Upcoming Seminars & Events

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March 29, 2017
2:30pm - 3:30pm
Applications of Noncommutative Harmonic Analysis in Engineering and the Applied Sciences

This talk will focus more on both the mathematics and the structural biology.

Speaker: Gregory Chirikjian, Johns Hopkins University
Location:
Fine Hall 224
March 29, 2017
3:00pm - 4:00pm
Embeddedness and convexity for hypersurfaces in hyperbolic space

I will talk a proof of the conjecture of Alexander and Currier on the embeddedness of a nonnegatively curved hypersurfaces in hyperbolic space. I will also discuss some recent works on hypersurfaces with nonnegative Ricci curvature in hyperbolic space.

Speaker: Jie Qing , UC Santa Cruz
Location:
Fine Hall 314
March 29, 2017
3:00pm - 4:00pm
Singular values of random band matrices: Marchenko-Pastur law and more

We consider the limiting spectral distribution of matrices of the form (R+X)(R+X)^∗/(2b_n+1), where X is an n by n band matrix of bandwidth b_n and R is a non random band matrix of bandwidth b_n. We show that the Stieltjes transform of spectrum of such matrices converges to the Stieltjes transform of a non-random measure. And the limiting Stieltjes transform satisfies an integral equation. For R=0, the integral equation yields the Stieltjes transform of the Marchenko-Pastur law. This is a joint work with Indrajit Jana.

Speaker: Alexander Soshnikov , UC Davis
Location:
Fine Hall 214
March 29, 2017
4:30pm - 5:30pm
Tableaux combinatorics of hopping particles and Koornwinder polynomials

The asymmetric simple exclusion process (ASEP) is a Markov chain describing particles hopping on a 1-dimensional finite lattice. Particles can enter and exit the lattice at the left and right boundaries, and particles can hop left and right in the lattice, subject to the condition that there is at most one particle per site. The ASEP has been cited as a model for traffic flow, protein synthesis, the nuclear pore complex, etc. In my talk I will discuss joint work with Corteel and with Corteel-Mandelshtam, in which we describe the stationary distribution of the ASEP and the 2-species ASEP using staircase tableaux and rhombic tilings. We also link these models to Askey Wilson polynomials and Macdonald-Koornwinder polynomials, which allows us to give combinatorial formulas for their moments.

Speaker: Lauren Williams , IAS/UCBerkeley
Location:
Fine Hall 314
March 30, 2017
9:30am - 10:30am
The many forms of rigidity for symplectic embeddings

We look at the following chain of symplectic embedding problems in dimension four.

$E(1,a) \to Z^4(A) E(1,a) \to C^4(A) E(1,a) \to P(A,bA) (b \in \mathbb{N}_{\geq 2}) E(1,a) \to T^4(A)$

Here $E(1,a)$ is a symplectic ellipsoid, $Z^4(A)$ is the symplectic cylinder $D^2(A) \times \mathbb{R}^2, C^4(A) = D^2(A) \times D^2(A)$ is the cube and $P(A,bA) = D^2(A) \times D^2(bA)$ the polydisc, and $T^4(A) = T^2(A) \times T^2(A)$, where $T^2(A)$ is the 2-torus of area $A$. In each problem we ask for the smallest $A$ for which $E(1,a)$ symplectically embeds. The answer is very different in each case: total rigidity, total flexibility with a hidden rigidity, and a two-fold subtle transition between them. The talk is based on works by Cristofaro-Gardiner, Frenkel, Latschev, McDuff, Muller, and myself.

Speaker: Felix Schlenk , Universite de Neuchatel, Switzerland
Location:
IAS Room S-101
March 30, 2017
10:45am - 11:45am
The stabilized symplectic embedding problem

I will discuss some recent work (mostly joint with Dan Cristofaro-Gardiner and Richard Hind) on the stabilized symplectic embedding problem for ellipsoids into balls. The main tools come from embedded contact homology.

 

Speaker: Dusa McDuff , Columbia University
Location:
IAS Room S-101
March 30, 2017
12:30pm - 1:30pm
What simple curves on surfaces know

 Simple closed curves on Riemann surfaces carry an interesting structure. An example is a novel symmetry satisfied by generating series for collections of such curves. This symmetry reflects the global geometry of certain moduli spaces (containing Teichmuller spaces), whose Diophantine study reveals new insight on mapping class groups. In this talk, we give an elementary discussion of some of these ideas.

Speaker: Peter (Jun Ho) Whang, Princeton University
Location:
Fine Hall 110
March 30, 2017
2:00pm - 3:30pm
The ground state construction of bilayer graphene

We consider a model of weakly-interacting electrons in bilayer graphene. Bilayer graphene is a 2-dimensional crystal consisting of two layers of carbon atoms in a hexagonal lattice. Our main result is an expression of the free energy and two-point Schwinger function as convergent power series in the interaction strength. In this talk, I discuss the properties of the non-interacting model, and exhibit three energy regimes in which the energy bands are qualitatively different. I then sketch how this decomposition may be used to carry out the renormalization group analysis used to prove our main result.  Joint work with Alessandro Giuliani. 

