Georgia Institute of Technology College of Sciences

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.

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.

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.