# Seminars & Events for Special Events and Conferences

##### Celebration of the Life and Work of John F. Nash, Jr.

**Lectures on Nash’s work:**(All talks will be in McDonnell A02)

##### Finite Simple Groups: Thirty Years of the Atlas and Beyond Celebrating the Atlases and Honoring John Conway

Please see the conference webpage at http://math.arizona.edu/~grouptheory/princeton/ for more information.

#### Schedule of talks:

**Monday, November 2**

##### New directions in statistical mechanics and dynamical systems A conference dedicated to 80th birthday of D. Ruelle and Y. Sinai.

The meeting is supported by Princeton University and IAMP.

**Wednesday, December 16, 2015 (talks in McDonnell A01):**

**10:00 am: **Hillel Furstenberg (Hebrew University of Jerusalem) - Affine Representations and Harmonic Functions

**11:10 am: **Peter Sarnak (Princeton University and the Institute for Advanced Study) - The Mobius flow,entropy and complexity

**2:00 pm:** Charles Fefferman (Princeton University) - Formation of singularities in fluid interfaces

**3:20 pm: **Senya Shlosman (Centre de Physique Théorique) - Spin glass transition as roughening phenomenon

**4:20 pm: **Hugo Duminil-Copin (Université de Genève) - Pirogov-Sinai type arguments for some planar loop models

##### Analysis, PDEs, and Geometry: A Conference in Honor Sergiu Klainerman

##### Analysis, PDEs, and Geometry: A Conference in Honor of Sergiu Klainerman

##### Analysis, PDEs, and Geometry: A Conference in Honor of Sergiu Klainerman

##### Analysis, PDEs, and Geometry: A Conference in Honor of Sergiu Klainerman

##### TBA - Introduction; Nick Sheridan

**This is a joint Geometry/Topology day.** There will be a short introduction of the RTG event, followed by Nicholas Sheridan's talk at 12:10.

##### Hyperbolic 3-manifolds with low cusp volume

**This is a joint Geometry/Topology day. **The past fifteen years have seen a great deal of progress towards a complete picture of hyperbolic manifolds of low volume. The volume of a hyperbolic manifold is a topological invariant and can be viewed as a measure of complexity. In fact, there are only finitely many hyperbolic manifolds of a given volume. For hyperbolic 3-manifolds with cusps, one can also consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results and techniques for understanding the infinite families of hyperbolic 3-manifolds of low cusp volume. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings.

##### TBA - Marco Aurelio Mendez Guaraco

**This is a joint Geometry/Topology day.**

##### Knot Concordance Invariants and Heegaard Floer Homology

**This is a joint Geometry/Topology day. ** I will define knot concordance and discuss a set of concordance invariants that are constructed using Heegaard Floer homology.

##### Min-max theory and least area minimal hypersurfaces

**This is a joint Geometry/Topology day. ** Min-max theory is a powerful way for constructing minimal hypersurfaces and has numerous geometric applications. In this talk, I will present one of them due to Calabi and Cao: on a convex sphere, any closed geodesic of least length is simple. I will explain how to extend this result to higher dimensions.

##### Cube of resolutions complexes for Khovanov-Rozansky homology and knot Floer homology

**This is a joint Geometry/Topology day. **I will compare the oriented cube of resolutions constructions for Khovanov-Rozansky homology and knot Floer homology. Manolescu conjectured that for singular diagrams (or trivalent graphs) the HOMFLY-PT homology and knot Floer homology are isomorphic - I will show that this conjecture is equivalent to a certain spectral sequence collapsing. This will also lead to a recursion formula for the HOMFLY-PT homology of singular diagrams that categorifies Jaeger's composition product formula.

##### Sparsification of Graphs and Matrices

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. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor. Our algorithms follow from the solution of a problem in linear algebra. Given an expression for a rank-n symmetric matrix A as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.

##### The solution of the Kadison-Singer Problem

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. I will explain how we solve the Kadison-Singer Problem by proving Weaver's Conjecture in Discrepancy Theory. I will explain the "method of interlacing polynomials" that we introduced to solve this problem, and sketch the major steps in the proof. These are the introduction of "mixed characteristic polynomials"---the expected characteristic polynomials of a sum of random symmetric rank-1 matrices, the proof that these polynomials are real rooted, and the derivation of an upper bound on their largest roots. These techniques are elementary, and should be understandable by a broad mathematical audience.

##### Ramanujan Graphs and Free Probability

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. Ramanujan graphs are defined in terms of the eigenvalues of their adjacency or Laplacian matrices. In this spectral perspective, they are the best possible expanders. Infinite families of Ramanujan graphs were first shown to exist by Margulis and Lubotzky, Phillips and Sarnak using Deligne's proof of the Ramanujan conjecture. These constructions were sporadic, only producing graphs of special degrees and numbers of vertices. In this talk, we outline an elementary proof of the existence of bipartite Ramanujan graphs of very degree and number of vertices.

##### 31st Annual Geometry Festival

##### Lecture Series on “Arithmetic in Geometry" - #1

Quotients of symmetric spaces of semi-simple Lie groups by torsion-free arithmetic subgroups are particularly nice Riemannian manifolds which can be studied by using diverse techniques coming from the theories of Lie Groups, Lie Algebras, Algebraic Groups and Automorphic Forms. One such manifold is a "fake projective plane" which is, by definition, a smooth projective complex algebraic surface with same Betti-numbers as the complex projective plane but which is not isomorphic to the latter. The first example of a fake projective plane (fpp) was constructed by David Mumford in 1978, and it has been known that there are only finitely many of them. In the theory of algebraic surfaces, it was an important problem to construct them and determine their geometric properties.