Title. Geometry and turbulent dissipation in 3D fluid flows

Abstract. Experiments, as well as computational simulations of turbulent

flows indicate that the regions of intense fluid activity organize

in coherent vortex structures, and in particular, in vortex

filaments.

Identifying the role that the vortex filaments play in the theory of

turbulent cascades and turbulent dissipation in 3D (incompresible,

viscous) flows, modeled by the 3D Navier-Stokes equations, has

been one of the central problems in turbulence since G.I. Taylor's

fundamental work in the 1930's. Mainly based on measurements of

the wind tunnel turbulence past a uniform grid, Taylor concluded

his observations with the following,

"It seems that the stretching of vortex filaments must be

regarded as the principal mechanical cause of the high rate

of dissipation which is associated with turbulent motion".

The goal of this lecture is to present an overview of a recent

work featuring several rigorous, mathematical results--derived

directly from the 3D Navier-Stokes equations--supporting

Taylor's view on turbulent dissipation.

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Title. Can Mathematicians Improve the Teaching of Mathematics?

Abstract. What we as mathematicians choose to do at the university and college level affects the teaching of mathematics at the K-12 level – because we educate the teachers. Two documents that represent a fairly broad and widely shared consensus among mathematicians at all levels have emerged in the past 5 years: The “Common Core State Standards – Mathematics” (CCSS-M) and the “Mathematical Education of Teachers II” (MET II). For mathematics to be high on the agenda of schools of education, mathematicians must put it there. In preparation for this colloquium, I suggest you read two recent short articles that appeared in the AMS Notices:

1. Jason Zimba, “The Common Core and the Potential for Mathematicians to Improve the Teaching of School Mathematics,” AMS Notices, Vol. 63, No. 2 (Communications Column, February 2016).

2. Jill Newton, et alia, “How Well Are Secondary Mathematics Teacher Education Programs Aligned with the Recommendations Made in MET II?” AMS Notices , Vol. 61, No. 3 (Doceamus Column, March 2014).

This colloquium is an invitation to join a movement. We cannot escape making a choice. But first you have to satisfy yourself that something needs to be done. Doing nothing is also a choice.

Please find the slides for the talk here.

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Title. Topology of quadrature domain and the valence of harmonic polynomials

Abstract. A quadrature domain is the domain whose Schwarz reflection has the analytic continuation over the domain. The topology of the quadrature domains can be studied by considering the dynamics under the iteration of the Schwarz reflections. The problem is related to the open problem of finding the maximal number of roots for a harmonic polynomial of a given degree. This is a joint work with Nikolai Makarov. ]]>

Title. The Ground State Energy of Heavy Atoms

Abstract. We begin with a review of the mathematical description of heavy atoms, i.e., of atoms with large atomic number $Z$: The Hamiltonian was suggested in physics on the basis of an approximation of quantum electro-dynamics and takes relativistic effects into account which is essential since the innermost electrons move very fast. It is the so called no-pair-operator in the Furry picture.We show that to leading order (large $Z$ and fixed quotient of $Z$ and velocity of light) the lowest spectral point (ground state energy) is -- as for the non-relativistic Schrödinger operator -- is $-c_1 Z^{7/3}$ (Lieb and Simon 1977) $. This is corrected $c_2 Z^2$ (Scott correction). Contrary to the leading term, this term differs from the non-relativistic case and yields quantitative reasonable results when compared with measured values.

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Title. An eigensystem approach to localization and three related problems in matrix theory

Abstract. We will formulate three problems in matrix theory related to diagonalization of special banded matrices. Their solutions illustrate the new approach to Anderson localization at high disorder, based on finite volume eigensystems, which we will describe in the second part of the talk.

(Joint work with A. Klein.)]]>

Title. Bigeodesics in first-passage percolation.

Abstract. First-passage percolation is a model for a random metric space, produced by assigning i.i.d. non-negative weights (t_e) to edges of Z^2 and considering the weighted graph metric. A number of longstanding conjectures exist regarding the behavior of infinite geodesics (infinite paths whose finite subsegments are point-to-point geodesics). Notable among them is the claim that, under mild assumptions on the distribution of t_e, there should a.s. be no doubly infinite geodesic ("bigeodesic"). In the 90's, Licea and Newman showed that, under an unproven curvature assumption on the model's "limiting shape" (which describes the shape of large balls in the random metric), every infinite geodesic a.s. has asymptotic direction, and there is a full-measure set D of [0, 2 \pi) such that for any \theta in D, there is no bigeodesic with an end directed in direction \theta. We will discuss new results on the bigeodesic conjecture showing, under the assumption that the limiting shape's boundary is differentiable, there is a.s. no bigeodesic with one end directed in any deterministic direction. This rules out existence of ground state pairs whose interface has a deterministic direction in the related disordered ferromagnet model.

