# Mathematics Colloquium Winter 2014

For further information please contact the Mathematics Department at 459-2969

**Tuesday, January 7, 2014**

**Thin groups: arithmetic and beyond**

**Elena Fuchs, University of California, Berkeley**,

**Special Colloquium Speaker**

In 1643, Rene Descartes discovered a formula relating curvatures of circles in Apollonian circle packings, constructed by Apollonius of Perga in 200 BC. This formula has recently led to a connection between the construction of Apollonius and orbits of a certain so-called thin subgroup G of GL_4(Z). This connection is key in recent results on the arithmetic of Apollonian packings, which I will describe in this talk. A crucial ingredient in the proofs is the spectral gap coming from families of expander graphs associated to G -- this gap is far less understood in the case of thin groups than that of non-thin groups. Motivated by this problem, I will then discuss the ubiquity of thin groups and present results on thinness of monodromy groups of hypergeometric equations in the case where these groups act on hyperbolic space.

**Thursday, January 9, 2014 **

**Arithmetic invariant theory and applications****Wei Ho, Columbia University,** **Special Colloquium Speaker**The origins of "arithmetic invariant theory" come from the work of Gauss, who used integer binary quadratic forms to study ideal class groups of quadratic fields. The underlying philosophy---parametrizing arithmetic and geometric objects by orbits of group representations---has now been used to study higher degree number fields, curves, and higher-dimensional varieties. We will discuss some of these constructions and highlight the applications to topics such as statistics for ideal class groups, bounding ranks of elliptic curves, and dynamics on K3 surfaces. This talk is intended for a general mathematical audience.

**Tuesday, January 14, 2014**

** L^p norms of eigenfunctions on locally symmetric spacesSimon Marshall, Northwestern University, Special Colloquium Speaker**Let M be a compact Riemannian manifold, and f an L^2-normalised Laplace eigenfunction on M. If p > 2, a theorem of Sogge tells us how large the L^p norm of f can be in terms of its Laplace eigenvalue. For instance, when p is infinity this is asking how large the peaks of f can be. I will present an analogue of Sogge's theorem for eigenfunctions of the full ring of invariant differential operators on a locally symmetric space, and discuss some links between this result and number theory.

**Thursday, January 16, 2014 **A question of Serre (which also appeared in Deligne's work on the Weil conjectures) asks whether the odd-degree Betti numbers of any proper smooth algebraic variety are necessarily even. We review the origin (from topology and differential geometry) and the significance of the question, and then give an answer in the affirmative

*Parity in the cohomology of algebraic varieties*

**Junecue Suh, Harvard University,**

**Special Colloquium Speaker**

**Tuesday, **January 21, 2014

No Colloquium**Tuesday,**January 21,

**Thursday January 23, 2014**

*Frobenius-Schur Indicators: From finite groups to tensor categories*

*Frobenius-Schur Indicators: From finite groups to tensor categories*

**Siu-Hung Ng, Iowa State University and Louisiana State University, Special Colloquium Speaker**Frobenius-Schur indicators were introduced a century ago for the representations of finite groups. The second indicators have been playing an important role in representation theory but the higher indicators are generally more obscure. In the last decade, Frobenius-Schur indicators have been extended to conformal field theory, Hopf algebras, quasi-Hopf algebras and fusion categories; they are invariants of tensor categories. Moreover, these indicators have constituted the integral components for a proof of the congruence subgroup theorem of modular tensor categories. In this talk, we will give an introduction of these indicators from an historical point of view to their current developments and some applications.

**Tuesday, February 4, 2014**

*Slowly decaying perturbations of Jacobi and Schrodinger operators*

*Slowly decaying perturbations of Jacobi and Schrodinger operators*

**Milivoje Lukic, Rice University, Special Colloquium Speaker**

**Milivoje Lukic, Rice University, Special Colloquium Speaker**

The spectrum of a linear operator generalizes the notion of the set of eigenvalues of a finite matrix. We are concerned with spectral properties of certain classes of operators, such as Schrodinger operators (central to quantum mechanics) and Jacobi matrices (tied to orthogonal polynomials). Work of Deift--Killip and Killip--Simon shows that L^2 perturbations of the free Jacobi matrix preserve a.c. (absolutely continuous) spectrum. This result is optimal on the L^p scale, so spectral properties of slower decaying perturbations can only be established under additional assumptions. In this talk, we will discuss several recent results on slowly decaying perturbations. Some of our results solve an open problem about a class of oscillatory decaying perturbations which includes almost periodic times decaying sequences. In another approach, we describe the spectral consequences of L^2 bounded variation conditions. Finally, we discuss our recent contributions to higher-order Szego theorems; this includes the disproving of a conjecture of Simon and the first equivalence result in the regime of arbitrarily slow decay.

**Thursday, February 6, 2014**

**Hamiltonian dynamical systems with infinitely many periodic orbits and symplectic topology**

**Hamiltonian dynamical systems with infinitely many periodic orbits and symplectic topology**

**Basak Gurel, University of Central Florida, Special Colloquium Speaker**

Ever since the Conley-Zehnder proof of the Arnold conjecture for tori, the study of periodic orbits has arguably been the most important interface between Hamiltonian dynamical systems and symplectic (and contact) topology and geometry. A general feature of Hamiltonian systems is that they tend to have numerous periodic orbits. In fact, for a broad class of closed symplectic manifolds, every Hamiltonian diffeomorphism has infinitely many simple periodic orbits. There are, however, notable exceptions such as the two-dimensional sphere.

