I am a postdoctoral fellow at the International Centre for Theoretical Physics in Trieste, Italy. My mentor is Prof. Alina Marian. I received my Ph.D. from the department of mathematics at the University of California San Diego in June 2023. My advisor was Prof. Dragos Oprea.
Here is my CV.
Research Area: Algebraic geometry; Enumerative Geometry; Moduli Theory; Algebraic combinatorics.
Email: ssinha1 at ictp.it
Office: LB-108, ICTP
Using a Quot scheme compactification, we calculate the virtual count of maps of degree d from a smooth projective curve of genus g to a hypersurface in a Grassmannian, sending specified points of the curve to special Schubert subvarieties restricted to the hypersurface. We study the question of whether this virtual count is in fact enumerative under suitable conditions on the hypersurface, in the regime when the map degree d is large.
We study the virtual Euler characteristics of sheaves over Quot schemes of curves, establishing that these invariants fit into a topological quantum field theory (TQFT) valued in Z[[q]]. Utilizing Quot scheme compactifications alongside the TQFT framework, we derive presentations of the small quantum K-ring of the Grassmannian. Our approach offers a new method for finding explicit formulas for quantum K-invariants.
We derive a K-theoretic analogue of the Vafa--Intriligator formula, computing the (virtual) Euler characteristics of vector bundles over the Quot scheme that compactifies the space of degree d morphisms from a fixed projective curve to the Grassmannian Gr(r,N). As an application, we deduce interesting vanishing results, used in Part I to study the quantum K-ring of Gr(r,N). In the genus-zero case, we prove a simplified formula involving Schur functions, consistent with the Borel-Weil-Bott theorem in the degree-zero setting. These new formulas offer a novel approach for computing the structure constants of quantum K-products.
We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial vector bundle, we obtain answers in genus zero. We also study the Euler characteristics of the symmetric powers of the tautological bundles, for rank zero quotients.
Isotropic Quot schemes parameterize rank r isotropic subsheaves of a vector bundle equipped with symplectic or symmetric quadratic form. We define a virtual fundamental class for isotropic Quot schemes over smooth projective curves. Using torus localization, we prescribe a way to calculate top intersection numbers of tautological classes, and obtain explicit formulas when r=2. These include and generalize the Vafa-Intriligator formula. In this setting, we compare the Quot scheme invariants with the invariants obtained via the stable map compactification.
For a positive integer t≥2, the t-core of a partition plays an important role in modular representation theory and combinatorics. We initiate the study of t-cores of partitions contained in an r×s rectangle. Our main results are as follows. We first give a simple formula for the number of partitions in the rectangle that are themselves t-cores and compute its asymptotics for large r,s. We then prove that the number of partitions inside the rectangle whose t-cores are a fixed partition ρ is given by a product of binomial coefficients. Finally, we use this formula to compute the distribution of the t-core of a uniformly random partition inside the rectangle extending our previous work on all partitions of a fixed integer n (Ann. Appl. Prob. 2023). In particular, we show that in the limit as r,s→∞ maintaining a fixed aspect ratio, we again obtain a Gamma distribution with the same shape parameter α=(t−1)/2 and rate parameter β that depends on the aspect ratio.
We show that the Bergman metric of the ball quotients B2/Γ, where Γ is a finite and fixed point free group, is Kähler-Einstein if and only if Γ is trivial. As a consequence, we characterize the unit ball B2, among 2 dimensional Stein spaces with isolated normal singularities, proving an algebraic version of Cheng's conjecture for 2 dimensional Stein spaces.
Fix t≥2. We first give an asymptotic formula for certain sums of the number of t-cores. We then use this result to compute the distribution of the size of the t-core of a uniformly random partition of an integer n. We show that this converges weakly to a gamma distribution after dividing by n−−√. As a consequence, we find that the size of the t-core is of the order of n−−√ in expectation. We then apply this result to show that the probability that t divides the hook length of a uniformly random cell in a uniformly random partition equals 1/t in the limit. Finally, we extend this result to all modulo classes of t using abacus representations for cores and quotients.
When molecular transitions strongly couple to photon modes, they form hybrid light-matter modes called polaritons. Collective vibrational strong coupling is a promising avenue for control of chemistry, but this can be deterred by the large number of quasi-degenerate dark modes. The macroscopic occupation of a single polariton mode by excitations, as observed in Bose-Einstein condensation, offers promise for overcoming this issue. Here we theoretically investigate the effect of vibrational polariton condensation on the kinetics of electron transfer processes. Compared with excitation with infrared laser sources, the condensate changes the reaction yield significantly due to additional channels with reduced activation barriers resulting from the large accumulation of energy in the lower polariton, and the many modes available for energy redistribution during the reaction. Our results offer tantalizing opportunities to use condensates for driving chemical reactions, kinetically bypassing usual constraints of fast intramolecular vibrational redistribution in condensed phase.