Partial Differential Equations seminar
Speaker: Aritro Pathak, visiting fellow, TIFR Mumbai
Host: Harsha Hutridurga
Title: Uniform domains with Ahlfors-David regular boundaries, and uniform rectifiability.
Time, day and date: 11:30:00 AM – 12:30:00 PM, Wednesday, August 19
Venue: Ramanujan Hall
Abstract: Uniform rectifiability is a quantitative notion of rectifiability that is closely related to the (L^2) boundedness of Riesz transforms. Uniform domains satisfy quantitative openness and quantitative path connectivity conditions. When the Ahlfors-David regular boundary of a uniform domain is uniformly rectifiable, one obtains solvability of the Dirichlet problem for the Laplacian with rough boundary data, as well as quantitative absolute continuity of harmonic measure. We will discuss some recent results in this active area at the interface of geometric measure theory and harmonic analysis. In particular, we will present a new result concerning the Weak Geometric Lemma and uniform rectifiability for Ahlfors-David regular sets whose complements are uniform domains.

Mathematics Colloquium
Speaker: Prof(Retd.). Sadanand D. Agashe, Electrical Engineering, IIT Bombay
Host: Sivaji Ganesh Sista
Title: New Formulas and Results for 3-Dimensional Vector Fields
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, August 19
Venue: Ramanujan Hall
Abstract: New formulas are derived for once-differentiable 3-dimensional fields, using the operator (x d/dx + y d/dy + z d/dz). This new operator has a property similar to that of the Laplacian operator; however, unlike the Laplacian operator, the new operator requires only once-differentiability. A simpler formula is derived for the classical Helmholtz decomposition. Orthogonality of the solenoidal and irrotational parts of a vector field, the uniqueness of the familiar inverse-square laws, and the existence of solution of a system of first-order PDEs in 3 dimensions are proved. New proofs are given for the Helmholtz Decomposition Theorem and the Divergence theorem. The proofs use the relations between the rectangular-Cartesian and spherical-polar coordinate systems. Finally, an application is made to the study of Maxwell’s equations.
Keywords: Spherical-Polar Coordinates, Helmholtz Decomposition, Divergence Theorem, Orthogonality, Maxwell’s Equations

Talk
Speaker: Arkadev Ghosh, Chennai Mathematical Institute (CMI)
Host: Saurav Bhaumik
Title: GIT quotients of Schubert varieties and Hessenberg varietiesmodulo a one-dimensional torus
Time, day and date: 5:00:00 PM – 6:00:00 PM, Wednesday, August 19
Venue: Online (https://meet.google.com/imq-fzbd-vez)
Abstract: In this talk, we present some results on Geometric Invariant Theory (GIT) quotients of Schubert varieties and Hessenberg varieties in flag varieties, for the action of a one-dimensional torus.

We begin with GIT quotients of Schubert varieties in partial flag varieties, for ample line bundles linearized with respect to a one-parameter subgroup dual to a simple root. We establish a criterion for the existence of semistable points. In the case of the Grassmannian Gr(r,n), we identify the unique minimal-dimensional Schubert variety admitting semistable points. Further, we describe the corresponding GIT quotient as a projective space of matrices, with the natural action of a product of two special linear groups arising as the semisimple part of a Levi subgroup.

We then extend these techniques to study the action of a higher-dimensional subtorus on certain Schubert varieties in Gr(r,n), determined by the peak set of the associated permutation. Under a natural divisibility condition on r and n, we describe the semistable locus explicitly and show that the GIT quotient is an iterated projective space bundle over a projective space.

In another direction, we study the action of a one-parameter subgroup on the irreducible components of semisimple Hessenberg varieties associated with a diagonal matrix having two distinct eigenvalues. We show that, for suitable line bundles, the resulting GIT quotients are isomorphic to homogeneous fiber bundles over a flag variety whose fibers are products of projective lines. Furthermore, we describe the space of global sections of the descended line bundle as a module over a suitable Levi subgroup.