Background
I have been very passionate about mathematics since I was little. When I was 8 years old, I began taking undergraduate mathematics and physics courses at University of California, Davis, while simultaneously being homeschooled. I had finished most of the undergraduate courses when I was 11 years old, and moved to taking the graduate course sequences in algebra, topology, analysis, and algebraic geometry. I attended all of these courses in person, alongside regularly enrolled undergraduate and graduate students, and received official transcripts for most of them via the UC Davis Open Enrollment system (where my grades were given as they would be for any regularly enrolled student). Thereafter, when I was 14 years old (in 10th grade) and homeschooled, I moved to Harvard to work with Prof. Joe Harris on his book '3264 and all that'; I also attended graduate courses on Algebraic Number Theory and Quantum Field Theory. I joined Harvard as an undergraduate student the next year, and due to my preparation, skipped all undergraduate courses and started taking graduate courses in my first year; most of the mathematics courses that I took at Harvard were special topics courses and reading/research courses to build up my foundation in elliptic curves. In Fall 2022, I found a research problem in the area of Heegner points on elliptic curves, worked on it (under the guidance of Prof. Barry Mazur) for my senior thesis. I persisted on it after I graduated, which led to an arXiv paper this July (for more details on my research, see the research section).
Coursework
Harvard (courses taken while an undergraduate student)
Fall 2023
Advisor: Prof. Barry Mazur
Topics: Worked on writing senior thesis.
Topics: Worked on writing senior thesis.
Professor: Alex Cowan
Topics: Perron's formula, applications to sum-of-divisors functions, Mobius function, totient function, explicit formula for von Mangoldt function, automorphic forms, Eisenstein series, Fourier expansions, Hecke operators, Petersson inner product, Maass forms, space of modular forms for unfolding, shifted convolutions, spectral decomposition, spectral large sieve, modular periods.
Topics: Perron's formula, applications to sum-of-divisors functions, Mobius function, totient function, explicit formula for von Mangoldt function, automorphic forms, Eisenstein series, Fourier expansions, Hecke operators, Petersson inner product, Maass forms, space of modular forms for unfolding, shifted convolutions, spectral decomposition, spectral large sieve, modular periods.
Professor: Stephen McKean
Topics: Genera, cobordism, Thom spaces, modular forms, moduli stack of elliptic curves, derived algebraic geometry, ring spectra, elliptic cohomology, Witten genus, tmf.
Topics: Genera, cobordism, Thom spaces, modular forms, moduli stack of elliptic curves, derived algebraic geometry, ring spectra, elliptic cohomology, Witten genus, tmf.
Spring 2023
Advisor: Prof. Barry Mazur
Topics: Worked on senior thesis problem of finding Heegner points on the elliptic curve with equation for any prime satisfying appropriate conditions.
Topics: Worked on senior thesis problem of finding Heegner points on the elliptic curve with equation for any prime satisfying appropriate conditions.
Professor: Alexander Betts
Topics: Chabauty’s theorem, pro-unipotent étale fundamental groupoid, Tannakian formalism, Selmer schemes, non-abelian Chabauty method, application to the S-unit equation, quadratic Chabauty.
Topics: Chabauty’s theorem, pro-unipotent étale fundamental groupoid, Tannakian formalism, Selmer schemes, non-abelian Chabauty method, application to the S-unit equation, quadratic Chabauty.
Professor: Fabian Gundlach
Topics: Squarefree integers, Prime Number Theorem for arithmetic progressions, random polynomials, lattices, Minkowski’s first and second theorems, Davenport’s lemma, fundamental domains, class number formula, Iwasawa decomposition, binary quadratic forms, average size of class groups, counting number fields: general techniques, Malle’s conjecture, étale algebras, -adic integration, mass formulas, binary cubic forms, abelian extensions.
