Undergraduate research in mathematics (StemForAll) never stops at the University of Rochester. In addition to summer programs described on this website, research group are working year-around on topics they are interesting in. When the Fall semester begins, we are going to decide on a meeting time. If you are interested in joining one of the research groups or starting another one, please come to my office next Friday. If you have any questions beforehand, please feel free to email me at iosevich@gmail.com. This program is co-organized by me and Azita Mayeli (CUNY).

Several of my colleagues, both from Rochester and beyond, played a crucial role in the success of this program over the years. Charlotte Aten, Steven Kleene, Azita Mayeli, Sevak Mkrtchyan, Jonathan Pakianathan and several others made this program possible and continue to make it possible to this day.

The topics we will be research during the Fall 2024 semester are yet to be determined, but with very high probability we are going to run a research group on signal recovery. The basic idea is the following. Suppose that $f:\; \{\backslash mathbb\; Z\}\_N^d\; \backslash to\; \{\backslash mathbb\; C\}$ and we wish to transmit it via its Fourier transform $$f(x)=\backslash sum\_\{m\; \backslash in\; \{\backslash mathbb\; Z\}\_N^d\}\; \backslash chi(x\; \backslash cdot\; m)\; \backslash widehat\{f\}(m).$$However, if some of the frequencies are missing, for example if $\{\backslash \{\backslash widehat\{f\}(m)\backslash \}\}\_\{m\; \backslash in\; S\}$ are unobserved for some $S\subset {\mathbb{Z}}_{N}^{d}.$$The\; question\; we\; ask\; is,\; under$what reasonable assumptions can we still recover the function (or the signal, as electrical engineers tend to call it) exactly? We are also going to explore some applications of signal recovery to data science. This project ran during the StemForAll2024 workshop this summer and is going to continue into the Fall. Next summer, the entire StemForAll2025 workshop will be dedicated to the pure and applied aspects of signal recovery.

Another project that is likely to run during the Fall semester is the investigation of the Erdos distance problem on Riemannian manifolds. The basic question is the following Let $E$ be a finite subset of $M$, a d-dimensional compact Riemannian manifold without a boundary. Let $\backslash Delta(E)$ denote the distance set of $E$ with respect to the Riemannian metric, i.e $\backslash Delta(E)=\backslash \{\backslash rho(x,y):\; x,y\; \backslash in\; E\backslash \}$, where $\backslash rho$ is the Riemannian metric on $M$. In Euclidean space it is conjectured that $|\backslash Delta(E)|\; \backslash ge\; c\{|E|\}^\{\backslash frac\{2\}\{d\}\}$, up to logarithmic terms, with the example provided by the $n^\{\backslash frac\{1\}\{d\}\}\; \backslash times\; n^\{\backslash frac\{1\}\{d\}\}\; \backslash times\; \backslash dots\; \backslash times\; n^\{\backslash frac\{1\}\{d\}\}$ piece of the integer lattice. On general manifolds, such rich arithmetic does not typically exist, which makes it unclear what the manifold variant of the Erdos distance conjecture should even say. This is a fascinating topic with lots to explore!