Title of Seminar: Infosys Chandrasekharan Random Geometry Colloquium
Title of Talk: Ratner's theorem on SL(2,R)-invariant measures
Speaker: Gaurav Aggarwal, TIFR, Mumbai
Date: September 2, 2024
Time: 15:00:00 Hours
Venue: A-369
Abstract: Let G be a Lie group, Γ a discrete subgroup of G, and H a connected subgroup of G generated by unipotent elements. A fundamental result by M. Ratner [Ann. of Math.] asserts that every ergodic H-invariant probability measure on G/Γ is the L-invariant volume on a closed orbit Lx of some subgroup L of G containing H. In this talk, we focus on the special case where H is isomorphic to SL(2,R). We will provide motivation and background to introduce key concepts for those not familiar with the area. The proof illustrates some of the important dynamical ideas involved in understanding the general case, but avoids many technical difficulties. This is based on a paper of M. Einsiedler, titled "Ratner's theorem on SL(2, R)-invariant measures"
Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Non-existence of phantoms on some rational surfaces
Speaker: Swarnava Mukhopadhyay, TIFR, Mumbai
Date: September 2, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: A phantom on a smooth projective variety is a non-trivial admissible subcategory of the derived category of coherent sheaves that is invisible to invariants like the Grothendieck group or Hochschild homology. There had been a general expectation that varieties admitting a full exceptional collection should not have phantoms however this was upended by the work of Efimov. More recently, Krah?s elegant example exhibits phantoms on the blowup of the projective plane at ten points in a general position. It is natural to wonder whether phantoms still exist if one takes the blowup at points that are not in general position. In this talk, following the recent work of Borisov-Kimboi (arXiv.2405.01683), we consider the above question for surfaces that are blowups of the projective plane at a finite set of points on a smooth cubic curve, in a very general position on the curve.
Title of Talk: Ratner's theorem on SL(2,R)-invariant measures
Speaker: Gaurav Aggarwal, TIFR, Mumbai
Date: September 2, 2024
Time: 15:00:00 Hours
Venue: A-369
Abstract: Let G be a Lie group, Γ a discrete subgroup of G, and H a connected subgroup of G generated by unipotent elements. A fundamental result by M. Ratner [Ann. of Math.] asserts that every ergodic H-invariant probability measure on G/Γ is the L-invariant volume on a closed orbit Lx of some subgroup L of G containing H. In this talk, we focus on the special case where H is isomorphic to SL(2,R). We will provide motivation and background to introduce key concepts for those not familiar with the area. The proof illustrates some of the important dynamical ideas involved in understanding the general case, but avoids many technical difficulties. This is based on a paper of M. Einsiedler, titled "Ratner's theorem on SL(2, R)-invariant measures"
Title of Seminar: Algebraic Geometry Preprint Seminar
Title of Talk: Non-existence of phantoms on some rational surfaces
Speaker: Swarnava Mukhopadhyay, TIFR, Mumbai
Date: September 2, 2024
Time: 16:00:00 Hours
Venue: AG-77
Abstract: A phantom on a smooth projective variety is a non-trivial admissible subcategory of the derived category of coherent sheaves that is invisible to invariants like the Grothendieck group or Hochschild homology. There had been a general expectation that varieties admitting a full exceptional collection should not have phantoms however this was upended by the work of Efimov. More recently, Krah?s elegant example exhibits phantoms on the blowup of the projective plane at ten points in a general position. It is natural to wonder whether phantoms still exist if one takes the blowup at points that are not in general position. In this talk, following the recent work of Borisov-Kimboi (arXiv.2405.01683), we consider the above question for surfaces that are blowups of the projective plane at a finite set of points on a smooth cubic curve, in a very general position on the curve.