Final Paper: The Average Size of Selmer Groups of Elliptic Curves
Topics: Squarefree integers, Prime Number Theorem for arithmetic progressions, random polynomials, lattices, Minkowski’s first and second theorems, Davenport’s lemma, fundamental domains, class number formula, Iwasawa decomposition, binary quadratic forms, average size of class groups, counting number fields: general techniques, Malle’s conjecture, étale algebras, -adic integration, mass formulas, binary cubic forms, abelian extensions.
Final Paper: The Average Size of Selmer Groups of Elliptic Curves
Fall 2022
Advisor: Barry Mazur
Topics: Heegner points, Gross-Zagier theorem, examples for the curves . This is the course in which I found the research problem to take up for my senior thesis.
Topics: Heegner points, Gross-Zagier theorem, examples for the curves . This is the course in which I found the research problem to take up for my senior thesis.
Professor: Peter Kronheimer
Topics: Numerous techniques for understanding 4-manifolds: signature, cobordism, Gluck construction, Thurston norm, instanton Floer homology, foliations.
Topics: Numerous techniques for understanding 4-manifolds: signature, cobordism, Gluck construction, Thurston norm, instanton Floer homology, foliations.
Professor: Yuriy Drozd
Topics: Classification of vector bundles over the projective line, over elliptic curves, over nodal cubics, and over cuspidal cubics, bunches of chains, classification of Cohen-Macaulay modules, hypersurface singularities and minimal resolutions, Gorenstein and Bass rings.
Topics: Classification of vector bundles over the projective line, over elliptic curves, over nodal cubics, and over cuspidal cubics, bunches of chains, classification of Cohen-Macaulay modules, hypersurface singularities and minimal resolutions, Gorenstein and Bass rings.
Spring 2022
Advisor: Prof. Barry Mazur
Topics: The maximal abelian extension of an imaginary quadratic field, integrality of the -invariant, complex multiplication, heights on elliptic surfaces (and varieties in general), specialization theorem, group schemes, Néron models, special fibers.
Textbook: Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, §2.4-2.10, 3.9-3.12, 4.1-4.8
Topics: The maximal abelian extension of an imaginary quadratic field, integrality of the -invariant, complex multiplication, heights on elliptic surfaces (and varieties in general), specialization theorem, group schemes, Néron models, special fibers.
Textbook: Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, §2.4-2.10, 3.9-3.12, 4.1-4.8
Professor: Fabian Gundlach
Topics: Abel Summation Formula, Euler-Maclaurin formulas, convolution, Gauss circle problem, Dirichlet series, Wiener-Ikehara theorem and Kato’s extension, Dirichlet L-series, Prime Number Theorem for arithmetic progressions, Hadamard product formula, Riemann zeta function, Perron’s formula, Selberg sieve, large sieve, circle method, equidistribution.
Topics: Abel Summation Formula, Euler-Maclaurin formulas, convolution, Gauss circle problem, Dirichlet series, Wiener-Ikehara theorem and Kato’s extension, Dirichlet L-series, Prime Number Theorem for arithmetic progressions, Hadamard product formula, Riemann zeta function, Perron’s formula, Selberg sieve, large sieve, circle method, equidistribution.
Professor: Mark Shusterman
Topics: Projective morphisms, vector bundles, affine morphisms, finite morphisms, proper morphisms, normalization, Zariski’s Main Theorem, flat morphisms, faithfully flat descent, sheaf cohomology.
Topics: Projective morphisms, vector bundles, affine morphisms, finite morphisms, proper morphisms, normalization, Zariski’s Main Theorem, flat morphisms, faithfully flat descent, sheaf cohomology.
Spring 2021
Advisor: Prof. Noam Elkies
Topics: Height functions, Mordell-Weil theorem for elliptic curves and elliptic surfaces, minimal discriminants, integral points on elliptic curves, Siegel’s theorem, fibered surfaces, intersection numbers.
Textbook: Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Ch. 8-9; Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, §3.1-3.9
Topics: Height functions, Mordell-Weil theorem for elliptic curves and elliptic surfaces, minimal discriminants, integral points on elliptic curves, Siegel’s theorem, fibered surfaces, intersection numbers.