Speaker: Ian Jauslin , Institute for Advanced Study
Location:
Jadwin Hall 111
March 30, 2017
2:30pm - 3:30pm
Matrix Optimal Mass Transport: a Quantum Mechanical Approach

Please note special day and location (Thursday, McDonnell 102A).  Optimal mass transport (OMT) is a rich area of research with applications to numerous disciplines including econometrics, fluid dynamics, statistical physics, shape optimization, expert systems, and meteorology. The problem was originally formulated on the space of scalar probability densities. In the present talk, we describe a non-commutative generalization of OMT, to the space of Hermitian matrices with trace one, and to the space of matrix-valued probability densities. Our approach follows a fluid dynamics formulation of OMT, and utilizes certain results from the quantum mechanics of open systems, in particular the Lindblad equation. The  non-commutative OMT introduces a natural distance metric for matrix-valued distributions. Our original motivation aimed at developing tools for multivariate time series modeling and matrix-valued power spectral analysis. However, the emergent theory turned out to have immediate applications in diffusion tensor imaging (DTI) where images are now tensor fields representing orientation and shape of brain white-matter. Thus, the framework we have developed allows us to compare, interpolate and fuse DTI images in a disciplined manner and, thereby, may lead to high resolution advances that in turn promise improved in vivo imaging of important brain structures. In addition, our formulation allows determining the gradient flow for the quantum entropy relative to the matricial Wasserstein metric induced by the non-commutative OMT. This may have implications to some key issues in quantum control and information theory.

Speaker: Yongxin Chen, Department of Medical Physics, Memorial Sloan Kettering Cancer Center
Location:
McDonnell Hall 102A
March 30, 2017
4:30pm - 5:30pm
Brief survey of computer assisted proofs for partial differential equations

 I will present a brief survey of computer assisted methods of studying partial differential equations that I have worked on. The methods I am going to discuss allow for obtaining proofs of the existence of particular solutions of a certain class of PDEs in a prescribed range of parameters.  I will discuss opportunities  and limitations of the presented approach. In particular most of the presented results have not been obtained using known techniques of 'classical analysis'.  I will focus on two particular examples from my research, namely 1) a proof of the existence of globally attracting solutions for the 1d viscous Burgers equation (with non-autonomous forcing) https://arxiv.org/abs/1403.7170, and 2) recent proof of the heteroclinic connections in the 1d Ohta-Kawasaki (diblock copolymers) model https://arxiv.org/abs/1703.01022

Speaker: Jacek Cyranka , Rutgers University
Location:
Fine Hall 322
March 30, 2017
4:30pm - 5:30pm
Galois Representations for the general symplectic group

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.

Speaker: Arno Kret , University of Amsterdam
Location:
IAS Room S-101
March 30, 2017
4:30pm - 5:30pm
Knot traces and concordance

A conjecture of Akbulut and Kirby from 1978 states that the concordance class of a knot is determined by its 0-surgery. In 2015, Yasui disproved this conjecture by providing pairs of knots which have the same 0-surgeries yet which can be distinguished in (smooth) concordance by an invariant associated to the four-dimensional traces of such a surgery. In this talk, I will discuss joint work with Lisa Piccirillo in which we construct many pairs of knots which have diffeomorphic 0-surgery traces yet some of which can be distinguished in smooth concordance by the Heegaard Floer d-invariants of their double branched covers. If time permits, I will also discuss the applicability of this work to the existence of interesting invertible satellite maps on the smooth concordance group.

Speaker: Allison Miller , University of Texas, Austin
Location:
Fine Hall 314
March 31, 2017
9:30am - 5:00pm
Columbia-Princeton Probability Day 2017

Speakers:

  • Marek Biskup (UCLA)
  • Mark Rudelson (Michigan)
  • Vladas Sidoravicius (NYU)
  • Fabio Toninelli (Lyon)

 Junior Speakers:

  • Tatyana Shcherbyna (Princeton)
  • Yi Sun (Columbia)

PLEASE CLICK HERE FOR CONFERENCE LINK.  

Contact Ramon van Handel for further details.

 

Speaker: ,
Location:
Lewis Library 120
March 31, 2017
5:00pm - 6:00pm
Solving packing problems by linear programming

Part 2: Cohn-Elkies linear programming bounds and modular forms.

The sphere packing problem asks which biggest portion of the euclidean d-dimensional space can be covered by non-overlapping unit balls. In most dimensions d this question is believed be an extremely difficult combinatorial geometric problem. However, in dimensions 8 and 24 the sphere the sphere packing problem has a surprisingly simple solution based on linear programming bounds.The goal of this series of talks is to explain the ideas behind this solution. 

