Georgia Institute of Technology College of Sciences

We consider classical scalar fields in dimension 1+1 with a
self-interaction potential being a symmetric double-well. Such a model
admits non-trivial static solutions called kinks and antikinks. A kink
cluster is a solution approaching, for large positive times, a
superposition of alternating kinks and antikinks whose velocities
converge to 0 and mutual distances grow to infinity. Our main result is
a determination of the asymptotic behaviour of any kink cluster at the
leading order.
Our results are partially inspired by the notion of "parabolic motions"
in the Newtonian n-body problem. I will present this analogy and mention
its limitations. I will also explain the role of kink clusters as
universal profiles for formation of multi-kink configurations.
This is a joint work with Andrew Lawrie.

In many applications across science and engineering it is common to have access to disparate types of data or models with different levels of fidelity. In general, low-fidelity data are easier to obtain in greater quantities, but it may be too inaccurate or not dense enough to accurately train a machine learning model. High-fidelity data is costly to obtain, so there may not be sufficient data to use in training, however, it is more accurate.  A small amount of high-fidelity data, such as from measurements or simulations, combined with low fidelity data, can improve predictions when used together. The important step in such constructions is the representation of the correlations between the low- and high-fidelity data. In this talk, we will present two frameworks for multifidelity machine learning. The first one puts particular emphasis on operator learning, building on the Deep Operator Network (DeepONet). The second one is inspired by the concept of model reduction. We will present the main constructions along with applications to closure for multiscale systems and continual learning. Moreover, we will discuss how multifidelity approaches fit in a broader framework which includes ideas from deep learning, stochastic processes, numerical methods, computability theory and renormalization of complex systems.

In 1973, Nambu published an article entitled "Generalized Hamiltonian dynamics". For that purpose, he constructed multilinear brackets - equivalent to Poisson brackets - with some interesting properties reminiscent of the Jacobi identity.
These brackets found some applications in fluid mechanics, plasma physics and mathematical physics with superintegrable systems.

In this seminar, I will recall some basic elements on Nambu mechanics in finite dimension with an n-linear Nambu bracket in dimension larger than n. I will discuss all possible Nambu brackets and compare them with all possible Poisson brackets. I will conclude that Nambu mechanics can hardly be considered a generalization of Hamiltonian mechanics.

Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this talk we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We first study the problem of estimating community memberships, and we show that a tensor generalization of a matrix algorithm can consistently estimate communities at a rate that improves relative to the matrix setting, provided one takes the tensor structure into account. Next, we study the problem of testing whether two nodes have the same community memberships, and we show that a tensor analogue of a matrix test statistic can yield consistent testing with a tighter local power guarantee relative to the matrix setting. If time permits we will also examine the performance of our estimation procedure on flight data. This talk is based on two recent works with Anru Zhang.