Patrick Lei | Other Notes

This page contains notes from UMass as well as short expositions that I wrote either for class or for talks. If you find any mistakes in the notes, please email me or submit an issue (or pull request) on the git repository. You can use the checkboxes to filter the notes by location and subject.

Filter: All alg-geo math-ph rep-thy alg/nt topology analysis tcs

Notes from UMass

singular spaces (S20)

topology

Course taught by Paul Gunnells.

Topics: singularities, Whitney stratification, intersection (co)homology, links of singular plane curves, symmetric spaces, compactifications.

symplectic topology (S20)

topology

Course taught by R. Inanç Baykur.

Topics: symplectic manifolds, almost complex structures, Kähler manifolds, geography of complex surfaces and symplectic 4-manifolds, isotopy, symplectic operations on 4-manifolds.

lie algebras (F19)

rep-thy

Course taught by Eric Sommers.

Topics: classification and representation theory of semisimple Lie algebras.

manifolds (F19)

topology

Course taught by Mike Sullivan.

Topics: smooth manifolds, vector fields, vector bundles, differntial forms, de Rham cohomology, Lie derivatives.

varieties (S19)

alg-geo

Course taught by Jenia Tevelev.

Topics: affine and projective varieties, blowups, normalization, local properties, divisors, differntial forms.

algebra 2 (S19)

alg/nt

Course taught by Paul Gunnells.

Topics: fields, Galois theory, commutative algebra.

analysis 2 (S19)

analysis

Course taught by Robin Young.

Topics: compactness, Banach and Hilbert spaces, Fourier theory.

representation theory (F18)

rep-thy

Course taught by Ivan Mirković.

Topics: representation theory of finite groups, Springer theory, representation theory of semisimple Lie algebras.

Warning: These notes were transcribed from handwritten notes taken during lecture.

Note: The instructor has more detailed notes here.

analysis 1 (F18)

analysis

Course taught by Robin Young.

Topics: Lebesgue measure, integration, differentiation, general measures.

Warning: These notes were transcribed from handwritten notes taken during lecture.

complex analysis (S18)

analysis

Course taught by Paul Hacking.

Topics: holomorphic functions, analytic continuation, singularities, Riemann mapping theorem, elliptic functions.

Warning: These notes were transcribed from handwritten notes taken during lecture.

Short expositions

Witten's conjecture

alg-geo
math-ph

Notes for a talk at the Columbia Undergraduate Math Society.

Abstract: In the 1990s, a remarkable correspondence was discovered between the geometry of algebraic curves and infinite-dimensional systems of differential equations. The correspondence has its origin in the study of two-dimensional quantum field theories and is related to many different areas of mathematics. After introducing the relevant objects, I will then state the first result in this story, which was conjectured by Witten and proved by Kontsevich.

lines on cubic surfaces

alg-geo

Notes for a talk at the Columbia Undergraduate Math Society.

Abstract: I will tell you why there are 27 lines on a cubic surface. Along the way, we will meet useful geometric notions such as Grassmannians, Schubert calculus, Chern classes, and intersection theory.

derived categories

alg-geo

Notes from introductory lectures by Rachel Webb and Tudor Pǎdurariu about Bondal-Orlov reconstruction and Fourier-Mukai transforms at the Derived categories and moduli workshop at Cornell.

complex cobordism

topology

Final paper for Algebraic Topology 2 course taught by Francesco Lin. Constructs the spectrum MU and computes its value on a point using the Adams spectral sequence.

artin l-functions

alg/nt

Final paper for Algebraic Number Theory course taught by Chao Li. Defines Artin L-functions and states the functional equation.

rational blowdowns

alg-geo
topology

Final paper for Symplectic Topology course taught by R. Inanç Baykur. Explains the construction of a simply connected surface of general type using rational blowdowns and its realization as a smoothing of a related surface.

permanent vs determinant

Final paper for Complexity Theory course taught by David Mix Barrington. Explains why the permanent is #P-hard.

geometric complexity theory

alg-geo
rep-thy
tcs

Notes for a talk at the UMass CS theory seminar. (Attempts to) explain geometric complexity theory to CS people.