Every multi-soliton manifold of the Benjamin–Ono equation on the line is invariant under the Benjamin–Ono flow. Its generalized action–angle coordinates allow to solve this equation by quadrature and we have the explicit expression of every multi-solitary wave.
We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.
Please Note: Joint Topology Seminar @ GaTech
There exist many different diagrammatic descriptions of 4-manifolds, with the usual claim that "such and such a diagram uniquely determines a smooth 4-manifold up to diffeomorphism". This raises higher order questions: Up to what diffeomorphism? If the same diagram is used to produce two different 4-manifolds, is there a diffeomorphism between them uniquely determined up to isotopy? Are such isotopies uniquely determined up to isotopies of isotopies? Such questions become important if one hopes to use "diagrams" to study spaces of diffeomorphisms between manifolds. One way to achieve these higher order versions of uniqueness is to ask that a diagram uniquely determine a contractible space of 4-manifolds (i.e. a 4-manifold bundle over a contractible space). I will explain why some standard types of diagrams do not do this and give at least one type of diagram that does do this.
Please Note: Joint Topology Seminar @ GaTech
Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.
This talk is part of the Atlanta Combinatorics Colloquium. Note the time (4pm) and location (Instructional Center 105).
It is easy to partition the vertices of any graph into two sets where each vertex has at least as many neighbours across as on its own side; take any maximal cut! Can we do the opposite? This is not possible in general, but Füredi conjectured in 1988 that it should nevertheless be possible on a random graph. I shall talk about our recent proof of Füredi's conjecture: with high probability, the random graph $G(n,1/2)$ on an even number of vertices admits a partition of its vertex set into two parts of equal size in which $n−o(n)$ vertices have more neighbours on their own side than across.
Please Note: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
In this talk, I will present results concerning the existence and the precise description of periodic solutions of the Navier-Stokes equations with a time- independent forcing, obtained in collaboration with Jan Bouwe van den Berg (VU Amsterdam), Jean-Philippe Lessard (McGill) and Lennaert van Veen (Ontario TU).
These results are obtained by combining numerical simulations, a posteriori error estimates, interval arithmetic, and a fixed point theorem applied to a quasi-Newton operator, which yields the existence of an exact solution in a small and explicit neighborhood of the numerical one.
I will first introduce the main ideas and techniques required for this type of approach on a simple example, and then discuss their usage in more complex settings like the Navier-Stokes equations.
One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.
Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error or fairness. In this talk, I describe two papers analyzing federated learning through the lens of cooperative game theory (both joint with Jon Kleinberg).
In the first paper, we discuss fairness in federated learning, which relates to how error rates differ between federating agents. In this work, we consider two notions of fairness: egalitarian fairness (which aims to bound how dissimilar error rates can be) and proportional fairness (which aims to reward players for contributing more data). For egalitarian fairness, we obtain a tight multiplicative bound on how widely error rates can diverge between agents federating together. For proportional fairness, we show that sub-proportional error (relative to the number of data points contributed) is guaranteed for any individually rational federating coalition. The second paper explores optimality in federated learning with respect to an objective of minimizing the average error rate among federating agents. In this work, we provide and prove the correctness of an efficient algorithm to calculate an optimal (error minimizing) arrangement of players. Building on this, we give the first constant-factor bound on the performance gap between stability and optimality, proving that the total error of the worst stable solution can be no higher than 9 times the total error of an optimal solution (Price of Anarchy bound of 9).
Relevant Links: https://arxiv.org/abs/2010.00753, https://arxiv.org/abs/2106.09580, https://arxiv.org/abs/2112.00818
Bio:
Kate Donahue is a fifth year computer science PhD candidate at Cornell advised by Jon Kleinberg. She works on algorithmic problems relating to the societal impact of AI such as fairness, human/AI collaboration and game-theoretic models of federated learning. Her work has been supported by an NSF fellowship and recognized by a FAccT Best Paper award. During her PhD, she has interned at Microsoft Research, Amazon, and Google.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Operational weather and ocean forecasting proceeds as a sequence of time intervals. During each interval numerical models produce a forecast, observations are collected and a comparison between the two is made. This comparison is used, in a process called data assimilation (DA), to construct observation-informed initial conditions for the forecast in the next time interval. Many DA algorithms are in use, but they all share the need to solve a high-dimensional (>1010) system of linear equations. Constructing and solving this system in the limited amount of time available between the reception of the observations and the start of the next time interval is highly non-trivial for three reasons. 1) As the numerical models are computationally demanding, it is generally impossible to construct the full linear system. 2) Its high dimensionality makes it impossible to store the system as a matrix in memory. Consequently, it is not possible to directly invert it. 3) The operational time-constraints strongly limit the number of iterations that can be used by iterative linear solvers. By adapting DA algorithms to use parallelization, it is possible to leverage the computational power of superclusters to construct a high-rank approximation to the linear system and solve it using less then ~20 iterations. In this talk, I will first introduce the two most popular families of DA algorithms: Kalman filters and variational DA. After this, I will discuss some of the adaptations that have been developed to enable parallelization. Among these are ensemble Kalman filters, domain localization, the EVIL (Ensemble Variational Integrated Localized) and saddle point algorithms.
