I’m at this talk by Dan Spielman of Yale University, who’s trying to give us an introduction to spectral and algebraic graph theory. I’m here because he was my friend’s undergraduate advisor and my friend said that “Professor Dan” is great! Dan has won a ton of fancy prizes and there are so many people in the audience to watch him.
This talk wove together computer science, graph theory, physics in a very engaging manner. It was a lot of fun, and any errors in this blog post are mine alone, which is embarrassing because I should know a fair amount of spectral graph theory and group theory (I even did my senior math seminar at Yale in spectral graph theory in 2010. Woof.)
In spectral graph theory, we relate graphs to matrices. The first example is an adjacency matrix, where you label the vertices of a graph and then use those labels as row/column labels for a square matrix, and put a “0” when there is no edge between the corresponding vertices, and a “1” if there is such an edge.
Dan: “I think it’s an accident that the adjacency matrix is useful.” He’ll go on to talk about associated matrices, linear equations, quadratic forms, and operators that are less accidental.
Quadratic forms give us some beautiful theorems about graphs.
Shoot, I accidentally sat right next to Anna Haensch, who was also planning on blogging and who co-writes the great AMS Blog on Math Blogs. Well, I’m taking it. Also, it was nice to meet Anna! We’ve talked on the internet but haven’t met before. And Adriana Salerno, the editor of this blog, live-tweeted the talk.
Benedict Gross, this year’s AMS Colloquium series lecturer.
Benedict Gross kicked off his series of talks in the AMS Colloquium Lectures on Tuesday by speaking about the past, with a plan to reach the future of Number Theory by Friday. Gross, former MacArthur Fellow and winner of the Cole Prize in Number Theory is the George Vasmer Leverett Professor of Mathematics, Department of Mathematics, Harvard University. The series, entitled “Complex Multiplication: Past, Present, Future,” considers the interplay between imaginary quadratic fields and the theory of elliptic curves. The area “has a long and twisted history,” according to Gross. The first talk covered the two hundred years from 1751 to 1951, beginning with Euler reviewing Fagnano’s work on the lemniscate, and beginning his investigations of “elliptic integrals” of the form
which lead to elliptic curves. Legendre and Gauss studied positive definite binary forms up to equivalence under the special linear group SL_2(Z). The number of equivalence classes of forms with a given discriminant is called the class number of the discriminant. The connection between these class numbers (and their modern variants) and elliptic curves becomes the story of complex multiplication.
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