Textbook: Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Ch. 8-9; Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, §3.1-3.9
Professor: Laura DeMarco
Topics: Bilu’s equidistribution theorem, potential theory, complex dynamics, -adic dynamics, the Berkovich space over , Laplacian on finite graphs, adelic equidistribution theorem, global dynamics.
Topics: Bilu’s equidistribution theorem, potential theory, complex dynamics, -adic dynamics, the Berkovich space over , Laplacian on finite graphs, adelic equidistribution theorem, global dynamics.
Fall 2020
Professor: Melanie Wood
Topics: Class groups and binary quadratic forms, genus theory, Cohen-Lenstra heuristics for number and function fields, random matrix theory, Davenport-Heilbronn theorem, moments of random groups, counting number fields, topological component counting for class groups of function fields.
Topics: Class groups and binary quadratic forms, genus theory, Cohen-Lenstra heuristics for number and function fields, random matrix theory, Davenport-Heilbronn theorem, moments of random groups, counting number fields, topological component counting for class groups of function fields.
Spring 2020
Advisor: Prof. Barry Mazur
Topics: Elliptic curves over finite and local Fields, formal groups, torsion points, weak Mordell-Weil theorem.
Textbook: Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Ch. 4, 5, 7, §8.1-8.2
Topics: Elliptic curves over finite and local Fields, formal groups, torsion points, weak Mordell-Weil theorem.
Textbook: Silverman, The Arithmetic of Elliptic Curves, 2nd ed., Ch. 4, 5, 7, §8.1-8.2
Fall 2019
Professor: Peter Kronheimer
Topics: Contour integration, Cauchy's residue formula, Poisson summation formula, Hadamard product formula, gamma and zeta Functions, prime number theorem, Riemann surfaces, uniformization theorem, differential forms.
Topics: Contour integration, Cauchy's residue formula, Poisson summation formula, Hadamard product formula, gamma and zeta Functions, prime number theorem, Riemann surfaces, uniformization theorem, differential forms.
Harvard (courses taken while in 10th grade)
Spring 2019
Professor: Alison Miller
Topics: Global Class Field Theory, reciprocity laws, Brauer groups and central simple algebras, Dirichlet L-functions, Chebotarev density theorem, complex multiplication, elliptic functions, modular functions, imaginary quadratic fields with class number 1.
Topics: Global Class Field Theory, reciprocity laws, Brauer groups and central simple algebras, Dirichlet L-functions, Chebotarev density theorem, complex multiplication, elliptic functions, modular functions, imaginary quadratic fields with class number 1.
Professor: Joe Harris
Topics: Linear systems of varieties, compactification of parameter spaces, projective bundles, counting high-order contact points and flexes.
Textbook: Harris, 3264 and all that, Ch. 7-9, 11.
Topics: Linear systems of varieties, compactification of parameter spaces, projective bundles, counting high-order contact points and flexes.
Textbook: Harris, 3264 and all that, Ch. 7-9, 11.
Fall 2018
Professor: Alison Miller
Topics: Overview of Class Field Theory, valuations on fields, local fields, Hensel’s lemma, ramification, group cohomology, Galois cohomology, Hilbert’s Theorem 90, Local Class Field Theory, formal groups.
Final Paper: Bounding the rank of rational elliptic curves
Topics: Overview of Class Field Theory, valuations on fields, local fields, Hensel’s lemma, ramification, group cohomology, Galois cohomology, Hilbert’s Theorem 90, Local Class Field Theory, formal groups.
Final Paper: Bounding the rank of rational elliptic curves
Professor: Joe Harris
Topics: Chow rings, cycles in , Grassmannians, Schubert calculus, Chern classes, Hilbert and Fano schemes.
Textbook: Harris, 3264 and all that, Ch. 1-6.
Topics: Chow rings, cycles in , Grassmannians, Schubert calculus, Chern classes, Hilbert and Fano schemes.
