# Intensive Training in Algebra, Analysis, Topology, and Applications May 27 -- June 01, 2024

## Objectives

1. Review and build upon the fundamentals of higher mathematics to develop research skills.
2. Train research scholars for problem-solving in advanced mathematics (Develop problem-solving and thinking skills in advanced mathematics).
3. Support in preparation for NET/SLET examinations.

## Topics

1. Analysis
2. Algebra
3. Topology

## Syllabus Outline

### Analysis

1. Continuity, types of discontinuity, oscillation, Cauchy continuity, uniform continuity.
2. Functions of several variables, directional derivative, partial derivative, derivative as a linear transformation, inverse and implicit function theorems.

### Algebra

1. Group Theory -- Groups, homomorphisms, subgroups and quotients, Sylow's theorems, direct and semidirect products, composition series, nilpotent groups.
2. Ring Theory -- Rings, homomorphisms, ideals, ED, PID, UFD.
3. Category Theory -- Categories, universal objects, functors, natural transformations, Yoneda lemma.

### Topology

1. A quick revision of topological and metric spaces, meagre sets, Baire spaces.
2. Choquet games and spaces, the Banach-Mazur game, applications of games to Baire spaces, other games in abstract analysis.

## Daily Schedule (sample)

| 9:30 |11.00 | 11:15 | 12:45 |14.00| 15:30 | 15:45|
| -------- | -------- | -------- | -------- | -------- | -------- | -------- |
| Talk 1 | Break | Talk 2 | Lunch | Tutorial 1| Break| Tutorial 2 |