Algebraic Geometry Seminar

Monday 30th Jan, 11:30 am

Venue: Ramanujan hall

Host: Sudarshan Gurjar

Speaker: Arusha

Affiliation: TIFR, Mumbai

Title: Poincare and Picard bundles for moduli spaces of vector bundles over nodal curves

Abstract: Poincare and Picard bundles and their different variants have been a topic of interest ever since the quest for moduli spaces of vector bundles was initiated, owing to their universality. Though a great deal is known about these objects in the case of smooth curves, the study on singular curves has been relatively slow. Interestingly, the results for irreducible nodal curves are very similar to those for smooth curves; however, the proofs are different and difficult. It was known since late 1960s that there does not exist a Poincar´e bundle (a universal family) for the moduli problem of vector bundles on smooth curves if the rank and degree are not coprime. The primary aim of the talk is to discuss the non-existence of a Poincare bundle in the non-coprime case for nodal curves. There has also been ample interest to understand the stability of Poincar´e and projective Poincare bundles as well as Picard and projective Picard bundles. The secondary aim is to discuss the stability of projective Poincar´e and Picard bundles, again when the degree and rank are not relatively prime to each other in the context of nodal curves. On the way to achieve these goals, we compute the codimension of a few closed subsets of the moduli spaces. They are of independent interest and have other applications; we discuss a few of them. This is a joint work with Prof. Usha Bhosle and Dr. Sanjay Singh.

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PDE seminar

Monday 30 January, 02:15 PM to 03:45 PM

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Host: Harsha Hutridurga
Venue: Ramanujan Hall, Department of Mathematics

Speaker: Bishnu Prasad Lamichhane
Affiliation: University of Newcastle, UK


Title: A finite element method for a biharmonic equation using biorthogonal systems.
Abstract: In this talk we will discuss applications of biorthogonal systems in a finite element method for the biharmonic equation with clamped and simply supported boundary conditions. We also discuss the construction of biorthogonal systems and their approximation properties.

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Lecture series on Lie groups 

Date and Time: Six Mondays at 4 pm
Tea: 3.50 pm
Venue: A1-A2, CDEEP, Mathematics Department
Host: Dipendra Prasad

Speaker: M. S. Raghunathan, CEBS, Mumbai

Title: Compact Lie groups and their representations

Abstract: In this course I will first talk about the structure theory of compact Lie groups, beginning with the fact that a compact connected Lie group is an almost direct product of the identity connected component of its centre and its commutator subgroup (which is closed subgroup) conjugacy of maximal tori and the fact that every element is contained in a maximal torus. In the course of proving these results, some results on the topology of compact Lie groups which will also be proved. I will then establish Weyl's theorem which asserts that if G is a compact connected Lie group and [G, G]=G, π_1(G,e) is finite (and hence the universal covering of a compact group whose abelianisation is trivial is  compact. 

Then I will introduce roots and weights and the Dynkin diagram of the compact group and sketch a proof of the fact that the Dynkin diagram determines the group locally. The remaining lectures will be devoted to representation theory. I will establish the bijective correspondence between 'Dominant Weights' and irreducible representations. The course will end with the Weyl Character Formula for the character of an irreducible representation corresponding to a 'dominant' weight. The entire theory is essentially the same as the representation theory of reductive algebraic groups. I will off and on indicate how the two are related.  

I will be assuming some familiarity with basic theory of Lie groups such as the correspondence between Lie sub-algebras of the Lie group and Lie subgroups of the Lie groups, also with some basic results from algebraic topology.