Upcoming Seminars & Events

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February 2, 2015
3:15pm - 4:30pm
A model for studying double exponential growth
Analysis Seminar

We discuss a model for studying spontaneous phenomena in the 2d Euler equations for incompressible fluid flow. We tie the behaviour of the model to the behavior of the actual Euler equations.  (Joint work with A. Tapay.)

Speaker: Nets Katz, Caltech
Location:
Fine Hall 314
February 2, 2015
4:30pm - 5:30pm
Efficient numerical methods for wave scattering in periodic geometries
PACM/Applied Mathematics Colloquium

A growing number of technologies (communications, imaging, solar energy, etc) rely on manipulating linear waves at the wavelength scale, and accurate numerical modeling is key for device design.  I will focus on diffraction problems where time-harmonic scalar waves scatter from piecewise-uniform periodic media.  After reviewing the integral equation method, I explain two innovations that allow our solvers to be high-order, robust, and optimal O(N) complexity: 1) new surface quadrature schemes in 2D and 3D compatible with the fast multipole method, and 2) a simple way to "periodize" the integral equation so that the unknowns live on only a single period of the geometry.  The resulting linear systems are solved iteratively or in a fast direct fashion.  I will present efficient solvers for gratings with up to a thousand periodic interfaces or inclusions.  We are extending the techniques to other boundary value problems such as Stokes flow.  This is joint work with: Min Hyung Cho, Jun Lai, Adrianna Gillman, Leslie Greengard, Andreas Kloeckner, Mike O'Neil, Zydrunas Gimbutas.

Speaker: Alex Barnett, Dartmouth College
Location:
Fine Hall 214
February 3, 2015
4:30pm - 5:30pm
Height Fluctuations in Interacting Dimers
Mathematical Physics Seminar

Perfect matchings of Z^2(also known as non-interacting dimers on the square lattice) are an exactly solvable 2D statistical mechanics model. It is known that the associated height function behaves at large distances like a massless gaussian field, with the variance of height gradients growing logarithmically with the distance. As soon as dimers mutually interact, via e.g. a local energy function favoring the alignment among neighboring dimers, the model is not solvable anymore and the dimer-dimer correlation functions decay polynomially at infinity with a non-universal (interaction-dependent) critical exponent. We prove that, nevertheless, the height fluctuations remain gaussian even in the presence of interactions, in the sense that all their moments converge to the gaussian ones at large distances. The proof is based on a combination of multiscale methods with the path-independence properties of the height function. Joint work with V. Mastropietro and F. Toninelli.

Speaker: Alessandro Giuliani, University of Rome 3
Location:
February 3, 2015
4:30pm - 5:30pm
On the rationality of the logarithmic growth filtration of solutions of $p$-adic differential equations
Princeton University/IAS Number Theory Seminar

Please note special day.  We consider an ordinary linear $p$-adic differential equation Dy=d^ny/dx^n+a_{n-1}d^{n-1}y/dx^{n-1}+\dots+a_0y=0, a_i\in\mathbb{Z}_p[p^{-1}] whose formal solutions in $\mathbb{Q}_p$ converge in the open unit disc $|x|<1$. In 1973, Dwork proved that $y$ has a log-growth $n-1$, that is,|y|_{\rho}=O((\log{1/\rho})^{1-n}) as $\rho\uparrow 1$, where $|y|_{\rho}$ denotes the $\rho$-Gaussian norm of $y$. Moreover, Dwork defined the log-growth filtration of the solution space of $Dy=0$ by measuring the log-growth of $y$. Then, Dwork conjectured that the log-growth filtration can be compared with the Frobenius slope filtration when $Dy=0$ admits a Frobenius structure. Recently, some partial results on Dwork's conjecture have been obtained by Andr\'e, Chiarellotto-Tsuzuki, and Kedlaya. In this talk, we discuss the rationality of breaks of the log-growth filtration.

