Course

MATH-467: Probabilistic methods in combinatorics

Lectures in this course (56)

Interlacing Family and Matrix Determinant Lemma

Explores interlacing family, matrix determinant lemma, and random matrix characteristics.

Partial Differential Equations and Hessians

Covers partial differential equations, Hessians, and the Implicit Function Theorem, with a focus on exam question resolution.

Partial Differential Equations

Introduces partial differential equations, covering derivatives, special cases, transformations, and the Jacobian matrix.

Symmetry in Integrals

Explores how symmetries can simplify integrals with interesting cases and examples.

Probabilistic Methods in Combinatorics

Covers probabilistic methods in combinatorics, monochromatic edges, 2-colorable graphs, and good 2-colorings.

Probabilistic Methods in Combinatorics

Covers the Erdős-Ko-Rado problem, intersecting set families, and selecting valid winners.

Probability Theory: Markov's Theorem

Explores Markov's theorem, Chernoff bound, and probability theory fundamentals, including good coloring, 2-colorable graphs, and rare events.

Markov's Inequality and Hoeffding's Lemma

Introduces Markov's inequality and Hoeffding's lemma to analyze random variables' behavior.

Linearity of Expectation

Covers Linearity of Expectation, Markov's inequality, random variables, and transitive tournaments.

Partial Differential Equations

Covers the basics of Partial Differential Equations, including the Laplace equation, heat equation, and wave equation.

Graph Coloring: Basics and Applications

Covers the basics and applications of graph coloring, including balancing vectors and achieving perfect fairness.

Linearity of Expectation: First Moment Method

Introduces Linearity of Expectation and the First Moment Method, explores probability theory problems like Buffon's Needle, and discusses transitive tournaments and Ham paths.

Ramsey Theory: Alterations and Colorings

Explores Ramsey theory, alterations, colorings in graphs, monochromatic matchings, and the significance of large cliques.

Stirling's Formula: Applications and Embeddings

Explores Stirling's formula applications, polarization in vectors, and graph embeddings in Euclidean space.

Graph Theory Basics

Introduces graph theory basics, Ramsey theory, and graph coloring concepts.

Graph Theory: Girth and Independence

Covers girth, independence, probability, union bound, sets, and hypergraph recoloring.

Angle Conjectures and False Claims

Discusses angle conjectures, false claims, geometric configurations, and finding collections of sets.

Graph Theory Fundamentals

Explores fundamental graph theory concepts, Erdős' results, Chromatic Lemma, and Union Bound theorem in graph theory.

Random Variables and Covariance

Covers random variables, variances, and covariance, as well as the probability in random graphs.

Second Moment Method

Explores the Second Moment Method and variance of random variables, including covariance and independence.