For such manifolds as the sphere, it has been conjectured that a Hamiltonian diffeomorphism with “more than necessary” fixed points must have infinitely many simple periodic orbits. Here the threshold is usually interpreted as a lower bound arising from some version of the Arnold conjecture. The assertion for the two-sphere is a special case of a theorem due to Franks: every area preserving homeomorphism of the sphere has either two or infinitely many periodic points. Moreover, a similar dichotomy has also been conjectured for a class of Reeb flows. This conjecture is one of the most important open questions at the interface of symplectic topology and dynamics.

In this talk I will discuss various aspects of the existence question for periodic orbits of Hamiltonian dynamical systems, focusing on recent results obtained by Floer or contact homological methods.

**Tuesday, February 11, 2014**

**Sharp Gagliardo--Nirenberg--Sobolev inequalities and conformal geometry **

**Sharp Gagliardo--Nirenberg--Sobolev inequalities and conformal geometry**

**Jeffrey Case, Princeton University, Special Colloquium Speaker**

**Jeffrey Case, Princeton University, Special Colloquium Speaker**

There is a deep connection between curvature prescription problems in conformal geometry and sharp Sobolev inequalities. The most famous example arises in the Yamabe Problem, where one observes a connection between constructing conformal metrics with constant scalar curvature and the norm inequality giving the Sobolev embedding $W^{1,2}\subset L^{2n/(n-2)}$. Another example arises in Perelman's work on the Ricci flow, where one observes a connection between prescribing Perelman's weighted scalar curvature and the sharp logarithmic Sobolev inequality. In this talk, I will describe some of my work on the conformal geometry of smooth metric measure spaces. In particular, I will describe how this work gives a natural geometric interpretation to the interpolating family of sharp Gagliardo--Nirenberg--Sobolev inequalities discovered by Del Pino and Dolbeault and also how it provides new insights into aspects of Perelman's work on the Ricci flow.

**Thursday, February 13, 2014**

Cancelled**Tuesday, February 18, 2014**

**Uniformity of harmonic map heat flow at infinite time**

**Uniformity of harmonic map heat flow at infinite time**

**Longzhi Lin, Rutgers University, Special Colloquium Speaker**

The theory of harmonic maps and harmonic map heat flows has been an intensely researched field in geometric analysis and PDE. They have numerous important applications not only in geometry and topology such as minimal surfaces and deformations of Riemannian surfaces, but also in many other areas such as nematic liquid crystals. In this talk we will show an energy convexity along the harmonic map heat flow with small initial energy on the unit 2-disk and we prove that such harmonic map heat flow converges uniformly in time strongly in the W^{1,2}-topology to the unique limiting harmonic map as time goes to infinity. The key ingredients of the proof are Wente's compensation compactness techniques and Riviere's conservation law for critical points of a class of conformally invariant variational problems. If time permits I will talk about the connection of the results to other interesting problems.

**Thursday, February 20, 2014**

*A blow-up analysis for calibrated currents.*

*A blow-up analysis for calibrated currents.*

**Costante Bellettini, Princeton University, Special Colloquium Speaker**

Calibrated currents naturally appear when dealing with several geometric questions, some aspects of which require a deep understanding of regularity properties of calibrated currents. After a brief review of some of these issues, we will focus on the pseudo holomorphic case. Here we will perform a blow up analysis, where we will be interested in the uniqueness of tangent cones and in the rate of decay for the mass ratio. We will sketch a proof relying on an analysis implementation of the algebro-geometric blowing up of a point.

**Tuesday, February 25, 2014**

*Fourier splitting and applications*

*Fourier splitting and applications*

**Tomas Schonbek, Florida Atlantic University**

In this talk I will discuss some of Maria Schonbek's work, especially the Fourier splitting method, and its applications, including applications to some exterior problems.

**Tuesday, March 4, 2014**

TBA

TBA

**Tuesday, March 11, 2014**

*The work of Jesse Douglas on Minimal Surfaces and the first Fields Medal*

Mario Micallef, University of Warick

The demonstration of the existence of a least area surface spanning a given contour has a long history. Through the soap-film experiments of Plateau, this existence problem became known as the Plateau Problem. *The work of Jesse Douglas on Minimal Surfaces and the first Fields Medal*

Mario Micallef, University of Warick

In the 1860s, Riemann and Weierstrass independently started a program for constructing a minimal surface bounded by a given polygonal contour; they sought the pair of holomorphic functions $f$ and $g$ required in the so-called Weierstrass-Enneper representation of the minimal surface. Following on from important contributions by Darboux, Garnier completed this programme in 1928.

Garnier's work was soon eclipsed by the works of Tibor Rado and Jesse Douglas about which there is considerable amount of inaccurate information in the literature and on which I hope to shed some light in this talk. In particular, in a joint work with the mathematical historian Jeremy Gray, we challenge the popular belief that Douglas arrived at his functional for solving the Plateau Problem by direct consideration of Dirichlet's integral and its relation to the area functional. I shall describe how, by looking at abstracts of Jesse Douglas in the Bulletin of the American Mathematical Society, I have been able to infer how Douglas MAY have arrived at his functional.

Douglas was awarded one of the first Fields Medals for his work on the Plateau problem. I shall relate some amusing aspects of the Fields Medal ceremony at which Douglas was awarded his prize. I shall also give a brief biography of Douglas.

**Thursday, March 13, 2014**

**Mixed Tate motives and higher homotopy theory**

**Mixed Tate motives and higher homotopy theory**

**Deepam Patel, IHES, Paris**

The category of mixed Tate motives over a number field is an extremely rich sub-category of the category of mixed motives. For instance, its Ext groups are given by the K-theory of the corresponding number field. Furthermore, periods of mixed Tate motives give rise to special values of L-functions. An important problem in the theory is to give explicit constructions of mixed Tate motives. In this talk, after recalling the relevant background, we describe how higher homotopy theory can be used to give new examples of mixed Tate motives.