Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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Abstract. xxxx
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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Abstract. xxxx
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

Title. xxxx

Abstract. xxxx
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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)
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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Abstract. xxxx
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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. 

 

Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Tue at 3.30pm-4.20pm, CH 205

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 "agreeable"societies.  This talk also features research with undergraduates.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.
Math Colloquium: Fri at 2:30pm-3:20pm, CH 443

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.