Lectures related to Constructive set theory

Functional Analysis I: November 18, 2021

Covers partially ordered sets, maximal elements, upper bounds, and the Zorn lemma in functional analysis.

Regularity Lemmas and Density Theorems

Explores Regularity Lemmas and Density Theorems for graph partitioning and structure identification.

Categories

Introduces categories as collections of objects with morphisms and identity morphisms.

Cartesian Product

Covers the concept of the Cartesian product, emphasizing the order of elements in pairs.

Two Definitions of Group Action

Explores two definitions of group action on a set, focusing on properties and applications.

Cartesian Product in Linear Algebra

Explores the Cartesian product in linear algebra and the method of induction for proving propositions.

Abstract Concepts: Semi-Ring

Explores the concept of a commutative semi-ring based on set theory properties.

Commutation Groups: Euler's Totient Function

Explores commutative groups, Euler's Totient Function, and Cartesian products in group theory.

Lifting properties and model categories

Covers the study of lifting properties in categories, focusing on the left and right lifting properties.

Limits and colimits: Two examples

Focuses on the pushout and pullback constructions in sets, illustrating colimits and limits with explicit examples.

Hausdorff Dimension and Brownian Motion

Explores Hausdorff dimension and its application to Brownian motion sets, emphasizing the importance of understanding set dimensions in stochastic processes.

Set Difference: Definition and Examples

Explains set difference and complement calculation with clear examples.

Recap of Group Theory

Provides a recap of group theory, defining a group as a set with a multiplication operation.

The Pigeonhole Principle: Basics and Applications

Covers the Pigeonhole Principle, its applications, and examples of its guarantees.

Counting Basics: Set Theory and Permutations

Covers the main rules of counting for combinatorial objects and introduces various counting techniques.

Real Numbers: Sets and Operations

Covers the basics of real numbers and set theory, including subsets, intersections, unions, and set operations.

Linear Algebra: Injective Functions

Focuses on injective functions in linear algebra, demonstrating how to verify properties and prove injectivity.

Crystallography: Describing Periodic Structures

Covers crystallography, lattice structures, vectors, and diffraction patterns.

Set Theory: Introduction and Operations

Covers the foundation of mathematics through set theory concepts like membership and unions.

Set Theory: Operations and Functions

Covers set theory operations and functions, including injections and surjections.