Algebraic Groups Seminar

 Wednesday, 08/03/2023, 10.30 am

Venue: https://meet.google.com/jcn-nwpx-nmq

Host:  Shripad M. Garge

Speaker: Arpita Nayek

Affiliation: IIT Bombay

Title: On the torus quotients of Schubert varieties in Grassmannians

Abstract: Let G=SO(8n, C) or SO(8n+4, C) and T be a maximal torus of G. Let P be the maximal parabolic subgroup of G corresponding to the simple root \alpha_{4n} (respectively, \alpha_{4n+2}). In the first part of the talk, we will discuss the projective normality of the GIT quotients of certain Schubert varieties in G/P with respect to a T-linearized very ample line bundle on G/P. Let G_{r,n} be the Grassmannian of all r-dimensional subspaces of C^n. For r and n coprime, let X(w_{r,n}) be the unique minimal dimensional Schubert variety in G_{r,n} admitting semi-stable points. Let X^v_{w_{r,n}} be the Richardson variety in G_{r,n} corresponding to the Weyl group elements v and w_{r,n}. In the second part of the talk, we will discuss the sufficient conditions on v such that the GIT quotient of X^v_{w_{r,n}} is the product of projective spaces. The first part of my talk is based on a joint work with Pinakinath Saha and the second part of my talk is based on a joint work with Somnath Dake and Shripad Garge.

Algebraic Geometry Seminar

Wednesday, 8 March 2023,  12.00pm
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Venue: Ramanujan Hall, Department of Mathematics

Host: Sudarshan Gurjar

Speaker: Nitin Nitsure

Affiliation: TIFR, Mumbai (retd)

Title: Local Criterion of Flatness-II

Analysis Seminar

Wednesday, 08/03/2023, 4 pm

Host: B. K. Das

Venue: Ramanujan Hall

Speaker: Haripada Sau

Affiliation:  IISER Pune

Title: Multivariable Operator Theory: Rational Dilation,

         Realization Formula, and Distinguished Varieties.

Abstract: In 1951, John von Neumann proved a pathbreaking inequality involving contractive linear transformations acting on Hilbert spaces and bounded analytic functions on the unit disk. The inequality -- now referred to as the von Neumann inequality -- has an extraordinary influence on the development of operator theory. A couple of years later, a simpler proof of the von Neumann inequality emerged when Sz.-Nagy proved his dilation theorem - the genesis of the dilation theory. Sz.-Nagy's discovery prompted several mathematicians to consider a multivariable generalization of the classical dilation theory. This is called the rational dilation problem -- the Holy Grail of spectral theory. In this talk, we shall discuss in brief the rational dilation problem corresponding to the unit disk, the bidisk, the symmetrized bidisk, and the tetrablock. We shall also discuss in brief a realization formula for bounded analytic functions on the disk, the bidisk, and the symmetrized bidisk. We shall see how a special class of algebraic varieties - the so-called distinguished varieties - becomes a natural object of study in the context of multivariable von Neumann inequality. A new characterization of such varieties will be presented. The last part of the talk will be about an ongoing project concerning a constrained two-variable dilation problem.