Lecture

Interlacing Polynomials

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

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

Introduces abstract concepts in linear algebra, focusing on operations with vectors and matrices.

Explores polynomial operations, properties, and subspaces in vector spaces.

Explores the Stein algorithm for polynomial identity testing and the minimization of a cut problem.

Covers the properties and operations of vector spaces, including addition and scalar multiplication.

Introduces linear combinations in vector spaces, operations, and polynomials of degree 2.

Covers topics related to probability and statistics, including finding specific values and significant events.

Explores interlacing families, Ramanujan graphs, and their construction using signed adjacency matrices.

Explores the theory of martingales, including conditional expectations, Chernoff bounds, and Azuma's inequality.

Covers examples of vector spaces and the concept of subspaces, emphasizing key properties and verification methods.

Covers linear transformations between polynomial spaces and explores examples of linear independence and bases.

Covers vector spaces, subspaces, kernel, image, linear independence, and bases in linear algebra.

Introduces vector spaces, subspaces, linear maps, and evaluation maps, with examples and exercises for better comprehension.

Explores changing bases in vector spaces using matrices and the significance of preserving structure within vector spaces.

Explores complex roots, polynomials, and factorizations, including roots of unity and the fundamental theorem of algebra.

Explores vector subspaces in R4, symmetric matrices, basis vectors, and canonical forms.

Delves into the concept of averaging polynomials and the significance of pairing up roots.

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

Covers the Poisson Paradigm, including the First/Second Moment Method and Martingales, discussing dependency graphs and Chernoff bounds.