the location of critical points of complex polynomials. In the first paper we prove that Sendov's conjecture is true for polynomials of degree 6 and determine the so-called extremal polynomials in this case. We also prove the conjecture for polynomials with at most 6 different zeros and generalize this result to polynomials of degree n with at most N(n) distinct roots, where N(n) is an increasing and unbounded function of n. In the process we get an easy proof of the conjecture for polynomials of degree at most 4. In the second paper we show that Sendov's conjecture is valid for polynomials of degree 7 and determine the extremal polynomials in this case as well. We also check the conjecture for polynomials with at most 7 distinct roots and discuss some properties of the extremal polynomials in the general case. Finally, in the last paper we investigate two different ways of studying the Sendov Conjecture: a variational method and an approach based on apolarity theory. We use the former to verify a special case of a conjectured property of extremal polynomials, and the latter to give an equivalent reformulation of Sendov's conjecture. This leads in particular to several sufficiency conditions and generalizations of known results, as well as to a possible inductive approach to the conjecture.