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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

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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 loose 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.]]>