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for events the day of Monday, April 20, 2015.

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Monday, April 20, 2015

12:00 pm in 341 AH,Monday, April 20, 2015

A C^0-characterization of symplectic and contact embeddings

Stefan Müller (UIUC Math)

Abstract: Symplectic (and anti-symplectic) embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of C^0-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a C^0-characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C. Sikorav and Y. Eliashberg). The aforementioned rigidity is again an easy consequence. The shape invariant has two immediate advantages: it avoids the cumbersome distinction between symplectic and anti-symplectic, and the results can be adapted to contact embeddings via coisotropic embeddings (of tori). The adaptation to the contact case is (partly) work in progress. In the talk, I will give proofs of the main results and explain what makes them work (J-holomorphic disc techniques for all deep results).

2:00 pm in 141 CSL,Monday, April 20, 2015

Algorithmic Improvements in Correlation Clustering

Gregory Puleo (Postdoctoral Research Associate, CSL, Univ of Illinois)

Abstract: We consider the Correlation Clustering problem: given a collection of objects together with labels indicating which pairs of objects are "similar" and which are "dissimilar", we wish to cluster the objects so that "similar" objects lie in the same cluster and "dissimilar" objects lie in different clusters. Since a perfect clustering may not exist, the goal is to (approximately) minimize the number of "mistakes", that is, the number of similar pairs that end up in different clusters together with the number of dissimilar pairs that end up within clusters. We consider a weighted version of this problem and give an approximation algorithm that accepts a broader range of input weights than previous algorithms. We also study some variations on the basic problem.

Bio: Gregory Puleo is a Postdoctoral Research Associate at CSL, working in the research group of Olgica Milenkovic. In 2009 he earned a BS in Applied Mathematics at the Rochester Institute of Technology, and in 2014 he earned a PhD in Mathematics under the supervision of Douglas B. West. His research interests include extremal graph theory and optimization problems on graphs.

4:00 pm in Altgeld Hall 145,Monday, April 20, 2015

b, Scattering, and Edge Symplectic Geometry

Melinda Lanius   [email] (UIUC Math)

Abstract: A symplectic manifold is a manifold $M$ equipped with a non-degenerate closed two form $\omega$. Symplectic manifolds are a special case of what is called a Poisson manifold. A Poisson structure gives a smooth partition of a manifold into even-dimensional symplectic submanifolds, which are not necessarily of the same dimension. These two geometries, symplectic and Poisson, each have benefits; while symplectic structures are fairly well understood, Poisson structures are much more general. We will consider the questions: can we define a generalized symplectic structure that will correspond to a mildly degenerate Poisson structure? If so, can we employ the techniques of symplectic geometry to better understand these?

5:00 pm in 241 Altgeld Hall,Monday, April 20, 2015

Norm estimates on almost Mathieu operators

Florin Boca (UIUC Math)

Abstract: The talk will discuss upper and lower bound estimates on the norm of the almost Mathieu operators (Hλ,θu)(n) = u(n + 1) + u(n - 1) + λcos(2πnθ)u(n) on 2(), known to coincide with the norm of the operator hλ,θ = uθ + uθ* + (λ∕2)(vθ + vθ*), where uθ and vθ are canonical generators of the rotation C*-algebra Aθ.