## Harmonic Analysis and Differential Equations Seminar: Multi-solitary waves of the Benjamin–Ono equation on the line
The Benjamin–Ono (BO) equation, which describes internal long waves of deep stratified fluids, has multi-soliton solutions. I shall prove the invariance of every multi-soliton manifold under the BO flow and construct global (generalized) action–angle coordinates in order to solve this equation by quadrature for any such initial datum. The complete integrability of the BO equation on every N-soliton manifold constitutes a first step towards the soliton resolution conjecture of the BO equation on the line. The construction of such coordinates relies on the Lax pair structure, the inverse spectral transform and the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the shift semigroup acting on some Hardy spaces. Furthermore, the N-soliton manifold of the BO equation on the line can be interpreted as the universal covering of the manifold of N-gap potentials for the space-periodic BO equation as described by Gérard–Kappeler. |
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## Graph Theory and Combinatorics Seminar: Independence number of random subgraphs of the hypercube
The discrete hypercube of dimension d is a d-regular bipartite graph on 2^d vertices whose maximum size independent set has size 2^(d-1). In this talk, we will prove that if we keep each edge with constant probability p > 1/2, the independence number is still 2^(d-1) with probability tending to 1 as d tends to infinity. In fact, much more about the independence number of this random subgraph of the cube is known for general p. We will also discuss the same theorem but for the random induced subgraph of the cube, where we include each vertex with probability p. These proofs are simplifications of earlier graph container-like arguments concerning random versions of the Erdos-Ko-Rado theorem, which will be discussed if time permits. This is joint work with Jozsef Balogh. |
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## Student Cluster Algebra Seminar
We will give the final installment from a series of talks about the cluster scattering diagram construction of Gross, Hacking, Keel, and Kontsevich.
Abstract: We will begin by defining scattering diagrams, which were first introduced in two dimensions by Kontsevich and Soibelman and then in arbitrary dimension by Gross and Siebert as a tool for constructing mirror spaces. Using concrete examples, we will then walk through the cluster scattering diagram construction given by Gross, Hacking, Keel, and Kontsevich and give definitions of cluster varieties, broken lines, theta functions, and other relevant objects. If time allows, we will then briefly sketch how cluster scattering diagrams are used to prove important results for ordinary cluster algebras, including positivity and the existence of the theta basis. |
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## Mathematics Colloquium: Optimal control for management of aquatic population models
Abstract: Optimal control techniques of ordinary and partial differential equations will be introduced to consider management strategies for aquatic populations. In the first example, managing invasive species in rivers can be assisted by adjustment of flow rates. Control of a flow rate in a partial differential equation model for a population in a river will be used to keep the population from moving upstream. The second example represents a food chain on the Turkish coast of the Black Sea. Using data from the anchovy landings in Turkey, optimal control of the harvesting rate of the anchovy population in a system of three ordinary differential equations (anchovy, jellyfish and zooplankton) will give management strategies. |
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## Graduate Analysis Seminar: Prekopa-Leindler inquality and the sharp Sobolev inequality
Prekopa-Leindler inequality is a functional version of Brunn-Minkowski inequality, and has a close relation to many interesting inequalities. We discuss the proof of Prekopa-Leindler inequality using optimal transport technique. As an application, we derive the sharp Sobolev inequality. |
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## Graduate Student Homotopy Seminar: Introduction to the Stolz-Teichner Program
The Stolz-Teichner program is a far-reaching research program that aims to connect QFT to the cohomology of manifolds. Importantly for homotopy theorists, it is expected to provide a cochain model for TMF. In this talk I will sketch some of the broad strokes of the program and describe some of the partial results currently known. |