Textbook: Harris, 3264 and all that, Ch. 1-6.
UC Davis (courses taken while in 9th grade and earlier)
Spring 2018
Professor: Brian Osserman
Topics: Divisors, divisor class group, Cartier divisors, invertible sheaves.
Textbook: Hartshorne, Algebraic Geometry, §2.6.
Topics: Divisors, divisor class group, Cartier divisors, invertible sheaves.
Textbook: Hartshorne, Algebraic Geometry, §2.6.
Winter 2018
Professor: Adrian Zahariuc
Topics: Sheaves, affine schemes, morphisms of schemes, finite, finite type, proper, separated, and integral morphisms, quasi-coherent sheaves and vector bundles.
Textbook: Hartshorne, Algebraic Geometry, §2.1-2.5.
Topics: Sheaves, affine schemes, morphisms of schemes, finite, finite type, proper, separated, and integral morphisms, quasi-coherent sheaves and vector bundles.
Textbook: Hartshorne, Algebraic Geometry, §2.1-2.5.
Fall 2017
Professor: Brian Osserman
Topics: Affine and projective varieties, morphisms, rational maps, nonsingularity, properties of curves, abstract varieties, complete varieties.
Textbook: Hartshorne, Algebraic Geometry, Ch. 1.
Topics: Affine and projective varieties, morphisms, rational maps, nonsingularity, properties of curves, abstract varieties, complete varieties.
Textbook: Hartshorne, Algebraic Geometry, Ch. 1.
Spring 2017
Professor: Eric Babson
Topics: Cohomology, universal coefficients and Künneth formulas, Eilenberg-Steenrod axioms, K(G,1)-spaces, cup and cap product, Poincare duality, obstruction theory.
Textbook: Hatcher, Algebraic Topology, Ch. 3, notes on obstruction theory.
Topics: Cohomology, universal coefficients and Künneth formulas, Eilenberg-Steenrod axioms, K(G,1)-spaces, cup and cap product, Poincare duality, obstruction theory.
Textbook: Hatcher, Algebraic Topology, Ch. 3, notes on obstruction theory.
Professor: Michael Kapovich
Topics: Tangent spaces, differential forms, manifolds, tangent bundle, vector fields, manifolds with boundary.
Topics: Tangent spaces, differential forms, manifolds, tangent bundle, vector fields, manifolds with boundary.
Winter 2017
Professor: Albert Schwarz
Topics: Higher homotopy groups, singular and simplicial homology, relative homology, examples and computations.
Textbook: Hatcher, Algebraic Topology, Ch. 2 and parts of Ch. 4.
Topics: Higher homotopy groups, singular and simplicial homology, relative homology, examples and computations.
Textbook: Hatcher, Algebraic Topology, Ch. 2 and parts of Ch. 4.
Professor: Andre Kornell
Topics: Ordered sets, nets, Banach spaces, Baire category theorem, Hahn-Banach extension theorem, weak* topology, Banach-Alaoglu theorem, Krein-Milman theorem, Hilbert spaces, operators, compact operators, trace and Hilbert-Schmidt operators, spectral theory.
Textbook: Pedersen, Analysis Now.
Topics: Ordered sets, nets, Banach spaces, Baire category theorem, Hahn-Banach extension theorem, weak* topology, Banach-Alaoglu theorem, Krein-Milman theorem, Hilbert spaces, operators, compact operators, trace and Hilbert-Schmidt operators, spectral theory.
Textbook: Pedersen, Analysis Now.
Fall 2016
Professor: John Sullivan
Topics: Fundamental groups, , Van Kampen's theorem, covering spaces.
Textbook: Hatcher, Algebraic Topology, Ch. 1.
Topics: Fundamental groups, , Van Kampen's theorem, covering spaces.
Textbook: Hatcher, Algebraic Topology, Ch. 1.
