Rice University Math Department Calendar

     October 2017          November 2017          December 2017    
 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  6  7             1  2  3  4                   1  2
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    3  4  5  6  7  8  9
 15 16 17 18 19 20 21   12 13 14 15 16 17 18   10 11 12 13 14 15 16
 22 23 24 25 26 27 28   19 20 21 22 23 24 25   17 18 19 20 21 22 23
 29 30 31               26 27 28 29 30         24 25 26 27 28 29 30
                                               31

4:00 pm Tuesday, November 7, 2017
AGNT: Reduction of dynatomic curves: The good, the bad, and the irreducible
by Andrew Obus (University of Virginia) in HBH 227

The dynatomic modular curves parameterize one-parameter families of dynamical systems on P^1 along with periodic points (or orbits). These are analogous to the standard modular curves parameterizing elliptic curves with torsion points (or subgroups). For the family x^2 + c of quadratic dynamical systems, the corresponding modular curves are smooth in characteristic zero. We give several results about when these curves have good/bad reduction to characteristic p, as well as when the reduction is irreducible. We will also explain some motivation from the uniform boundedness conjecture in arithmetic dynamics.Host Department: Rice University-Mathematics
Submitted by jb93@rice.edu

4:00 pm Wednesday, November 8, 2017
Colloquium: Complexity of triviality in topology with applications to geometric variational problems
by Alexander Nabutovsky (University of Toronto) in HBH 227

The first theme of the talk is that some geometric objects have trivial topology, but it is very difficult to see that this, indeed, is the case. The second theme is that this phenomenon implies the ruggedness of some moduli spaces in differential and combinatorial geometry, and that it even forces the existence of non-trivial solutions of some problems in geometric calculus of variations. These phenomena were previously known for dimensions >4 (joint work with Shmuel Weinberger). However, recently we managed to prove that they already exist in dimension 4 (joint work with Boris Lishak). In particular, I will show that (1) it is very difficult to untie some trivial 2-knots in the four-dimensional space; (2) there exist ``many" contractible 2-dimensional complexes, which are extremely difficult to contract and for each pair of them it is also very difficult to see that they are homotopy equivalent to each other; (3) for each n>3 there exist infinitely many distinct local minima of curvature-pinching sup |K| diam^2 (C^0 norm of the sectional curvature normalized by the square of the diameter) on the space of Riemannian structures on the n-sphere; (4) in high-dimensions there exist many non-trivial ``thick" knots of codimension one (unlike the usual knot theory).Host Department: Rice University-Mathematics
Submitted by gchambers@rice.edu

4:00 pm Thursday, November 9, 2017
Colloquium: Lengths of periodic geodesics and related questions
by Regina Rotman (University of Toronto) in HBH 227

Thirty five years ago M. Gromov asked if it is true that the length of a shortest periodic geodesic on a closed Riemannian manifold does not exceed c(n)Vol^{1/n}, where Vol denotes the volume of the manifold, and c(n) is a constant that depends only on its dimension n. This question and a similar question with the diameter of the manifold instead of Vol^{1/n} are still open. I will discuss the solutions to these and related questions in dimension 2, as well as the upper bounds for periodic geodesics, stationary geodesic nets, loops and minimal surfaces in higher dimensions.Host Department: Rice University-Mathematics
Submitted by gchambers@rice.edu

4:00 pm Tuesday, November 14, 2017
AGNT: Graph formulas for tautological cycles
by Renzo Cavalieri (Colorado State University) in HBH 227

The tautological ring of the moduli space of curves is a subring of the Chow ring that, on the one hand, contains many of the classes represented by "geometrically defined" cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By this I mean that we can explicitly describe a set of additive generators, which are indexed by suitably decorated graphs. The study of the tautological ring was initiated by Mumford in the '80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that we are making progress in a much needed development of a "calculus on the tautological ring", i.e. a way to effectively compute and compare expressions in the tautological ring. An example of such a "calculus" consists in describing formulas for geometrically described classes (e.g. the hyperelliptic locus) via meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense "graph formulas" give a good example of what the adjective "meaningful" meant in the previous sentence, and present a few examples of graph formulas. The original work presented is in collaboration with Nicola Tarasca and Vance Blankers.Host Department: Rice University-Mathematics
Submitted by jb93@rice.edu

3:00 pm Wednesday, November 15, 2017
Joint Math/CAAM Seminar: Stability of Minkowski space and asymptotics of the metric
by Peter Hintz (University of California, Berkeley) in DH 1064

I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a suitable compactification of R^4 to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity; I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. This talk is based on joint work with András Vasy.Host Department: Rice University-Mathematics
Submitted by mathweb@rice.edu

4:00 pm Wednesday, November 15, 2017
Geometry-Analysis Seminar: Stability results of generalized Beltrami fields and applications to vortex structures in the Euler equations
by David Poyato (University of Granada) in HBH 227

