Eric Tal Myzelev

We develop a topological framework for understanding where zero-mean functions vanish or change sign. A key observation is that function families from Fourier analysis can sometimes be described in a unified geometric way. For example, a (real) trigonometric polynomial of degree \(d\) can be written as

\[ f(t)=\sum_{k=1}^d \big(a_k \cos(2\pi k t)+ b_k \sin(2\pi k t)\big), \]

where $a_k,b_k\in \mathbb R$ for all $k$. Rather than viewing \(f\) in isolation, we package the sine and cosine terms into a single map

\[ \gamma(t)=(\cos(2\pi t),\sin(2\pi t),\dots,\cos(2\pi d t),\sin(2\pi d t))\in\mathbb{R}^{2d}. \]

This map is equivariant with respect to the natural rotational symmetry of the circle. Every trigonometric polynomial of degree at most \(d\) can then be written as the composition of \(\gamma\) with a linear functional on \(\mathbb{R}^{2d}\), namely

\[ f(t)=\langle a,\gamma(t)\rangle \]

for a suitable coefficient vector \(a\).

This perspective allows us to apply tools from equivariant topology and configuration-space methods inspired by Borsuk--Ulam-type theorems. Using these ideas, we obtain sharp bounds on the size of intervals or regions on which all functions in the family must vanish or change sign. Our results recover and strengthen several classical theorems on one- and multi-dimensional periodic functions, and extend to compositions of linear functionals with arbitrary equivariant maps. We also show how our framework yields existence results for efficient numerical integration formulas.

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 $\mathbb R^d$. 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.