Professor: John Hunter
Topics: Metric spaces, normed spaces, continuous functions on metric spaces, -spaces, contraction mapping theorem, Banach spaces, Hilbert spaces.
Textbook: Hunter, Nachtergaele, Applied Analysis.
Topics: Metric spaces, normed spaces, continuous functions on metric spaces, -spaces, contraction mapping theorem, Banach spaces, Hilbert spaces.
Textbook: Hunter, Nachtergaele, Applied Analysis.
Spring 2016
Professor: Dmitry Fuchs
Topics: Homological algebra: chain complexes and homology, projective, injective, and free resolutions, derived functors, Tor, Ext, universal coefficients and Künneth formulas, category theory formulation of homological algebra, spectral sequences, sheaves, sheaf cohomology.
Topics: Homological algebra: chain complexes and homology, projective, injective, and free resolutions, derived functors, Tor, Ext, universal coefficients and Künneth formulas, category theory formulation of homological algebra, spectral sequences, sheaves, sheaf cohomology.
Winter 2016
Professor: Monica Vazirani
Topics: UFDs, Noetherian and Aritinian rings, Zorn's Lemma, varieties and the Nullstellensatz, categories and functors, projective, free, injective and flat modules, tensor products, representation theory, localization.
Textbook: Rotman, Advanced Modern Algebra, 2nd ed., Ch. 5-6, §7.1-7.5 and §10.2.
Topics: UFDs, Noetherian and Aritinian rings, Zorn's Lemma, varieties and the Nullstellensatz, categories and functors, projective, free, injective and flat modules, tensor products, representation theory, localization.
Textbook: Rotman, Advanced Modern Algebra, 2nd ed., Ch. 5-6, §7.1-7.5 and §10.2.
Fall 2015
Professor: Monica Vazirani
Topics: Isomorphism theorems, fundamental theorem of finite abelian groups, Sylow theorems, group presentations, rings, fields, ideals, Euclidean domains and PIDs, linear algebra over fields, finite fields, seperable field extensions, Galois theory.
Textbook: Rotman, Advanced Modern Algebra, 2nd ed., Ch. 1-4.
Topics: Isomorphism theorems, fundamental theorem of finite abelian groups, Sylow theorems, group presentations, rings, fields, ideals, Euclidean domains and PIDs, linear algebra over fields, finite fields, seperable field extensions, Galois theory.
Textbook: Rotman, Advanced Modern Algebra, 2nd ed., Ch. 1-4.
Professor: Javier Arsuaga
Topics: Random Variables, -algebras, probability distributions, measure theory, monotone convergence theorem, bounded convergence theorem, Fubini’s theorem, law of large numbers.
Textbook: Durrett, Probability: Theory and Examples, Ch. 1-2.
Topics: Random Variables, -algebras, probability distributions, measure theory, monotone convergence theorem, bounded convergence theorem, Fubini’s theorem, law of large numbers.
Textbook: Durrett, Probability: Theory and Examples, Ch. 1-2.
Spring 2015
Professor: John Hunter
Topics: Conformal transformations, Riemann mapping theorem, Arzela-Ascoli theorem, Riemann surfaces.
Topics: Conformal transformations, Riemann mapping theorem, Arzela-Ascoli theorem, Riemann surfaces.
Professor: Monica Vazirani
Topics: Random walks, Radon transform, Posets, Young diagrams, Young tableaux.
Textbook: Stanley, Algebraic Combinatorics, Ch. 1-6, 8.
Topics: Random walks, Radon transform, Posets, Young diagrams, Young tableaux.
Textbook: Stanley, Algebraic Combinatorics, Ch. 1-6, 8.
Winter 2015
Professor: Craig Tracy
Topics: Contour integration, gamma and zeta functions, prime number theorem, modular forms.
Textbook: Stein and Shakarchi, Complex Analysis, Ch. 2-7, 9.
Topics: Contour integration, gamma and zeta functions, prime number theorem, modular forms.
