Summer Minicourses 2025

Abstract. We cover the basics of category theory: limits, representability, the Yoneda Lemma, functors, natural transformations, adjunctions, and related concepts, with a focus on examples from topology and algebra. A secondary goal is to serve as background for Sai’s homological algebra course.

Abstract. (Tentative) We will learn about basic ideas in homalg the first week, (think topics from the first four chapters of weibel) and then we can do spectral sequences and or derived categories the next week depending on what people are interested in.

Abstract. Data Science is the study of extracting meaningful insights from data. It combines a variety of disciplines including mathematics, statistics, and computer science. Data science is a rich field and covers a variety of tasks including data storage and management, data engineering, machine learning, data visualization, and more. In this course we will focus on using Machine learning to make predictions or uncover patterns in data that can be translated into real world ideas. Machine learning itself is also vast and rich field so we will focus mainly on supervised and unsupervised machine learning algorithms and how they can be applied to tabular data. Though during our study, we may discuss ways that machine learning can be used to process text and images. The purpose of the course is 2-fold. We will spend the first half learning a general overview of the mathematics behind popular machine learning algorithms and data processing techniques. In the second part we will work in groups to implement what we’ve learned to analyze a data set.

Abstract. Structure theory of complex semisimple Lie algebras, Dynkin diagrams, classification of these, real forms, Cartan involution, maximally compact and non-compact Cartan subalgebras, Vogan diagrams, classification of real simple Lie algebras.

Abstract. Sheaf categories on spaces such as the flag variety and affine Grassmannian encode rich representation-theoretic structures, making sheaf theory an essential tool in the study of geometric representation theory. This minicourse will focus on explicit computations involving sheaves, each situated in important contexts such as the Kazhdan–Lusztig conjecture, the geometric Satake equivalence, and Soergel’s conjecture. The course has two main objectives:

1.To make the abstract formalism of sheaf theory more approachable by working through concrete, computation-based examples.

2.To provide exposure to foundational results in geometric representation theory by situating each computation within its relevant representation-theoretic context.

Although several types of sheaves arise in representation-theoretic applications, this course will focus exclusively on constructible and perverse sheaves.

Abstract. Going to try and cover the relevant topics in the first two chapters of "Spin Geometry" by Lawson and Michelsohn with examples as they relate to Seiberg-Witten theory. .