"Na hi jñānena sadṛśaṃ pavitram iha vidyate" “To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of thought.”
-Bhagwat Geeta -Henri Poincaré
Welcome to my Math world:
I am interested in Algebraic geometry, more precisely in Birational Geometry in positive characteristic. In various fields of Mathematics, we are often interested in the classification of the structures in the given category based on the notion of equivalence in there. One of the main goals of Birational Geometry is to classify Algebraic varieties up to birational equivalence. In Birational Geometry, we are looking for more than just a class: a good representative for that particular class. For curves, normalisation provides an excellent representative. In dimension 2, the situation is easy to handle as we contract (-1)-curves, which gives us the "Minimal Model" for that class. However, in higher dimensions, the situation becomes more challenging to handle. That is when vanishing theorems and several other tools come to the rescue. Vanishing theorems form a foundational tool for running the Minimal Model Program. We have vanishing theorems in char 0. They fail in positive characteristics in general. Fano varieties and their log generalizations are expected to behave better in this setting, and it is conjectured that Kodaira-type vanishing theorems hold for Fano varieties of a given dimension n, provided the characteristic is sufficiently large, say p(n) (see Open Problems in the paper.) In my recent preprint, I have proved Kawamata-Viehweg vanishing theorem for surfaces of del Pezzo type over imperfect fields of characteristic p > 5.
Invited Talks and Poster Presentations:
"On Kawamata-Viehweg Vanishing theorem on surfaces of del Pezzo type over imperfect fields characteristic p>5" at the University of Utah (Upcoming on November 18th)
(Poster)"On Kawamata-Viehweg vanishing for surfaces of del Pezzo type over imperfect fields", Summer Research Institute in Algebraic Geometry 2025 at Colorado State University, Fort Collins, Colorado(July 2025)
(Poster) On Kawamata-Viehweg vanishing for surfaces of del Pezzo type over imperfect fields, Singularities in Algebra and Geometry at CIMAT, Guanajuato City, Mexico( June 2025)
On Kawamata-Viehweg vanishing for surfaces of del-Pezzo type over imperfect fields, Michigan Algebraic Geometry Symposium, University of Michigan, Michigan (April 2025)
Seminars organised:
Fall 2024 & Spring 2025: Algebraic Geometry Paper Reading Seminar at Michigan State University.
Fall 2023, Spring 2024: Co-organized the Student Algebra Seminar
Conference Attended:
Western Algebraic Geometry Symposium at the University of Oregon(November 2025)
Notions of Singularities in different characteristics at Banff International Research Center(October 2025)
Summer Research Institute in Algebraic Geometry 2025 at Colorado State University, Fort Collins, Colorado(July 2025)
Singularities in Algebra and Geometry at CIMAT, Guanajuato City, Mexico(June 2025)
Moduli of Surfaces at the University of Michigan, Michigan (May 2025)
Michigan Algebraic Geometry Symposium at the University of Michigan, Michigan (April 2025)
Moduli of Varieties at the University of Utah, Utah (November 2024)
Algebraic Geometry North Eastern Series(AGNES) 2024. (March 2024)
Connections Workshop: Commutative Algebra at SLMath, Berkeley, California. (January 2024)
Higher Dimensional Algebraic Geometry at the University of California, San Diego, California.(January 2024)
Michigan Algebraic Geometry Seminar(MAGS), Michigan State University, Michigan. (November 2023)
Western Algebraic Geometry Symposium, Washington University in St. Louis, Missouri.(November 2023)
Special Month in Ann Arbor (May 2023)
Michigan Algebraic Geometry Symposium (MAGS), University of Michigan, Michigan.(March 2023)
Algebraic Geometry and Singularities Learning Workshop and Conference, University of Washington, Seattle. (June 2022)