Sep 03, Monday

An intuitive review of the concept of a knot group will be presented. Simple physics ideas will then be used, in this context, to derive an exact expression for a flat connection on the complement of a torus knot.

The calculation of Koszul complexes is a classical problem in commutative algebra. The Eagon-Northcott complex and the Lascoux resolution are examples of some generalizations of these complexes in the context of multilinear algebra.\par I will talk about one such complex which arises as a pushforward of a Koszul complex of vector bundles. I will show an example of how this complex can be calculated for studying certain varieties which arise in the context of quiver representations (read as \textit{representations of finite-dimensional algebras}).\par I will not assume familiarity with anything more than basic linear algebra, commutative algebra and a smattering of algebraic geometry.

In the theory of complex vector bundles, holomorphic connections play an important role. Unlike differential connections, holomorphic connections may not exist at all. In the case of holomorphic vector bundles over a complex abelian variety the existence of holomorphic connection and that of the stability (semistability) are interlinked. Such vector bundles are turn out to be homogeneous. Holomorphic connections in holomorphic bundles over a complex abelian variety were studied by Balaji, Biswas, Iyer, G\'{o}mez and Subramanian. In this talk we will give analogues, for real abelian varieties, of some of their results. We will determine various equivalent conditions for the presence of real holomorphic connections in a real holomorphic vector bundle over a real abelian variety.

The Jacobson-Morozov theorem (J-M theorem, for short) asserts that any nilpotent element in a complex semi-simple lie algebra can be embedded in a 3-dimensional subalgebra isomorphic to $sl_2$, the lie algebra of 2x2 trace-0 matrices; further this 3-dimensional subalgebra satisfies some nice properties. This theorem has played a very crucial role in lie theory. The J-M theorem is true for any semi-simple lie algebra over an algebraically closed field of characteristic 0. However, over fields of positive characteristic the theorem is not true, in general. There have been several attempts towards formulating various analogues of the J-M theorem in positive characteristics, which would play the same role as its characterisitic 0 counterpart. In this talk, we look at certain analogues due to Kempf-Rousseau, Premet. The aim is to study the behaviour of these analogues with regard to morphisms. This would be stated more precisely during the talk and some results due to Hesselink, Jantzen, McNinch and others in this direction will be indicated. Various examples and counter-examples shall be discussed.