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In 2021, Dan Reznik made a YouTube video demonstrating that power circles of Poncelet triangles have an invariant total area. He made a simulation based on this observation and put forward a few conjectures. One of these conjectures suggests that the sum of the areas of three circles, each centered at the midpoint of a side of the Poncelet triangle and passing through the opposite vertex, remains constant. In this paper, we provide a proof of Reznik's conjecture and present a formula for calculating the total sum. Additionally, we demonstrate the algebraic structures formed by various sets of products and the geometric properties of polygons and ellipses created by these products., Comment: 13 pages. This article was written as part of the Polymath Jr. program in the summer of 2022

Let $\Omega$ be a bounded convex domain in $\mathbb{C}^{n}$. We show that if $\varphi \in C^{1}(\overline{\Omega})$ is holomorphic along analytic varieties in $b\Omega$, then $H^{q}_{\varphi}$, the Hankel operator with symbol $\varphi$, is compact. We have shown the converse earlier, so that we obtain a characterization of compactness of these operators in terms of the behavior of the symbol relative to analytic structure in the boundary. A corollary is that Toeplitz operators with these symbols are Fredholm (of index zero)., Comment: 10 pages, to appear in J. Operator Theory, minor changes

Let $\Omega$ be a bounded convex domain in $\mathbb{C}^{n}$, $n\geq 2$, $1\leq q\leq (n-1)$, and $\phi\in C(\bar{\Omega})$. If the Hankel operator $H^{q-1}_{\phi}$ on $(0,q-1)$--forms with symbol $\phi$ is compact, then $\phi$ is holomorphic along $q$--dimensional analytic (actually, affine) varieties in the boundary. We also prove a partial converse: if the boundary contains only `finitely many' varieties, $1\leq q\leq n$, and $\phi\in C(\bar{\Omega})$ is analytic along the ones of dimension $q$ (or higher), then $H^{q-1}_{\phi}$ is compact., Comment: minor changes, to appear in Houston J. Math

We report on a mode-locked laser design using Yb-doped fiber lasers for low-noise frequency-comb generation. The frequency comb covers the spectral range from $700$ to $1400$ nm. Although this range is more practical for many measurements than that produced by the more commonly used Er-fiber lasers, it has been addressed in only a handful of reports, mainly due to the difficulty of generating a fully coherent supercontinuum at $1\;\mu$m. We overcome this difficulty by a tailored design of the mode-locking dynamics that succeeds in generating energetic 33 fs-long pulses without even using higher-order dispersion compensation, while ensuring that the laser operates with net zero cavity dispersion for low-noise supercontinuum generation. After locking to a Cs atomic clock, this frequency comb is used for absolute-frequency measurements of a Nd:YAG-I$_2$ laser to verify its accuracy by comparison with results from the International Committee for Weights and Measures. After this verification, it is further used to measure the absolute frequency of a 543-nm two-mode stabilized He-Ne laser, which is routinely used for length measurements in our institute, thus verifying its practical utility in metrology applications. The entire setup is built with readily available components for easy duplication by other researchers.

Let $1\leq q\leq (n-1)$. We first show that a necessary condition for a Hankel operator on $(0,q-1)$-forms on a convex domain to be compact is that its symbol is holomorphic along $q$-dimensional analytic varieties in the boundary. Because maximal estimates (equivalently, a comparable eigenvalues condition on the Levi form of the boundary) turn out to be favorable for compactness of Hankel operators, this result then implies that on a convex domain, maximal estimates exclude analytic varieties from the boundary, except ones of top dimension $(n-1)$ (and their subvarieties). Some of our techniques apply to general pseudoconvex domains to show that if the Levi form has comparable eigenvalues, or equivalently, if the domain admits maximal estimates, then compactness and subellipticity hold for forms at some level $q$ if and only if they hold at all levels., Comment: A typo in footnote 2 is fixed. 15 pages, in to appear in Proc. Amer. Math. Soc

Let $\Omega$ be a bounded pseudoconvex domain in $\mathbb{C}^2$ with Lipschitz boundary or a bounded convex domain in $\mathbb{C}^n$ and $\phi\in C(\overline{\Omega})$ such that $H_{\phi}$ is compact on $A^2(\Omega)$. Then $\phi\circ f$ is holomorphic for any holomorphic $f:\mathbb{D}\rightarrow b\Omega$., Comment: made minor changes. To appear in Integral Equations Operator Theory

There have been several attempts at modeling context in robots. However, either these attempts assume a fixed number of contexts or use a rule-based approach to determine when to increment the number of contexts. In this paper, we pose the task of when to increment as a learning problem, which we solve using a Recurrent Neural Network. We show that the network successfully (with 98\% testing accuracy) learns to predict when to increment, and demonstrate, in a scene modeling problem (where the correct number of contexts is not known), that the robot increments the number of contexts in an expected manner (i.e., the entropy of the system is reduced). We also present how the incremental model can be used for various scene reasoning tasks., Comment: The first two authors have contributed equally, 6 pages, 8 figures, International Conference on Intelligent Robots (IROS 2018)

We establish a connection between compactness of Hankel operators and geometry of the underlying domain through compactness multipliers for the $\overline{\partial}$-Neumann operator. In particular, we prove that any compactness multiplier induces a compact Hankel operator. We also generalize the notion of compactness multipliers to vector fields and matrices and then we use this generalization to generate compact Hankel operators., Comment: To appear in the Illinois Journal of Mathematics

In this note, we discuss the preservation of certain analytic properties of the $\overline{\partial}$-Neumann operator, Bergman projection and Hankel operators on the intersection of pseudoconvex domains., Comment: To appear in Contemporary Mathematics (Proceedings of the Conference on Analysis and Geometry in Several Complex Variables, Doha, Qatar, January 2015)