Strong Beltrami fields, that is, 3D velocity fields whose vorticity is the product of itself by a constant factor, are particular solutions to the Euler equations that have long played a key role in Fluid Mechanics. Its importance relies on the Lagrangian theory of turbulence as they were expected to exhibit chaotic configurations. Very recently, this ancient conjecture of Lord Kelvin was positively answered by Alberto Enciso and Daniel Peralta-Salas (ICMAT, Spain). Specifically, there are strong Beltrami fields exhibiting any type of linked vortex lines and tubes of arbitrarily complicated topology. Nevertheless, such existence result is quite tight in the sense that Beltrami fields with non-constant factor (generalized Beltrami fields) are “rare”. Thus, the existence of turbulent configurations is limited to Beltrami fields with a constant factor. The aim of this talk is twofold. First, we will review the state of the art in this topic. Second, we will show that although a full stability result is not possible, there are certain privileged ways to perturb a strong Beltrami field and obtain Beltrami fields with a non-constant factor that even realize arbitrarily complicated vortex structures. This partial stability will be captured in terms of an "almost global” and a “local" stability theorem. The proof relies on analyzing the well-posedness and propagation of compactness and regularity of an innovative iterative scheme of Grad-Rubin type inspired by some numerical methods coming from Astrophysics.Host Department: Rice University-Mathematics
Submitted by ctan@rice.edu

4:00 pm Tuesday, November 21, 2017
AGNT: Brauer p-dimensions of complete discretely valued fields
by Nivedita Bhaskhar (UCLA) in HBH 227

The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. One can analogously define the Brauer-p-dimension of F for p, a prime, by restricting to algebras with period, a power of p. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields. In this talk, we look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case. More specifically, let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p > 0 and p-rank n (= [k:k^p]). It was shown by Parimala and Suresh that the Brauer p-dimension of K lies between n/2 and 2n. We will investigate the Brauer p-dimension of K when n is small and find better bounds. For a general n, we will also construct a family of examples to show that the optimal upper bound for the Brauer-p-dimension of such fields cannot be less than n+1. These examples embolden us to conjecture that the Brauer p-dimension of K lies between n and n+1. The proof involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier. This is joint work with Bastian Haase.Host Department: Rice University-Mathematics
Submitted by ae22@rice.edu

4:00 pm Tuesday, November 28, 2017
AGNT: The unreasonable effectiveness of the Polya-Vinogradov inequality
by Leo Goldmakher (Williams College) in HBH 227

The Polya-Vinogradov inequality, an upper bound on character sums proved a century ago, is essentially optimal. Unfortunately, it's also not so useful in applications, since it's nontrivial only on long sums (while in practice one usually needs estimates on sums which are as short as possible). The best tool we have to handle shorter sums is the Burgess bound, discovered in 1957; this is generally considered to supersede Polya-Vinogradov, both because its proof is "deeper" (building on results from algebraic geometry) and because it is more applicable. In this talk I will introduce and motivate both of these bounds, and then describe the unexpected result (joint with Elijah Fromm, Williams '17) that even a tiny improvement of the (allegedly weaker) Polya-Vinogradov inequality would imply a major improvement of the (supposedly superior) Burgess bound. I'll also discuss a related connection between improving Polya-Vinogradov and the classical problem of bounding the least quadratic nonresidue (joint with Jonathan Bober, University of Bristol).Host Department: Rice University-Mathematics
Submitted by ae22@rice.edu

4:30 pm Friday, December 1, 2017
Rice Geometry Lab Poster Presentation: in HBH 438

Students from two projects of the Rice Geometry Lab will be doing a poster presentation to showcase what they have done this semester. Prof. Roy and Prof. Li have been directing teams of undergraduate students this semester on the projects 'Music and Geometry’ and 'Understanding the works by Nash on isometric embeddings'. Graduate students, Yikai Chen and Xian Dai, have been helping the teams out. Come take a look at what some of your fellow undergraduate students are doing! We will be looking for interested undergraduate students and people who are interested in directing projects for the participating students. All are welcome and encouraged to attend.Host Department: Rice University-Mathematics
Submitted by lpl1@rice.edu

4:00 pm Thursday, January 25, 2018
Colloquium: Volume, Renormalized
by Jeff Brock in HBH 227

Much of the appeal of Thurston's work on the geometrization of 3-manifolds lies in its hands-on simplicity: synthetic and combinatorial constructions involving geodesic loops, simplicial surfaces, and coarse "quasi-geodesics" gain their power from rigidity theorems due to Mostow and Sullivan that guarantee that rough geometric estimates ensure spot-on control. But considerable analytic work of Ahflors and Bers lies at the foundation, and only recently has work of Graham and Witten been found to give clues to an analytic framework for understanding Thurston's intuition and conjectures. In this talk I will describe history context and recent developments that use Graham and Witten's notion of "renormalized volume", as elaborated by Krasnov and Schlenker, to provide a satisfying analytic explanation for the connection between volumes of hyperbolic 3-manifolds that fiber over the circle and Weil-Petersson lengths of closed geodesics on moduli space. I'll discuss an array of applications to Weil-Petersson geometry as well as some new results. This talk describes joint work with Ken Bromberg and Martin Bridgeman.Host Department: Rice University-Mathematics
Submitted by ml28@rice.edu

4:00 pm Friday, January 26, 2018
Undergraduate Colloquium: Billiards in Polygons
by Ronen Mukamel (Rice University) in HBH 227

In the study of dynamical systems, mathematicians attempt to understand the long term behavior of systems which evolve in time by a mathematical set of rules. A polygon in the plane determines such a system via idealized billiard trajectories. The study of billiards in polygons gives rise to beautiful connections between the fields of dynamical systems, geometry, algebra, and number theory. In this talk, I will describe some of the motivating questions in the study of billiards, some of the celebrated results in this area, as well as some of the connections to other areas of mathematics.Host Department: Rice University-Mathematics
Submitted by sswang@rice.edu