**Info**

- I am an Associate Professor at Columbia University. I graduated from Harvard under the supervision of Professor Benedict Gross in 2015.
- My research interests: Number Theory, Arithmetic Geometry, Automorphic Forms.
- Address:
- Math Building 614, MC4430
- Department of Mathematics, Columbia University
- 2990 Broadway
- New York, NY 10027

- Email:
- I am on sabbatical leave in 2024.

**Research**

- Arithmetic Fundamental Lemma for the spherical Hecke algebra (with Michael Rapoport, Wei Zhang),
*preprint* - Geometric and arithmetic theta correspondences

Lectures at the IHES 2022 Summer School on the Langlands Program

*Proc. Symp. Pure. Math*, to appear - Degrees of unitary Deligne-Lusztig varieties

Contribution to the Kudla 70 volume

*Fields Inst. Commun.*, to appear - A proof of Kudla-Rapoport conjecture for Kramer models (with Qiao He, Yousheng Shi, Tonghai Yang)

*Invent. Math.*, 234 (2023), no. 2, 721-817. doi:10.1007/s00222-023-01209-1 - A note on Tate's conjectures for abelian varieties (with Wei Zhang)

*Essent. Number Theory*, 1 (2022), no. 1, 41-50. doi:10.2140/ent.2022.1.41 - From sum of two squares to arithmetic Siegel-Weil formulas (survey)

*Bull. Amer. Math. Soc.*, 60 (2023), no. 3, 327-370. doi:10.1090/bull/1786 - On the arithmetic Siegel-Weil formula for GSpin Shimura varieties (with Wei Zhang)

*Invent. Math.*, 228 (2022), no. 3, 1353-1460. doi:10.1007/s00222-022-01106-z - Chow groups and L-derivatives of automorphic motives for unitary groups, II (with Yifeng Liu)

*Forum Math. Pi*, 10 (2022), e5, 77pp. doi:10.1017/fmp.2022.2 - Chow groups and L-derivatives of automorphic motives for unitary groups (with Yifeng Liu)

*Ann. of Math.*, 194 (2021), no. 3, 817-901. doi:10.4007/annals.2021.194.3.6 - Kudla-Rapoport cycles and derivatives of local densities (with Wei Zhang)

*J. Amer. Math. Soc.*, 35 (2022), no. 3, 705-797. doi:/10.1090/jams/988 - Proof of (Fourier-Jacobi) arithmetic fundamental lemma for the minuscule case (with Yihang Zhu)

An appendix to Yifeng Liu, Fourier-Jacobi cycles and arithmetic relative trace formula

*Camb. J. Math.*, 9 (2021), No. 1, 90-95. doi:/10.4310/CJM.2021.v9.n1.a1 - Fine Deligne-Lusztig varieties and arithmetic fundamental lemmas (with Xuhua He and Yihang Zhu)

*Forum Math. Sigma*, 7 (2019), e47, 55pp. doi:/10.1017/fms.2019.45 - On the 2-part of the BSD conjecture for quadratic twists of elliptic curves (with Li Cai and Shuai Zhai)

*J. Lond. Math. Soc.*, 101 (2020), No. 2, 714-734. doi:10.1112/jlms.12284 - Recent developments on quadratic twists of elliptic curves (survey)

*Proceedings of the ICCM 2017, First Annual Meeting*, 381-400. ISBN 9781571463920 - Prime twists of elliptic curves (with Daniel Kriz)

*Math. Res. Lett.*, 26 (2019), No. 4, 1187-1195. doi:10.4310/MRL.2019.v26.n4.a10 - Goldfeld's conjecture and congruences between Heegner points (with Daniel Kriz)

*Forum Math. Sigma*, 7 (2019), e15, 80pp. doi:10.1017/fms.2019.9 - Arithmetic intersection on GSpin Rapoport-Zink spaces (with Yihang Zhu)

*Compos. Math.*, 154 (2018), No. 7, 1407-1440, doi:10.1112/S0010437X18007108 - Remarks on the arithmetic fundamental lemma (with Yihang Zhu)

*Algebra Number Theory*, 11 (2017), No. 10, 2425-2445, doi:10.2140/ant.2017.11.2425 - 2-Selmer groups, 2-class groups and rational points on elliptic curves

*Trans. Amer. Math. Soc.,*371 (2019), 4631-4653, doi:10.1090/tran/7373 - Level raising mod 2 and obstruction to rank lowering

*Int. Math. Res. Not. IMRN*(2019), no. 8, 2332-2355. doi:10.1093/imrn/rnx188 - Level raising mod 2 and arbitrary 2-Selmer ranks (With Bao V. Le Hung)

