Rice University Math Department Calendar

    November 2011          December 2011           January 2012    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
        1  2  3  4  5                1  2  3    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    4  5  6  7  8  9 10    8  9 10 11 12 13 14
 13 14 15 16 17 18 19   11 12 13 14 15 16 17   15 16 17 18 19 20 21
 20 21 22 23 24 25 26   18 19 20 21 22 23 24   22 23 24 25 26 27 28
 27 28 29 30            25 26 27 28 29 30 31   29 30 31

10:00 am Monday, December 12, 2011
Spectral Theory Brown Bag Seminar: Spectrum of the Fibonacci Hamiltonian as a Dynamically Defined Cantor Set
by Anton Gorodetski (UC Irvine) in HB 427
Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu

12:00 pm Wednesday, December 21, 2011
Thesis Defense: The Hopf differential and harmonic maps between branched hyperbolic structures
by Evelyn Lamb in HB427

Let M be a Riemann surface of genus g>1 with fundamental group π. Representations from π to PSL(2,R) occur in 4g-3 components indexed by Euler class. There is one component for each natural number e such that |e| < 2g-1. We use harmonic maps to study representations occurring in the non-maximal components of Hom(PSL(2,R))/PSL(2,R). Specifically, to a branched hyperbolic metric we associate a harmonic map. To this map we associate a quadratic differential φ. We show that the map between branched hyperbolic structures that takes a harmonic map to its Hopf differential φ is injective. This is a partial generalization of Wolf's parametrization of Teichmüller space by harmonic maps.Host Department: Rice University-Mathematics
Submitted by mwolf@rice.edu

4:00 pm Tuesday, January 10, 2012
Special Colloquium: Okounkov bodies: from algebraic to convex geometry
by David Anderson (University of Washington) in HB 227

Building on earlier work of Okounkov, in 2008 Kaveh, Khovanskii, Lazarsfeld, and Mustata showed how to construct a convex body in n-dimensional Euclidean space naturally associated to a line bundle on an n-dimensional algebraic variety, in such a way that the convex geometry of this body reflects algebro-geometric properties of the line bundle. This construction generalizes a well-understood correspondence between toric varieties and polytopes: when one starts with a toric variety and an equivariant line bundle, the associated convex body is the polytope arising from the yoga of toric geometry. After describing the history and construction of these so-called "Okounkov bodies" from an elementary point of view, I will explain how the toric correspondence can be made tighter: under the right conditions, the Okounkov body is a polytope, and the variety in question deforms to a toric variety with the same Okounkov body. The toric correspondence provides a remarkably useful bridge between several branches of mathematics, and we will see connections between geometry, algebra, combinatorics, and representation theory.Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu

4:00 pm Wednesday, January 18, 2012
Geometry-Analysis Seminar: Dynamics of the Fibonacci trace map, geometry of dynamical invariants and applications to spectral theory of 1D quasicrystals
by William Yessen (UC Irvine) in HB 227

Since the discovery of quasicrystals in the early 80's by Schechtman et. al., quasiperiodic models in mathematical physics have gained the status of an independent research field. In addition to spectral theory, a few (seemingly unrelated at first glance) tools from other areas have been successfully applied. In particular, a fascinating connection between smooth dynamical systems and spectral theory of 1D quasicrystals was established during the early years of development. After motivating the problem in terms of spectral theory of certain so-called quasiperiodic Hamiltonians, we shall discuss briefly a relation with the so-called Fibonacci trace map---a three-dimensional polynomial map on the Euclidean three-space---and then in some detail dynamics of the Fibonacci trace map and geometry of dynamical invariants (invariant sets, stable and unstable manifolds, etc.). We'll close the loop by applying this knowledge to the spectral problem, consequently establishing the following: 1) topology of the spectrum, 2) measure-theoretic properties, 3) fractal-dimensional properties. We'll conclude with some open problems and conjectures. Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu

4:00 pm Thursday, January 26, 2012
Stulken Lecture: Hot Topics in Cold Gases - a Mathematical Physics Perspective
by Robert Seiringer (McGill University) in HB 227

