Eric Tal Myzelev

We develop a novel topological framework that yields results constraining the distribution of zeros of certain zero mean real-valued maps, namely those obtained from composing a fixed equivariant map with linear functionals. We use this framework to establish upper bounds for the topology of set systems in the domain where (multivariate) trigonometric polynomials do not change their sign, generalizing and, in certain regimes, strengthening results in the literature. Our results more generally contain restrictions on the distribution of zeros of Chebyshev spaces as special cases. Lastly, we apply this framework to derive existence results for efficient cubature rules for compositions of affine functionals and equivariant maps.

There is a famous problem in geometric graph theory to find the chromatic number of the unit distance graph on Euclidean space; it remains unsolved. A theorem of Erdos and De-Bruijn simplifies this problem to finding the maximum chromatic number of a finite unit distance graph. Via a construction built on sequential finite graphs obtained from a generalization of this theorem, we have found a class of geometrically defined hypergraphs of arbitrarily large edge cardinality, whose proper colorings exactly coincide with the proper colorings of the unit distance graph on ${ℝ}^d$. We also provide partial generalizations of this result to arbitrary real normed vector spaces.

Szlam's Lemma is a powerful tool for obtaining upper bounds on the chromatic numbers of distance graphs in normed vector spaces. We introduce and provide a complete characterization for a regimented variety of colorings obtained by the lemma, which we called "ordered Szlam colorings".

Aortic dissection is a life-threatening condition where a tear occurs in the inner layer of the aorta. In this study, we propose a new computational framework called warm-start physics-informed neural networks (WS-PINNs). WS-PINNs use neural networks leveraging external MRI data to approximate solutions to the Navier-Stokes equations, thereby generating a detailed model of the hemodynamics inside the evolving false lumen. By incorporating transfer learning, our method efficiently assesses patient-specific risks and complex flow dynamics without the computational cost of modeling the entire vessel structure, offering a promising tool for clinical prognosis.