Lectures related to Interlacing Families and Ramanujan Graphs

Interlacing Families and Ramanujan Graphs

Explores interlacing families of polynomials and 1-sided Ramanujan graphs, focusing on their properties and construction methods.

Expander Graphs: Properties and Eigenvalues

Explores expanders, Ramanujan graphs, eigenvalues, Laplacian matrices, and spectral properties.

Building Ramanujan Graphs

Explores the construction of Ramanujan graphs using polynomials and addresses challenges with the probabilistic method.

Graph Theory and Network Flows

Introduces graph theory, network flows, and flow conservation laws with practical examples and theorems.

Polynomial Identity Testing

Covers polynomial identity testing using oracles and random point evaluation, with applications in graph theory and algorithmic aspects.

Graphical Models: Representing Probabilistic Distributions

Covers graphical models for probabilistic distributions using graphs, nodes, and edges.

Reformulating Problems: Tools and Intuition

Focuses on open problems and the importance of reformulating problems with better tools and intuition.

Sparsest Cut: ARV Theorem

Covers the proof of the Bourgain's ARV Theorem, focusing on the finite set of points in a semi-metric space and the application of the ARV algorithm to find the sparsest cut in a graph.

Spectral Graph Theory: Introduction

Introduces Spectral Graph Theory, exploring eigenvalues and eigenvectors' role in graph properties.

Convergence of Random Walks

Explores the convergence of random walks on graphs and the properties of weighted adjacency matrices.

Pseudorandomness: Theory and Applications

Explores pseudorandomness theory, AI challenges, pseudo-random graphs, random walks, and matrix properties.

Pseudorandomness: Expander Mixing Lemma

Explores pseudorandomness and the Expander Mixing Lemma in the context of d-regular graphs.

Networked Control Systems: Properties and Connectivity

Explores properties of matrices, irreducibility, and graph connectivity in networked control systems.

Cheeger's Inequalities

Explores Cheeger's inequalities for random walks on graphs and their implications.

Isogenic Graphs: Spectral Analysis and Mathematical Applications

Explores isogenic graphs, spectral properties, and mathematical applications in modular forms and cryptography.

Cheeger's Inequality

Explores Cheeger's inequality and its implications in graph theory.

Networked Control Systems: Opportunities

Explores coordination in networked control systems, graph theory, and consensus algorithms.

Graph Sketching: Connected Components

Covers the concept of graph sketching with a focus on connected components.

Statistical Analysis of Network Data

Introduces network data structures, models, and analysis techniques, emphasizing permutation invariance and Erdős-Rényi networks.

Spectral Clustering: Theory and Applications

Explores spectral clustering theory, eigenvalue decomposition, Laplacian matrix, and practical applications in identifying clusters.