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Title. Ridges and valleys in the high excursion sets of Gaussian random fields

Abstract. It is well known that normal random variables do not like taking large values. Therefore, a continuous Gaussian random field on a compact set does not like exceeding a large level. If it does exceed a large level at some point, it tends to go back below the level a short distance away from that point. One, therefore, does not expect the excursion set above a high for such a field to possess any interesting structure. Nonetheless, if we want to know how likely are two points in such an excursion set to be connected by a path ("a ridge") in the excursion set, how do we figure that out? If we know that a ridge in the excursion set exists (e.g. the field is above a high level on the surface of a sphere), how likely is there to be also a valley (e.g. the field going to below a fraction of the level somewhere inside that sphere)?

We use the large deviation approach. Some surprising results (and pictures) are obtained.

(Joint work with R. Adler and E. Moldavskaya)

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Title. Variable Lebesgue spaces: theory and applications

Abstract. In this talk we will give a general overview of the theory of variable Lebesgue spaces, emphasizing their similarities and differences with the classical Lebesgue spaces. We will also review some of these applications. We will end the talk with an open (though possibly trivial) question that might benefit from numerical exploration.

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Title. Mean field Ising models

Abstract. In this talk we consider the asymptotics of the log partition function of an Ising model on a sequence of finite but growing graphs/matrices. We give a sufficient condition for the mean field prediction to the log partition function to be asymptotically tight, which in particular covers all regular graphs with degree going to infinity. We show via several examples that our condition is "almost necessary" as well.

As application of our result, we derive the asymptotics of the log partition function for approximately regular graphs, and bi-regular bi-partite graphs. We also re-derive analogous results for a sequence of graphs convering in cut metric.

This is joint work with Anirban Basak from Duke University.

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Title. An integral formula on Riemannian and Semi-Riemannian manifolds

Abstract. In this talk, we will present a recent joint work with C. Xia. We first prove a general integral formula for bounded domains in Riemannian manifolds. This formula includes Reilly's integral formula and the recent work of Qiu-Xia as special cases. In the second part of the talk, we will apply this formula to prove 1) Heitnz-Karcher type inequalities, 2) Minkowski inequality, 3) two almost Schur type of Theorems. All these geometric inequalities hold for the substatic Riemannian manifolds which consists of a large family Riemannian manifolds including all the space forms. We note that Heitze-Karcher inequality naturally leads to an Alexandrov rigity theorem for substatic warped product spaces. Thus we recovered S. Brendle's recent work by a completely different approach. The results in this talk are focused on Riemannian manifolds, however it has deep roots from Semi-Riemannian spacetimes.]]>

Title. The Future of Money Lies on Elliptic Curves

Abstract. This talk explores the role elliptic curves over finite fields play in cyber security, with an emphasis on Bitcoin, a decentralized peer-to-peer digital payment system.]]>

Title. ARock: Asynchronous Parallel Coordinate Updates

Abstract. We propose ARock, an asynchronous parallel algorithmic framework for finding a fixed point to a nonexpansive operator. In the framework, a set of agents (machines, processors, or cores) updates a sequence of randomly selected coordinates of the unknown variable in a parallel asynchronous fashion. As special cases of ARock, novel algorithms in linear algebra, convex optimization, machine learning, distributed and decentralized optimization are introduced. We show that if the nonexpansive operator has a fixed point, then with probability one the sequence of points generated by ARock converges to a fixed point. Very encouraging numerical performance of ARock is observed on solving linear equations, sparse logistic regression, and other large-scale problems in recent data sciences. This is joint work with Zhimin Peng, Yangyang Xu, and Wotao Yin.]]>

Title. Any way you slice it, it comes up Julia sets

Abstract. Is it possible to visualize the graph of an analytic function f: C -> C? Unfortunately, any such graph would be four dimensional. In this talk, we consider different ways to project four dimensional graphs down to at most three dimensions. We use techniques from calculus as well as some results from complex analysis and complex dynamics to visualize the graphs of families of functions arising from iteration. In so doing, beautiful images of Julia sets appear out of nowhere. Therefore, this talk begins with an overview of Julia sets and two different ways to generate them. This is joint work with Beth Schaubroeck at the Air Force Academy.]]>

Title. Voting in Agreeable Societies

Abstract. When does a majority exist? How does the geometry of the political spectrum influence the outcome? What does mathematics have to say about how people behave? When mathematical objects have a social interpretation, the associated results have social applications. We will show how some classical mathematics about the geometry and intersection of convex sets can be used to model people's preferences and understand voting in "agreeabl

Title. Regularity for Extremal Problems in Bergman Spaces of Analytic Functions and the p-Harmonic Equation

Abstract. I will discuss Hardy spaces and Bergman spaces, two spaces of analytic functions. I will then talk about extremal problems in these spaces. In Bergman spaces, extremal problems are related to the complex analogue of the p-harmonic equation, which is a generalization of the Laplace equation used to define harmonic functions. Khavinson and Stessin used this relationship to prove Lipschitz regularity results for Bergman space extremal problems. A similar result known as Ryabykh's theorem gives Hardy space regularity results for Bergman space extremal problems. I will present an extension of Ryabykh's theorem and discuss the role Khavinson's and Stessin's result plays in its proof.]]>

Title. The instability of the two-dimensional fluid, 2d Euler equation.

