# Shubham Sinha

Email: shs074 at ucsd dot edu
Office: AP&M 5760, UC San Diego

I am a fourth-year Ph.D. student at the mathematics department at the University of California San Diego working with Prof. Dragos Oprea. I graduated from the Indian Institue of Science with a bachelor's degree in Mathematics in 2017. Here is my CV.

### Research Interest

I am interested in enumerative problems in Algebraic geometry, such as Moduli spaces parameterizing vector sub-bundles and Gromov-Witten Invariants. I am also interested in problems in Algebraic combinatorics.

### Papers

• The virtual intersection theory of isotropic Quot Schemes.[arXiv] (2021)
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.
• The size of t-cores and hook lengths of random cells in random partitions.[arXiv] (2019)
(with Arvind Ayyer) : Fix $t \geq 2$. 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 $\sqrt{n}$. As a consequence, we find that the size of the $t$-core is of the order of $\sqrt{n}$ 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.
• [2'] Random t-Cores and Hook Lengths in Random Partitions (2020)
Journal ref : Sém. Lothar. Combin. 84B (2020), Art. 58, 11 pp. 05A17 (05A15) [pdf]
• Driving chemical reactions with polariton condensates.[arXiv] (2021)
(with Sindhana Pannir-Sivajothi, Jorge A. Campos-Gonzalez-Angulo, Luis A. Martínez-Martínez, Joel Yuen-Zhou (1)) : 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.
• ### Research Talks

IMSc Algebraic Combinatorics Seminar : June 18, 2020;[Link] : Random t-cores and hook lengths in random partitions.

FPSAC 2020 : July 15,2020;[FPSAC] : Random t-cores and hook lengths in random partitions.
[Video] [Slides]

### Teaching:

I will be teaching Math 10B (Calculus) in Summer Session-2 2021 at UC San Diego. Here is the course webpage : Math 10B.

### Outreach

Olympiad : I was involved in training Indian IMO (International Mathematical Olympiad) team during the years 2014-2017 and 2021.

Mentoring at UCSD : I mentored a group of undergraduate students at UCSD, in Spring 2020, as part of UCSD's RTG grant. We discussed basics of algebraic geometry and classical theorem : '27 lines on a cubic surface'.

### Teaching Assistant:

I was a teaching assistant for the following courses:

1. Spring 2021 - Math 220C Complex Analysis ; Math 154 Discrete Math & Graph Theory
2. Winter 2021 - Math 220B Complex Analysis ; Math 103A Modern Algebra
3. Fall 2020 - Math 220A Complex Analysis
4. Spring 2020 - Math 20E Vector Calculus
5. Winter 2020 - Math 103B Modern Algebra
6. Fall 2019 - Math 103A Modern Algebra
7. Spring 2019 - 20E Vector Calculus
8. Winter 2019 - 104B Number Theory and 20A Calculus
9. Fall 2018 - 104A Number Theory
10. Spring 2018 - 184 Enumerative Combinatorics
11. Winter 2018 - 154 Discrete Math & Graph Theory
12. Fall 2017 - 10A Calculus