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### Episodes from Quantitative Topology: 1. Variational problems, Morse and Turing.

Shmuel Weinberger
University of Chicago
February 21, 2017

This lecture will begin the series of discussing how effective solutions of topological problems are: and in particular, how large solutions to geometric topological problems are with various measures of complexity.  Lecture one will show how one can use basic results about computability, algorit

### Episodes from Quantitative Topology: 2. Quantitative Nullcobordism

Shmuel Weinberger
University of Chicago
February 23, 2017

In the 50's, Rene Thom solved the problem of determining when a closed smooth manifold bounds a compact manifold.  Subsequent work of Milnor and Wall solved the analogous oriented problem.  These works comprise an important early example of the fundamental method of geometric topology via reduct

### Sparsification of Graphs and Matrices

Daniel Spielman
Yale University
March 21, 2016

Random graphs and expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We formalize this notion of approximation and ask how well an arbitrary graph can be approximated by a sparse graph.

### The solution of the Kadison-Singer Problem

Daniel Spielman
Yale University
March 23, 2016

In 1959, Kadison and Singer posed a problem in operator theory that has reappeared in many guises, including the Paving Conjecture, the Bourgain-Tzafriri Conjecture, the Feichtinger Conjecture, and Weaver's Conjecture.

### Ramanujan Graphs and Free Probability

Daniel Spielman
Yale University
March 25, 2016

We use the method of interlacing polynomials and a finite dimensional analog of free probability to prove the existence of bipartite Ramanujan graphs of every degree and number of vertices. No prior knowledge of Ramanujan graphs or free probability will be assumed.

### Classification of Bernoulli shifts

Lewis Bowen
University of Texas, Austin & Princeton University
November 20, 2015

Bernoulli shifts over amenable groups are classified by entropy (this is due to Kolmogorov and Ornstein for $Z$ and Ornstein-Weiss in general). A fundamental property is that entropy never increases under a factor map. This property is violated for nonamenable groups.