We present an overview of mathematical results on the low temperature properties of dilute quantum gases, which have been obtained in the past few years. The presentation includes results on the free energy in the thermodynamic limit, and on Bose-Einstein condensation, Superfluidity and quantized vortices in trapped gases. All these properties are intensely being studied in current experiments on cold atomic gases. We will give a brief description of the mathematics involved in understanding these phenomena, starting from the underlying many-body Schroedinger equation.Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu

4:00 pm Wednesday, February 1, 2012
Geometry-Analysis Seminar: On the Gross-Pitaevskii Hierarchies
by Natasa Pavlovic (UT Austin) in HB 227

The Gross-Pitaevskii (GP) hierarchy is an infinite system of coupled linear non-homogeneous PDEs, which appear in the derivation of the nonlinear Schr\"{o}dinger equation (NLS). Inspired by the PDE techniques that have turned out to be useful on the level of the NLS, we realized that, in some instances we can introduce analogous techniques at the level of the GP. In this talk we will discuss some of those techniques which we use to study well-posedness for GP hierarchies. Time permitting, we will also discuss a new derivation of the defocusing cubic GP hierarchy in dimensions $d=2,3$, from an $N$-body Schr\"{o}dinger equation describing a gas of interacting bosons in the GP scaling, in the limit $N\rightarrow\infty$. In particular, we prove convergence of the corresponding BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work on the well-posedness of the Cauchy problem for GP hierarchies, which are inspired by the solutions spaces based on space-time norms introduced by Klainerman and Machedon. We note that in $d=3$, this has been a well-known open problem in the field. The talk is based on joint works with Thomas Chen. Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu

4:00 pm Thursday, February 2, 2012
Special Lecture: Math at Top Speeds: Establishing and Breaking Myths in the Drag Racing Folklore
by Richard Tapia (Rice University (CAAM)) in McMurtry Auditorium in Duncan Hall

In this talk the speaker will identify elementary mathematical frameworks for the study of popular drag racing beliefs. In this manner some myths are validated, while others are destroyed. The speaker will explain why dragster acceleration is greater than the acceleration due to gravity, an age old inconsistency. His "Fundamental Theorem of Drag Racing" will be presented. The first part of the talk will be a historical account of the development of drag racing and will include several lively videos.Host Department: Rice University-Mathematics
Submitted by hausman@rice.edu

4:00 pm Monday, February 6, 2012
Topology Seminar: Asymptotic Geometry of Teichmuller Space and Divergence
by Harold Sultan (Columbia University) in HB 447

Using the combinatorial model of the pants complex I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. As a corollary, I will explain a new way to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller space in that it has a divergence function which is superquadratic yet subexponential.Host Department: Rice University-Mathematics
Submitted by shelly@rice.edu

4:00 pm Tuesday, February 7, 2012
Algebraic Geometry Seminar: Alternate compactifications of Hurwitz spaces
by Anand Deopurkar (Harvard) in HB 227

Moduli spaces of geometrically interesting objects are usually not compact. They need to be compactified by allowing certain carefully chosen degenerations. Often, this can be done in several ways, leading to different birational models that are related in interesting ways. I will describe a range of compactifications of the Hurwitz space H^d_g, which parametrizes d sheeted, simply branched, genus g covers of the projective line. These compactifications are constructed by allowing degenererations where the branch points can collide in a prescribed way, recovering as a particular case the standard compactification by admissible covers. After the general construction, I will focus on the case of d=3. In this case, the above construction gives a sequence of compactifications which contract the boundary divisors in the admissible cover compactification. I will construct a sequence of yet more compactifications that modify the interior, featuring the classical Maroni invariant of trigonal curves. Host Department: Rice University-Mathematics
Submitted by evanmb@gmail.com

4:00 pm Monday, February 13, 2012
Topology Seminar: Surgery and tight contact structures
by John Etnyre (Georgia Tech) in HB 447

One of the fundamental problems in 3-dimensional contact geometry is the construction of tight contact structures on closed manifolds. Two obvious ways to try to construct such structures are via Legendrian surgery and admissible transverse surgery. It was long thought that when performed on a closed tight contact manifold these operations would yield a tight contact manifold. We show that this is not true for admissible transverse surgery. Along the way we discuss the relations between these two surgery operations and construct some contact structures with interesting properties.Host Department: Rice University-Mathematics
Submitted by shelly@rice.edu

