Georgia Institute of Technology College of Sciences

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.

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.

The well-known tree packing conjecture of Gyárfás from 1976 says that, given any sequence of n trees in which the ith tree has i vertices, the trees can be packed edge-disjointly into the complete n-vertex graph. Packing even just the largest trees in such a sequence has proven difficult, with Bollobás drawing attention to this in 1995 by conjecturing that, for each k, if n is sufficiently large then the largest k trees in any such sequence can be packed. This has only been shown for k at most 5, by Zak, despite many partial results and much related work on the full tree packing conjecture.

I will discuss a result which proves Bollobás's conjecture by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture. This is joint work with Barnabás Janzer.

Vassiliev knot invariants, or finite-type invariants, are a broad class of knot invariants resulting from extending usual invariants to knots with transverse double points. We will show that the Conway and Jones polynomials are fully described by Vassiliev invariants. We will discuss the fundamental theorem of Vassiliev invariants, relating them to the algebra of chord diagrams and weight systems. Time permitting, we will also discuss the Kontsevich integral, the universal Vassiliev invariant.

Langhede and Thomassen conjectured in 2020 that there exists a positive constant c such that every planar graph G with 5-correspondence assignment (L,M) has at least 2^{c v(G)} distinct (L,M)-colourings. I will discuss a proof of this conjecture (which relies on the hyperbolicity of a certain family of graphs), a generalization of this result to some other embedded graphs (again, relying on a hyperbolicity theorem), and a few open problems in the area. Everything presented is joint work with Luke Postle.

In this talk, we study the long-term behaviour of Random Dynamical Systems (RDSs) conditioned upon staying in a region of the space. We use the absorbing Markov chain theory to address this problem and define relevant dynamical systems objects for the analysis of such systems. This approach aims to develop a satisfactory notion of ergodic theory for random systems with escape.