Welcome

It is by logic that we prove, but by intuition that we discover. ---- Henri Poincaré

Channel Islands National Park, by Yueling Zeng.

About Me

I am Yusen Xia (in Chinese: 夏雨森, which reads "hsia yü sen "). I’m a 4th year PhD student at Mathematics Department, UC Santa Barabra. I got my BSc in Mathematics degree (with First Class honors) from Hong Kong University of Science & Technology (HKUST). I was an exchange student at UCLA from Winter to Spring 2020.

My CV is here.

Research Interest

I am currently insterested in Geometric Analysis, in particular Eigenvalue Problems of Laplace Operator on manifolds and Fundamental Groups of manifolds with nonegative Ricci Curvature.

Contact:

Email: yusen@ucsb.edu

Address

Department of Mathematics, University of California, Santa Barbara

552 University Rd, South Hall, Santa Barbara, CA 93106

Course Notes

Undergraduate at HKUST

Tensor Calulus , an evolving short notes on basics of tensors based on Lecture Notes on Diff. Geo. by Prof Peter Petersen, Lecture notes on Diff. Mnflds. & Rie. Geo. by Prof Fong Tsz Ho, YouTube on tensor calculus.
Surface Theory in \mathbb{R}^3. These notes are about classical differential geometry, based on my review notes of MATH 120A UCLA. References are Lecture Notes on Diff. Geo. by Prof Peter Petersen, Lecture notes on Diff. Mnflds. & Rie. Geo. by Prof Fong Tsz Ho, ICTP’s video on Diff. Geo. and Elementary Diff. Geo. by Andrew Pressley.
An incomplete elementary notes on differential manifolds. It covers tangent & cotangent bundle, differential forms and Lie derivatives.
Real analysis. These are notes I took down in MATH 5011 Advanced Analysis at HKUST in Fall 2019. It covers Lebesgue Integration theory, Lpspaces, Lebesgue Diff. Thm. and Fubini Thm. Egorov, Luzin Thm and basics of Hausdorff measure at the level of Year 1 Graduate course.
PDEs. In Spring 2021, I am taking a course on PDEs taught by Prof Gunther Uhlmann at University of Washington, Seattle. These notes are what I dropped down in class. He started with theory of distributions and Fourier transforms. Then he covered Classical PDEs and Dirichlet Problems via theory of H^1 space.
Functional Analysis. These notes are based on a course on YouTube channel of IMPA, taught by Prof Claudio Landim. I am currently studying it hence new notes are uploaded periodically. It covers Vector Space, Normed Space, Examples of Normed Space, Finite Dimensional Space, Infinite Dimensional Space…
PDEs. These notes are from MATH6050H, Fall 2020 taught by Prof Li Dong, HKUST. It covers Chapter 5 (Sobolev Space) & 6 (Elliptic PDEs) of Evans’ PDE.
PDE course of 2020 & 2021 Summer Schools on Differential Geometry, Peking University. It covers the Schauder Estimates of elliptic linear PDE of 2nd order. Reference book: Elliptic Partial Differential Equations of Second Order, David Gilbarg & Neil S. Trudinger.
Elementary ODE class. Midterm Review and Linear Systems.

Graduate at UCSB

MATH 241A Topics in Differential Geometry, Fall 2021. This course is on comparison theorems in Riemannian Geometry, taught by Prof Guofang Wei. Here is my hand-written notes. Here is the typed notes contributed by the whole class.
MATH 220B Commutative Algebra by Prof Francesc Castella. Here are the notes after in-person teaching resumed (hence incomplete), which covers Noetherian modules, localization of modules, tensor products, flat modules. We covered Finitely Generated Modules over PID and Irreducibility of polynomial ring as well.
MATH 240C Differential forms and Hodge theory by Prof Rugang Ye.
MATH 241C, Spring 2022. Title: Minimal Surface, Scalar Curvature and Spin. Instructor: Prof Xianzhe Dai. Handwritten. Typed notes (contributed by the whole class). We talked about stable minimal submanifold, geometry of positive scalar curvature, Witten's proof of Positive Mass Theorem, Penrose inequality etc.
Useful formulae & Theorems in geometric analysis. This is a long-term dairy of mine. I basically drop down all formulae that might be useful one day. Updated it regularly.
Lectures on Gromov Hausdorff Convergence and Gromov's Precompactness Theorem. Part of the notes for MATH 241B, 2023 Winter.