4:00 pm Thursday, February 16, 2012
Colloquium: Actions of Higher Rank Abelian Groups: From Measure rigidity to Arithmeticity to Topology
by Anatole Katok (Penn State) in HB 227

While topology of a compact manifold allows to make inferences about periodic points of maps and, to a lesser extent, closed orbit of vector fields supported by the manifold, there is notoriously little relation between topology of a manifold and ergodic properties of diffeomorphisms and smooth flows supported by it. A somewhat related fact is that ergodic properties of such dynamical systems tell little about their geometric structure. All this changes in a dramatic way when one considers dynamical systems with multi-dimensional time, i.e. actions of higher rank abelian groups by diffeomorphisms of compact manifolds. Here is a representative case that will be discussed in the talk. Assume that k>1. Consider k commuting diffeomorphisms of a (k+1)- dimensional manifold M and assume that they preserve a measure that is sufficiently ``rich'' or ``stochastic''. This assumption will be explained in the talk, but for a person with some dynamical background one form of it states that all non-zero elements of the suspension action have positive entropy with respect to the measure. A striking topological conclusion is that the manifold is a ``slightly modified'' (finite factor of the) torus: there is homeomorphism from an open subset in finite cover of M that contains support of the lifted measure to the( k+1)-dimensional torus with a finite set removed. In particular the fundamental group of a finite cover of M contains a free abelian subgroup or rank (k+1). On the ergodic side we obtain precise information about the structure of the action. It is fully ``arithmetic'', i.e. exactly the same as an action by commuting hyperbolic matrices with integer entries and determinant 1 or -1 on the torus or its finite factor. Moreover, the correspondence is smooth the sense of Whithey on a set whose complement has arbitrary small measure. THis is the joint work with Federico Rodriguez Hertz based on our joint work with Boris Kalinin. Host Department: Rice University-Mathematics
Submitted by dani@rice.edu

4:00 pm Tuesday, February 21, 2012
Algebraic Geometry Seminar: Semi-algebraic horizontal subvarieties of Calabi-Yau type
by Radu Laza (Stony Brook) in HB 227

Except a few special cases (e.g. abelian varieties and K3 surfaces), the images of period maps for families of algebraic varieties satisfy non-trivial Griffiths' transversality relations. It is of interest to understand these images of period maps, especially for Calabi-Yau threefolds. In this talk, I will discuss the case when the images of period maps can be described algebraically. Specifically, I will show that if a horizontal subvariety Z of a period domain D is semi-algebraic and is stabilized by a large discrete group, then Z is automatically Hermitian symmetric with a totally geodesic embedding into the period domain D. Additionally, I will discuss the classification of the semi-algebraic cases for variations of Hodge structures of Calabi-Yau type. This is joint work with R. Friedman.Host Department: Rice University-Mathematics
Submitted by evanmb@gmail.com

4:00 pm Wednesday, February 22, 2012
Geometry-Analysis Seminar: Spectral Theory and Parreau-Widom Sets
by Jacob Stordal Christiansen (University of Copenhagen) in HB 227

In the talk, I'll discuss almost periodic Jacobi operators on $\ell^2(\Z)$. Such operators are selfadjoint and tend to have Cantor spectrum. Throughout, we shall focus on the class of operators whose spectrum (or essential spectrum) is an infinite gap set, $E$, of Parreau-Widom type. This notion is suitably defined via conformal mappings of $\C_+$ onto comb-like domains and it includes Cantor sets of positive measure. An all-important role will be played by the set $T_E$ of reflectionless operators on $E$. By a result of Remling, they form the natural limiting object for operators with a.c. spectrum on $E$. As will be explained, $T_E$ is topologically an infinite dimensional torus for most Parreau-Widom sets. We shall introduce the Szego class for E and show that all its elements are asymptotically almost periodic operators. It also follows that the associated orthogonal polynomials admit a power asymptotic behavior, aka Szego asymptotics.Host Department: Rice University-Mathematics
Submitted by damanik@rice.edu