1. Together with K. Fedosova and K. Klinger-Logan we have proved the conjecture of Chester, Green, Pufu, Wang, and Wen on sums of even powers of divisors. This conjecture was based on the Ads/CFT correspondence in string theory. The proof gives a large family of analogous identities involving coefficients of holomorphic cusp forms of level 1.
2. Together with A. Arman, A. Bondarenko, F. Nazarov, and A. Prymak we have constructed bodies of constant width 2 of volume exponentially smaller than the volume of a unit ball in high dimensions, resolving an old question asked by Oded Schramm.
3. Together with J. Ramos, we have constructed Heisenberg uniqueness pairs for a class of perturbed lattice crosses. This complements the results of Hedenmalm and Montes-Rodriguez on Heisenberg uniqueness pairs involving uniform lattice crosses, but uses a different set of ideas that are more flexible and give rise to a much wider class of uniqueness pairs.
4. Together with A. Bondarenko, J. Ortega-Cerdà, and K. Seip, we gave a very precise description of the Hormander-Bernhardsson extremal function, characterizing it via differential and functional equations, and giving a way to compute the optimal associated constant for point evaluation in the space PW^1 to any precision. As a byproduct of this investigation, we have constructed a large family of interpolation formulas for Paley-Wiener class functions, parameterized by eigenfunctions of a certain family of second order differential operators.
5. Together with Steven Charlton and Daniil Rudenko, we have proved a recent conjecture about multiple polylogarithms by establishing an unexpected link between them and cohomology of GL_n and Steinberg modules.
6. Together with Qihang Sun, we have generalized the Poincare sum construction of interpolation bases for square roots of integers for the space of radial Schwartz functions from dimensions >=5 to dimensions 3 and 4.
7. We have obtained a common generalization of the interpolation formula of Radchenko-Viazovska and the interpolation formula of Cohn-Kumar-Miller-Radchenko-Viazovska. The resulting three-parametric family of Fourier interpolation formulas clarifies the relationship between the two, is considerably simpler to prove and analyze, and allows a more direct proof of positivity of the modular kernel needed for the universal optimality in dimensions 8 and 24. A manuscript is in preparation.
