Ahmed Ashraf, The Tutte Polynomial of a Matroid
(Tuesday Feb 19, 2019)
The Tutte polynomial is a universal Tutte-Grothendieck invariant of a matroid. We introduce this polynomial for a matroid and show how it is a generalization of characteristic polynomial of a geometric lattice and chromatic polynomial of a graph. We use this topic to briefly introduce the audience to matroids, as a generalization of graphs.
Babak Beheshti Vadeqan, A Conceptual Introduction to Quantum Mechanics
(Friday Mar 15, 2019)
The aim of this talk is to discuss some basic concepts of quantum mechanics such as the concept of state space, observables, measurement, etc. with comparison to their classical counterparts. We then proceed to explore the logical structure underlying propositional calculus for quantum mechanics and we will see how it differs from classical logic.
Andrew Herring, Ramanujan Graphs
(Tuesday March 19, 2019)
Ramanujan graphs are an important class of graphs in combinatorics and number theory. In a certain sense they are optimal “expander graphs” (graphs which are sparse, yet highly connected), and are thus naturally of interest to network scientists. The original constructions of Ramanujan graphs come from number theory, and only produce Ramanujan graphs with prime power degrees of regularity. Until recently it was merely conjectured that for an arbitrary fixed degree of regularity, there exist an infinite number of Ramanujan graphs of that degree. Marcus, Spielman, and Srivastava proved the conjecture using degree 2 covering graphs and a new technique they call “interlacing families of polynomials.” In this talk we attempt to give an overview of this influential work. No prior background knowledge will be assumed.
Dinesh Valluri, Riemann-Roch Theorem for Graphs
(Tuesday Mar 26, 2019)
Graphs can be thought of as discrete analogs of Riemann Surfaces. In this talk we will see a 'Riemann-Roch theorem' for graphs in relation to a certain chip firing game played on them. The game starts with assigning arbitrary (possibly negative) integers on the vertices of a graph called 'dollars' or 'chips'. A move in the game is to choose a vertex v and lend a dollar each to its neighbors. Given that the sum of all the dollars assigned to all vertices is non-negative, one could ask if it is possible to make the dollars at each vertex to be non-negative in finite number of moves. We answer this question using the aforementioned Riemann-Roch theorem. All the results in this talk are due to Matthew Baker and Serguei Norine.