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1. The Lambda Invariants at CM Points

   https://doi.org/10.1093/imrn/rnz230  

   Yang, Tonghai; Yin, Hongbo; Yu, Peng (November 2019, International Mathematics Research Notices)

   Abstract In this paper, we show that $$\lambda (z_1) -\lambda (z_2)$$, $$\lambda (z_1)$$, and $$1-\lambda (z_1)$$ are all Borcherds products on $$X(2) \times X(2)$$. We then use the big CM value formula of Bruinier, Kudla, and Yang to give explicit factorization formulas for the norms of $$\lambda (\frac{d+\sqrt d}2)$$, $$1-\lambda (\frac{d+\sqrt d}2)$$, and $$\lambda (\frac{d_1+\sqrt{d_1}}2) -\lambda (\frac{d_2+\sqrt{d_2}}2)$$, with the latter under the condition $$(d_1, d_2)=1$$. Finally, we use these results to show that $$\lambda (\frac{d+\sqrt d}2)$$ is always an algebraic integer and can be easily used to construct units in the ray class field of $${\mathbb{Q}}(\sqrt{d})$$ of modulus $$2$$. In the process, we also give explicit formulas for a whole family of local Whittaker functions, which are of independent interest. 

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2. Optimal Small Scale Equidistribution of Lattice Points on the Sphere, Heegner Points, and Closed Geodesics

   https://doi.org/10.1002/cpa.22076  

   Humphries, Peter; Radziwiłł, Maksym (September 2022, Communications on Pure and Applied Mathematics)

   Abstract We asymptotically estimate the variance of the number of lattice points in a thin, randomly rotated annulus lying on the surface of the sphere. This partially resolves a conjecture of Bourgain, Rudnick, and Sarnak. We also obtain estimates that are valid for all balls and annuli that are not too small. Our results have several consequences: for a conjecture of Linnik on sums of two squares and a “microsquare”, a conjecture of Bourgain and Rudnick on the number of lattice points lying in small balls on the surface of the sphere, the covering radius of the sphere, and the distribution of lattice points in almost all thin regions lying on the surface of the sphere. Finally, we show that for a density 1. subsequence of squarefree integers, the variance exhibits a different asymptotic behaviour for balls of volume with . We also obtain analogous results for Heegner points and closed geodesics. Interestingly, we are able to prove some slightly stronger results for closed geodesics than for Heegner points or lattice points on the surface of the sphere. A crucial observation that underpins our proof is the different behaviour of weighting functions for annuli and for balls. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC. 

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3. Accumulation points of the edit distance function

   https://doi.org/10.1016/j.disc.2022.112857  

   Cox, Christopher; Martin, Ryan R.; McGinnis, Daniel (July 2022, Discrete Mathematics)
4. Tipping points in the dynamics of speciation

   https://doi.org/10.1038/s41559-016-0001  

   Nosil, Patrik; Feder, Jeffrey L.; Flaxman, Samuel M.; Gompert, Zachariah (January 2017, Nature Ecology & Evolution)
5. Tracking exceptional points above the lasing threshold

   https://doi.org/10.1038/s41467-023-43874-z  

   Ji, Kaiwen; Zhong, Qi; Ge, Li; Beaudoin, Gregoire; Sagnes, Isabelle; Raineri, Fabrice; El-Ganainy, Ramy; Yacomotti, Alejandro M. (December 2023, Nature Communications)

   Abstract Recent studies on exceptional points (EPs) in non-Hermitian optical systems have revealed unique traits, including unidirectional invisibility, chiral mode switching and laser self-termination. In systems featuring gain/loss components, EPs are commonly accessed below the lasing threshold, i.e., in the linear regime. In this work, we experimentally demonstrate that EP singularities in coupled semiconductor nanolasers can be accessed above the lasing threshold, where they become branch points of a nonlinear dynamical system. Contrary to the common belief that unavoidable cavity detuning impedes the formation of EPs, here we demonstrate that such detuning is necessary for compensating the carrier-induced frequency shift, hence restoring the EP. Furthermore, we find that the pump imbalance at lasing EPs varies with the total pump power, enabling their continuous tracking. This work uncovers the unstable nature of EPs above laser threshold in coupled semiconductor lasers, offering promising opportunities for the realization of self-pulsing nanolaser devices and frequency combs. 

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