My goal in this talk is to survey some of the emerging applications of polynomial methods in both learning and in statistics. I will give two examples from my own work in which the solution to well-studied problems in learning and statistics can be best understood through the language of algebraic geometry.
A classical theorem in Euclidean geometry asserts that if a set of points has the property that every line through two of them contains a third point, then they must all be on the same line. We prove several approximate versions of this theorem (and related ones), which are motivated from questions about locally correctable codes and matrix rigidity. The proofs use an interesting combination of combinatorial, algebraic and analytic tools.
Joint work with Boaz Barak, Zeev Dvir and Amir Yehudayoff
This talk will review some theorems and conjectures about phase transitions of interacting spin systems in statistical mechanics. A phase transition may be thought of as a change in a typical spin configuration from ordered state at low temperature to disordered state at high temperature. I will illustrate how the symmetry of a spin system plays a crucial role in its qualitative behavior. Of particular interest is the connection between supersymmetric statistical mechanics and the spectral theory of random band matrices.
I will describe the notions of strong and weak epsilon nets in range spaces, and explain briefly some of their many applications in Discrete Geometry and Combinatorics, focusing on several recent results in the investigation of the extremal questions that arise in the area, and mentioning some of the remaining open problems.
It is becoming more and more clear that many of the most exciting structures of our world can be described as large networks. The internet is perhaps the foremost example, modeled by different networks (the physical internet, a network of devices; the world wide web, a network of webpages and hyperlinks). Various social networks, several of them created by the internet, are studied by sociologist, historians, epidemiologists, and economists. Huge networks arise in biology (from ecological networks to the brain), physics, and engineering.
The modern theory of dynamical systems, as well as symplectic geometry, have their origin with Poincare as one field with integrated Ideas. Since then these fields developed quite independently. Given the progress in these fields one can make a good argument why the time is ripe to bring them closer together around the core area of Hamiltonian dynamics