Speaker: Shun Ohkubo, University of Tokyo
Location:
Fine Hall 214
February 4, 2015
4:30pm - 5:30pm
Stabilization of control systems: From the water clocks to the regulation of rivers
Department Colloquium

A control system is a dynamical system on which one can act by using controls. For these systems a fundamental problem is the stabilization issue: Is it possible to stabilize a given unstable equilibrium by using suitable feedback laws? (Think to the classical experiment of an upturned broomstick on the tip of one's finger.) On this problem, we present some pioneer devices and works (Ktesibios, Watt, Maxwell, Lyapunov...), some more recent results, and an application to the regulation of the rivers La Sambre and La Meuse in Belgium. A special emphasize is put on positive or negative effects of the nonlinearities.

Speaker: Jean-Michel Coron, Université Pierre et Marie Curie
Location:
Fine Hall 314
February 5, 2015
2:00pm - 3:30pm
The rare interaction limit in a fast-slow mechanical system
Ergodic Theory & Statistical Mechanics

In 2008 Gaspard and Gilbert suggested a two-step strategy to derive the 'macroscopic' heat equation from the 'microscopic' kinetic equation. Their model consisted of a chain of localized and rarely interacting hard disks. For a paradigm billiard model - realizing the first, truly dynamical part of the GG-strategy - we obtain the 'mesoscopic' master equation describing a Markov jump process for the energies of the particles.  Joint work with P. Bálint, P. Nándori and IP. Tóth.

Speaker: Domokos Szász, Budapest University of Technology
Location:
Fine Hall 601
February 5, 2015
3:00pm - 4:00pm
TBA - Alex Scott
Discrete Mathematics Seminar
Speaker: Alex Scott, Oxford University
Location:
Fine Hall 224
February 5, 2015
4:30pm - 5:30pm
On the formal degrees of square-integrable representations of odd　special orthogonal and metaplectic groups
Princeton University/IAS Number Theory Seminar

The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint gamma-factor of its L-parameter. We prove the formal degree conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality. This is joint work with Erez Lapid and Zhengyu Mao.

Speaker: Atsushi Ichino, Kyoto University
Location:
IAS Room S-101
February 6, 2015
1:30pm - 2:30pm
Path Products in Projective Space
Symplectic Geometry Seminar

We compute the mod 2 homology of the space of paths in CP^n with endpoints in RP^n, and its algebra structure with respect to the Pontryagin-Chas-Sullivan product . Our method combines Morse theory with geometry. Joint work with Alexandru Oancea.

Speaker: Nancy Hingston, College of New Jersey
Location:
IAS Room S-101
February 9, 2015
3:00pm - 4:00pm
Asymptotic behaviour of some CMC hypersurfaces of Minkowski space
Analysis Seminar

Please note special time.  We study time-like constant mean curvature hypersurfaces of Minkowski space from the point of view of their initial value problems. We focus on initial data close to those generating spherically symmetric expanding solutions, such as anti de Sitter space. Using intuitions from mathematical relativity, we explain how a global stability'' statement cannot be true due to the presence of cosmological horizons. Yet using the same intuition we discuss how to easily obtain a spatially localised stability result after a careful geometric reformulation of the problem.

Speaker: Willie Wong, EPFL
Location:
Fine Hall 314
February 9, 2015
4:00pm - 5:00pm
TBA - Andre' Lisibach
Analysis Seminar

Please note special time.

Speaker: Andre' Lisibach, ETH
Location:
TBD
February 9, 2015
4:30pm - 5:30pm
Disordered Hyperuniform Materials: New States of Matter
PACM/Applied Mathematics Colloquium

While there are four commonly observed states of matter (solid crystal, liquid, gas, and plasma), we have known for some time now that there exist many other forms of matter. For example, both quasicrystals and liquid crystals are states of matter that possess properties that are intermediate between those of crystals and conventional liquids. The focus of my talk will be disordered hyperuniform many-body systems [1], which can be regarded to be new distinguishable states of disordered matter in that they behave more like crystals or quasicrystals in the manner in which they suppress large-scale density fluctuations, and yet are also like liquids and glasses because they are statistically isotropic structures with no Bragg peaks. Thus, disordered hyperuniform systems can be regarded to possess a "hidden order" that is not apparent on short length scales while being structurally rotationally invariant. I will describe a variety of different examples of such disordered states of matter, both equilibrium and nonequilibrium varieties. I will demonstrate that there exist classical ground states that are hyperuniform and disordered in a high-density regime down to some critical density, below which the system undergoes a phase transition to ordered states [2]. Disordered hyperuniform systems appear to be endowed with novel physical properties, including complete photonic band gaps comparable in size to those in photonic crystals [3] and improved electronic band-gap properties. Moreover, we have recently shown that photoreceptor cell patterns (responsible for detecting light) in avian retina have evolved to be disordered and hyperuniform [4].

