Undergraduate Research Mini-course in Complex Variables and Complex Geometry

Practical Information:

Title: UCSD Undergraduate Research Mini-course in Complex Variables and Complex Geometry 

Dates: Monday, July 21, 2025 -  Saturday, July 26, 2025 

Description: This program aims to bring together undergraduate mathematics students at UCSD and nearby universities in Southern California for a six-day workshop on complex variables and geometry.  The goal of the workshop is to provide undergraduates with a platform where they can receive early preparation for graduate-level math, gain exposure to potential REU topics, interact with other students interested in Analysis and Geometry, and build connections with faculty that may lead to further study or collaborations.  The workshop will consist of lectures by faculty members, discussion sections under the supervision of a graduate student TA, and student presentations to be given at the end of the course. The target audience are those who have completed UCSD’s Math 140 sequence or equivalent (undergraduate real analysis at the level of Walter Rudin’s book Principles of Mathematical Analysis) and Math 120AB (the undergraduate complex analysis sequence). The program is supported by NSF grant DMS-2045104.

Lecture Series:  The two lead instructors Dr. Yunus Zeytuncu and Dr. Lihan Wang will lead two complementary lectures series to run approximately the first four to five mornings of the program.  A detailed description of the topics of their lectures can be found below.  Information about the instructors can be found at the bottom of the page.

Organizers:  Dr. Ming Xiao and Dr. John Treuer.  Biographical sketches of the organizers can be found at the bottom of the page.

Questions: Please contact the organizers Ming Xiao and John Treuer at the emails listed below

Emails: m3xiao AT ucsd DOT edu and jtreuer AT ucsd DOT edu

Application: Please fill out an application here through Ideal-Logic.  Application deadline is 2/28/25.

Lecture series

Lecture Series A: Spectral Theory and Geometry: The Kohn Laplacian
Yunus Zeytuncu

The spectrum of an operator reveals key information about both the operator and the underlying domain or manifold. In this lecture series, we will explore the Kohn Laplacian, a central operator in the analysis of CR manifolds and complex domains. We will examine how its spectral properties connect to geometric structures, drawing insights from results like Weyl's law and recent advances in the field.

Designed for participants with a background in linear algebra, real analysis, and complex analysis, this series will showcase the Kohn Laplacian’s role in bridging operator theory and geometry.

Lecture Series B: Eigenvalues and Structures of Kähler Manifolds at Infinity
Lihan Wang

How do eigenvalues reveal the shape and structure of a manifold? Inspired by Mark Kac’s famous question, “Can one hear the shape of a drum?”, this minicourse explores how the spectral properties of a manifold are linked to its geometry, focusing on Kähler manifolds and their behavior at infinity.

We will introduce key techniques such as comparison theorems and eigenvalue estimates, then apply them to the study of non-compact Kähler manifolds. Designed for students with a background in real and complex analysis, this series offers an introduction to the fascinating connections between spectral theory and geometric analysis.