# 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

(1) The size of t-cores and hook lengths of random cells in random partitions.[arXiv]
(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.

(1)' . Random t-Cores and Hook Lengths in Random Partitions
Journal ref : Sém. Lothar. Combin. 84B (2020), Art. 58, 11 pp. 05A17 (05A15) [pdf]

### 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]

### Outreach

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

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:

I am a TA for the gradutae course Complex Analysis (Math 220B) and Discrete Math & Graph Theory (Math 154).

Here is the course website : Math 220C

I was a teaching assistant for the following courses:

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