1. S. Torquato and F. H. Stillinger, "Local Density Fluctuations,
Hyperuniform Systems, and Order Metrics," Phys. Rev. E, 68, 041113 (2003).

2. R. D. Batten, F. H. Stillinger and S. Torquato, "Classical Disordered
Ground States: Super-Ideal Gases, and Stealth and Equi-Luminous
Materials," J. Appl. Phys., 104, 033504 (2008).

3. M. Florescu, S. Torquato and P. J. Steinhardt, "Designer Disordered
Materials with Large, Complete Photonic Band Gaps," Proc. Nat. Acad. Sci.,
106, 20658 (2009).

4. Y. Jiao, T. Lau, H. Haztzikirou, M. Meyer-Hermann, J. C. Corbo, and S.
Torquato, "Avian Photoreceptor Patterns Represent a Disordered
Hyperuniform Solution to a Multiscale Packing Problem," Phys. Rev. E, 89,
022721 (2014).

Speaker: Sal Torquato , Princeton University
Location:
Fine Hall 214
February 10, 2015
4:30pm - 5:30pm
TBA - Michael Kiessling
Mathematical Physics Seminar
Speaker: Michael Kiessling, Rutgers University
Location:
February 12, 2015
4:30pm - 6:00pm
TBA - Gautam Iyer
Analysis of Fluids and Related Topics
Speaker: Gautam Iyer, Carnegie Mellon
Location:
Fine Hall 322
February 12, 2015
4:30pm - 5:30pm
TBA - Baldwin
Topology Seminar
Speaker: John Baldwin, Boston College
Location:
Fine Hall 314
February 12, 2015
4:30pm - 5:30pm
Kottwitz-Rapoport conjecture on crystals with additional structure
Princeton University/IAS Number Theory Seminar

In 1972, Mazur showed that the Newton polygon of a crystal lies below the Hodge polygon of the associated isocrystal and the two polygons have the same end points. In 2003, Kottwitz and Rapoport showed that the converse is true, i.e., given two such polygons, there exists a crystal with given polygons as its Hodge polygon and Newton polygon respectively. Kottwitz and Rapoport conjectured a similar statement for crystals with additional structure. This conjecture plays an important role in the study of reduction of Shimura varieties. In this talk, I will explain this conjecture, its relation to the Shimura varieties, and I will discuss some ideas of the proof.

Speaker: Xuhua He, University of Maryland
Location:
Fine Hall 214
February 13, 2015
1:30pm - 2:30pm
TBA - Luis Diogo
Symplectic Geometry Seminar
Speaker: Luis Diogo, Columbia University
Location:
IAS Room S-101
February 16, 2015
3:15pm - 4:30pm
TBA - Yannis Angelopoulos
Analysis Seminar
Speaker: Yannis Angelopoulos, University of Toronto
Location:
Fine Hall 314
February 16, 2015
4:30pm - 5:30pm
Hypothesis Testing on Random Networks
PACM/Applied Mathematics Colloquium

I will discuss some new hypothesis testing problems on random graph models. A popular example is the problem of detecting community structure in a graph. Here we will consider more exotic situations, such as testing one of the basic assumption in social network analysis: whether the graph has "geometric structure". We will also talk about dynamical models of random graphs (such as preferential attachment), and how to test different hypotheses on the "history" of these graphs.

Speaker: Sebastien Bubeck, Princeton University - ORFE
Location:
Fine Hall 214
February 19, 2015
2:30pm - 3:30pm
Existence of Lefschetz fibrations on Stein/Weinstein domains
Symplectic Geometry Seminar

Please note special day and location.   I will describe joint work with E. Giroux in which we show that every Weinstein domain admits a Lefschetz fibration over the disk (that is, a singular fibration with Weinstein fibers and Morse singularities). We also prove an analogous result for Stein domains in the complex analytic setting. The main tool used to prove these results is Donaldson's quantitative transversality.

Speaker: John Pardon, Stanford University
Location:
TBD