Textbook: Stein and Shakarchi, Complex Analysis, Ch. 2-7, 9.
Fall 2014
Professor: Kevin Luli
Topics: Groups, homomorphisms, Lagrange's theorem, quotient groups, symmetry groups, Sylow theorems.
Textbook: Artin, Algebra, Ch. 2, 3, 6, 7.
Topics: Groups, homomorphisms, Lagrange's theorem, quotient groups, symmetry groups, Sylow theorems.
Textbook: Artin, Algebra, Ch. 2, 3, 6, 7.
Spring 2014
Professor: Michael Kapovich
Topics: Point-set topology: topological spaces, bases, product, subspace and quotient topologies, limits and continuity, metric spaces, connected, compact, and path-connected spaces.
Textbook: Munkres, Topology, §12-26.
Topics: Point-set topology: topological spaces, bases, product, subspace and quotient topologies, limits and continuity, metric spaces, connected, compact, and path-connected spaces.
Textbook: Munkres, Topology, §12-26.
Professor: Dirk-Andre Deckert
Topics: Real analysis of functions in many variables, Banach spaces, Hilbert spaces, Inverse Function theorem.
Topics: Real analysis of functions in many variables, Banach spaces, Hilbert spaces, Inverse Function theorem.
Winter 2014
Professor: Qinglan Xia
Topics: Complex numbers, holomorphic and meromorphic functions, contour integration, Cauchy's residue theorem.
Textbook: Marsden, Hoffman, Basic Complex Analysis, Ch. 1-4.
Topics: Complex numbers, holomorphic and meromorphic functions, contour integration, Cauchy's residue theorem.
Textbook: Marsden, Hoffman, Basic Complex Analysis, Ch. 1-4.
Fall 2013
Professor: Dirk-Andre Deckert
Topics: Real analysis of functions in one variable, limits, differentiation.
Topics: Real analysis of functions in one variable, limits, differentiation.
Summer 2013
Professor: Matthew Reed
Topics: Real analysis of sequences and series.
Topics: Real analysis of sequences and series.
Spring 2013
Professor: Michael Kapovich
Topics: Integration, -substitution, surfaces of revolution, integration by parts, applications.
Topics: Integration, -substitution, surfaces of revolution, integration by parts, applications.
Professor: John Chuchel
Topics: Matrices, row and column reduction, trace and determinant, eigenvectors and eigenvalues.
Topics: Matrices, row and column reduction, trace and determinant, eigenvectors and eigenvalues.
Winter 2013
Professor: Greg Kuperberg
Topics: Gaussian integers, quadratic reciprocity, primitive roots, -adic numbers.
Topics: Gaussian integers, quadratic reciprocity, primitive roots, -adic numbers.
Fall 2012
Professor: Dmitry Fuchs
Topics: Fundamental theorem of arithmetic, congruences, Chinese remainder theorem, Fermat's little theorem, Hensel's lemma, Möbius inversion formula.
Topics: Fundamental theorem of arithmetic, congruences, Chinese remainder theorem, Fermat's little theorem, Hensel's lemma, Möbius inversion formula.
Spring 2012
Professor: Motohico Mulase
Topics:
Topics:
Winter 2012
Professor: Abigail Thompson
Topics: Counting problems, binomial coefficients, Pascal’s triangle, Fibonacci numbers, graph theory: graphs, trees, Eulerian walks, Hamiltonian cycles, perfect matchings, planar graphs and bipartite graphs, five-color theorem.
Textbook: Lovász, Pelikán, Vesztergombi, Discrete Mathematics: Elementary and Beyond, Ch. 1-13.
Topics: Counting problems, binomial coefficients, Pascal’s triangle, Fibonacci numbers, graph theory: graphs, trees, Eulerian walks, Hamiltonian cycles, perfect matchings, planar graphs and bipartite graphs, five-color theorem.
Textbook: Lovász, Pelikán, Vesztergombi, Discrete Mathematics: Elementary and Beyond, Ch. 1-13.