Abstract. We will consider the 2d Euler equation that describes the inviscid and incompressible fluid in dimension two. The classical problem of the patch evolution will be discussed. In particular, we will study the scenarios for the boundary of the initially smooth patch to lose its regularity in time. Some sharp results will be presented along with open problems.]]>

Title. Asymptotics of Chebyshev Polynomials.

Abstract. Chebyshev polynomials are the unique monic polynomials that minimize

the sup-norm on a given compact subset of the complex plane. These

polynomials have important applications in approximation theory and

numerical analysis. H. Widom in his 1969 influential work initiated a

study of the Chebyshev polynomials on compact sets given by a finite

union of disjoint arcs in the complex plane. He obtained several

partial results on the norm and pointwise asymptotics of the

polynomials and made several long lasting conjectures. In this talk I

will present some of the classical results on the Chebyshev

polynomials as well as recent progress on Widom's conjecture on the

large n pointwise asymptotics for the Chebyshev polynomials on finite

gap subsets of the real line and a new upper bound for the norms of

the Chebyshev polynomials on positive measure Cantor-type sets.]]>

Title. Level Set Segmentation using Non-Negative Matrix Factorization – Application to Brain MRI

Abstract. We address the problem of robust image segmentation using a new deformable model based on the level set method (LSM) and non-negative matrix factorization (NMF). We describe the use of NMF to reduce the dimension of large images from thousands of pixels to a handful of “metapixels” or regions. In addition, the exact number of regions is discovered using the nuclear norm of the NMF factors. The proposed NMF-LSM characterizes the histogram of the image, calculated over the image blocks, as nonnegative combinations of “basic histograms”. We show that the matrix W represents the histograms of the image regions, whereas the matrix H provides the spatial clustering of the regions. We define two local clustering criteria in terms of the NMF factors taking into account the intensity inhomogeneity or the bias field present particularly in medical images. The first criterion defines a local intensity clustering property based on the matrix W by computing the average intensity and standard deviation of every region. The second criterion defines a local spatial clustering using the matrix H by computing the area of the regions in every block of the image. The local clustering is then summed over all regions to give a global criterion of image segmentation. In LSM, these criteria define an energy in terms of the level set functions (LSFs) that represent a partition of the image domain. Image segmentation is then achieved by minimizing this energy functional w.r.t. LSFs and the bias field. The proposed method is validated on synthetic binary and gray-scale images, then applied to real brain MRI images. We found the NMFLSM has the following advantages compared to other state-of-the-art approaches: i) it relies on the histogram information rather than the absolute intensity values (robustness to noise and model parameters), ii) it introduces a new spatial term that aids in delineating regions boundaries (higher accuracy) and iii) it does not introduce additional model parameters that have to be simultaneously and iteratively estimated with the LSFs (higher convergence rate). These properties provide a general approach for robust region discovery and segmentation in heterogeneous images.]]>

Title. The chemical distance in 2D percolation

Abstract. In 2D percolation, we consider the infinite square grid Z^2. Each nearest-neighbor edge e is declared occupied with probability p or vacant with probability 1-p, independently from edge to edge. In a large box B(n) = [-n,n]^2, we are interested in S_n, the length of the shortest occupied path that connects the left side of the box to the right side. For p large, S_n behaves linearly in n, but as p decreases to the critical value 1/2, S_n is predicted to grow to an unknown super-linear power of n. We will discuss the history and results on this "chemical distance" and explain our recent work on the critical case, which solves a question of Kesten-Zhang from '92. Joint with J. Hanson and P. Sosoe.]]>

Title. Thurston's combinatorial theory of rational maps.

Abstract. In 1980's, Thuston's proved a classification of rational maps giving a topological criterion of which combinatorial dynamics can be realized by a rational map. We will discuss this and more recent result in that direction.]]>

Title. The KdV equation with almost periodic initial data.

Abstract. Percy Deift posed the following problem in 2008: Show that the KdV equation with almost periodic initial data exhibits a global solution that is almost periodic in time. In this talk we sketch the history of this problem and describe recent progress, obtained in collaboration with Ilia Binder, Michael Goldstein, and Milivoje Lukic.]]>