*Compos. Math.*152 (2016), no. 8, 1576-1608, doi:10.1112/S0010437X16007454

**Teaching**

- I taught Math 4044 (Representations of finite groups), Fall 2022.
- I taught a minicourse at Elliptic Curves 2022 Summer School, Baskerville Hall.
- Lecture 1 BSD Conjecture, Modularity Theorem, Gross-Zagier-Kolyvagin Theorem
- Lecture 2 Gross-Zagier formula for X0(N), applications to the BSD conjecture and the Gauss class number problem
- Lecture 3 Gross-Zagier formula for Shimura curves, Waldspurger's formula, a proof outline of the Gross-Zagier formula

- I taught Math 6657 (Class field theory), Spring 2022
- I taught Math 4044 (Representations of finite groups), Fall 2021
- I taught Math 6657 (Class field theory), Spring 2021.
- I taught Math 4044 (Representations of finite groups), Fall 2020
- I taught Math 6657 (Class field theory), Spring 2020.
- I taught Math 1101 (Calculus I), Fall 2019.
- I taught Math 6657 (Class field theory), Spring 2019.
- I taught Math 8674 (Arithmetic of L-functions), Fall 2018. Course Notes by Pak-Hin Lee.
- I taught Math 1101 (Calculus I), Fall 2017 and Spring 2018.
- I taught Math 6657 (Class field theory), Spring 2017.
- I taught Math 1101 (Calculus I) and Math 4043 (Algebraic number theory), Fall 2016.
- I taught Math 1101 (Calculus I), Fall 2015 and Spring 2016. (Columbia Departmental Teaching Award)
- I CAed Math 123 (Theory of rings and fields) and Math 223b (Class field theory 2), Spring 2015. (Harvard Certificate of Distinction in Teaching)
- I taught Math 21a (Multivariable calculus), Fall 2013 and Fall 2014. (Harvard Certificate of Distinction in Teaching)
- I CAed Math 223b (Class field theory 2), Spring 2013. (Harvard Certificate of Distinction in Teaching)
- I CAed Math 223a (Class field theory), Fall 2012. (Harvard Certificate of Distinction in Teaching)
- I taught Knots and Primes, Summer 2012.
- I taught Math 21a (Multivariable calculus), Fall 2011.

**Expository Notes**

- By clicking the orange links below, you solemnly swear to send me any mistakes you find.

- Moonshine and the BSD conjecture (a Student Mathematics Colloquium talk)
- Basic functions and the arc space of L-monoids
- Quadratic polynomials and modular forms (a UMS talk)
- Higher Gross-Zagier formulas: Cohomological spectral decomposition and finish the proof
- Kato's explicit reciprocity laws
- The Hodge-Tate period map and the cohomology of Shimura varieties
- Vector bundles on the Fargues-Fontaine curve

- Lusztig's classification of irreducible representations of reductive groups over finite fields
- K2 and L-functions of elliptic curves (a Trivial Notions talk)
- Mixed Tate motives, algebraic K-theory and multiple zeta values
- Applications of Deligne's Weil II
- Kolyvagin's conjecture
- Almost mathematics
- Deligne-Lusztig curves (a Trivial Notions talk)
- What is the Tate conjecture?
- Basic Dieudonne theory
- Eisenstein descent and rational points on modular curves
- Mumford curves (a Trivial Notions talk)
- What is the Birch and Swinnerton-Dyer conjecture?
- Elliptic Surfaces and Mordell-Weil Lattices
- Heights and Finiteness theorems of abelian varieties
- Enriques classification of complex algebraic surfaces
- Stable vector bundles on curves
- Shimura Curves (a Trivial Notions talk)
- Endomorphism rings of elliptic curves and singular moduli — my minor thesis:

**Course Notes**

- Relative trace formulae and the Gan-Gross-Prasad conjectures
- Geometric aspects of p-adic Hodge theory
- Bloch-Kato conjectures for some polarized motives
- Langlands correspondence for general reductive groups over function fields
- Perverse sheaves and fundamental lemmas

- Intersection theory in algebraic geometry
- Galois Representations
- Pure Motives and Rigid Local Systems
- Topics in Automorphic Forms
- Class field theory: proofs
- Class field theory
- Abelian varieties
- Representation theory and number theory
- Perverse sheaves in representation theory
- Discrete subgroups of Lie groups and discrete transformation groups
- Arithmetic surfaces and successive minima
- Teichmuller Theory