Speaker: Maryna Viazovska , Humboldt University
Location:
Fine Hall 314
April 3, 2017
3:00pm - 4:00pm
Dispersive estimates for undergraduates

In this extremely elementary seminar, I discuss how to prove some scaling-sharp dispersive estimates for the linear Schrodinger equation using purely physical space methods. The main part of the argument uses only the fundamental theorem of calculus and is accessible to undergraduates, even those not enrolled at Princeton. 

Speaker: Willie Wong , Michigan State University
Location:
Fine Hall 314
April 3, 2017
4:00pm - 5:00pm
Sample-optimal inference, computational thresholds, and the methods of moments

We propose an efficient meta-algorithm for Bayesian inference problems based on low-degree polynomials, semidefinite programming, and tensor decomposition. The algorithm is inspired by recent lower bound constructions for sum-of-squares and related to the method of moments. Our focus is on sample complexity bounds that are as tight as possible (up to additive lower-order terms) and often achieve statistical thresholds or conjectured computational thresholds.  Our algorithm recovers the best known bounds for partial recovery in the stochastic block model, a widely-studied class of inference problems for community detection in graphs. We obtain the first partial recovery guarantees for the mixed-membership stochastic block model (Airoldi et el.) for constant average degree—up to what we conjecture to be the computational threshold for this model. Our algorithm also captures smooth trade-offs between sample and computational complexity, for example, for tensor principal component analysis. In contrast, we show that our algorithm exhibits a sharp computational threshold for the stochastic block model with multiple communities beyond the Kesten–Stigum bound—giving evidence that this task may require exponential time.  The basic strategy of our algorithm is strikingly simple: we compute the best-possible low-degree approximation for the moments of the posterior distribution of the parameters and use a robust tensor decomposition algorithm to recover the parameters from these approximate posterior moments.

Joint work with Samuel B. Hopkins (Cornell).

Speaker: David Steurer, Cornell University
Location:
Fine Hall 214
April 3, 2017
5:15pm - 6:15pm
Solving packing problems by linear programming

Part 3: Fourier interpolation.

The sphere packing problem asks which biggest portion of the euclidean d-dimensional space can be covered by non-overlapping unit balls. In most dimensions d this question is believed be an extremely difficult combinatorial geometric problem. However, in dimensions 8 and 24 the sphere the sphere packing problem has a surprisingly simple solution based on linear programming bounds.The goal of this series of talks is to explain the ideas behind this solution. 

Speaker: Maryna Viazovska , Humboldt University
Location:
Fine Hall 314
April 4, 2017
4:30pm - 5:30pm
Tropical Schemes

Tropical scheme theory is a method of describing tropical varieties with equations, in order to incorporate more foundations and constructions from modern algebraic geometry into the subject. I'll give an overview of this topic, emphasizing recent connections to matroid theory.

Speaker: Noah Giansiracusa , Swarthmore College
Location:
Fine Hall 322
April 5, 2017
2:30pm - 3:30pm
Demixing Sines and Spikes: Spectral Super-resolution in the Presence of Outliers

In this talk we consider the problem of super-resolving the line spectrum of a multisinusoidal signal from a finite number of samples, some of which may be completely corrupted. Measurements of this form can be modeled as an additive mixture of a sinusoidal and a sparse component. We propose to demix the two components and super-resolve the spectrum of the multisinusoidal signal by solving a convex program. Our main theoretical result is that-- up to logarithmic factors-- this approach is guaranteed to be successful with high probability for a number of spectral lines that is linear in the number of measurements, even if a constant fraction of the data are outliers. We show that the method can be implemented via semidefinite programming and explain how to adapt it in the presence of dense perturbations, as well as exploring its connection to atomic-norm denoising. In addition, we propose a fast greedy demixing method which provides good empirical results when coupled with a local nonconvex-optimization step.

Speaker: Carlos Fernandez-Granda, NYU
Location:
Fine Hall 224
April 5, 2017
3:00pm - 4:00pm
Stable CMC hypersurfaces

In a joint work with N. Wickramasekera (Cambridge) we develop a regularity and compactness theory for a class of codimension-1 integral n-varifolds with generalised mean curvature in L^{p}_{loc} for some p > n. Subject to suitable variational hypotheses on the regular part (namely stationarity and stability for the area functional with respect to variations that preserve the "enclosed volume") and two necessary structural assumptions, we show that the varifolds under consideration are "smooth" (and have constant mean curvature in the classical sense) away from a closed singular set of codimension 7. In the case that the mean curvature is non-zero, the smoothness is to be understood in a generalised sense, i.e. also allowing the tangential touching of two smooth CMC hypersurfaces (e.g. two spheres touching).

Speaker: Costante Bellettini , University College London
Location:
Fine Hall 314

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