Search

We study the harmonic locus consisting of the monodromy-free Schr\"odinger operators with rational potential and quadratic growth at infinity. It is known after Oblomkov that it can be identified with the set of all partitions via the Wronskian map for Hermite polynomials. We show that the harmonic locus can also be identified with the subset of the Calogero--Moser space introduced by Wilson, which is fixed by the symplectic action of $\mathbb C^\times.$ As a corollary, for the multiplicity-free part of the locus we effectively solve the inverse problem for the Wronskian map by describing the partition in terms of the spectrum of the corresponding Moser matrix. We also compute the characters of the $\mathbb C^\times$-action at the fixed points, proving, in particular, a conjecture of Conti and Masoero. In the Appendix written by N. Nekrasov there is an alternative proof of this result, based on the space of instantons and ADHM construction., Comment: Extended version with an Appendix written by N. Nekrasov containing an alternative proof of Conti-Masoero conjecture

We study a delay-differential analogue of the first Painlev\'e equation obtained as a delay periodic reduction of Shabat's dressing chain. We construct formal entire solutions to this equation and introduce a new family of polynomials (called Bernoulli-Catalan polynomials), which are defined by a nonlinear recurrence of Catalan type, and which share properties with Bernoulli and Euler polynomials. We also discuss meromorphic solutions and describe the singularity structure of this delay Painlev\'e-I equation in terms of an affine Weyl group of type $A_1^{(1)}$. As an application we demonstrate the link with the problem of calculation of the Masur-Veech volumes of the moduli spaces of meromorphic quadratic differentials by re-deriving some of the known formulas., Comment: Revisions to Introduction and Concluding Remarks. 26 pages, 6 figures

Real-time global illumination is key to enabling more dynamic and physically realistic worlds in performance-critical applications such as games or any other applications with real-time constraints.Hardware-accelerated ray tracing in modern GPUs allows arbitrary intersection queries against the geometry, making it possible to evaluate indirect lighting entirely at runtime. However, only a small number of rays can be traced at each pixel to maintain high framerates at ever-increasing image resolutions. Existing solutions, such as probe-based techniques, approximate the irradiance signal at the cost of a few rays per frame but suffer from a lack of details and slow response times to changes in lighting. On the other hand, reservoir-based resampling techniques capture much more details but typically suffer from poorer performance and increased amounts of noise, making them impractical for the current generation of hardware and gaming consoles. To find a balance that achieves high lighting fidelity while maintaining a low runtime cost, we propose a solution that dynamically estimates global illumination without needing any content preprocessing, thus enabling easy integration into existing real-time rendering pipelines.

In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the ``forgotten symmetric functions", which follows from our previous work on complex cobordisms. In particular, this leads to a simpler proof of the Hirzebruch formula and provides new interpretations for the Hirzebruch numbers., Comment: Slightly revised version, published in the Proceedings of the Steklov Institute of Mathematics, 2024, Vol. 325

We establish a link between the new theory of $q$-deformed rational numbers and the classical Burau representation of the braid group $\mathrm{B}_3$. We apply this link to the open problem of classification of faithful complex specializations of this representation. As a result we provide an answer to this problem in terms of the singular set of the $q$-rationals and prove the faithfulness of the Burau representation specialized at complex $t\in \mathbb{C}^*$ outside the annulus $3-2\sqrt2 \leq |t| \leq 3+2\sqrt2.$

A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group belonging to one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed., Comment: 67 pages; slightly amended English version of the paper published in Uspekhi Matematicheskikh Nauk (Russian Math Surveys)