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1. Integral representations of rank two false theta functions and their modularity properties

   https://doi.org/10.1007/s40687-021-00284-1  

   Bringmann, Kathrin; Kaszian, Jonas; Milas, Antun; Nazaroglu, Caner (December 2021, Research in the Mathematical Sciences)

   Abstract False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type $$A_2$$ A 2 and $$B_2$$ B 2 . This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze $${\hat{Z}}$$ Z ^ -invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing $$\mathtt{H}$$ H -graphs. Along the way, our method clarifies previous results on depth two quantum modularity. 

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2. Partial degeneration of finite gap solutions to the Korteweg–de Vries equation: soliton gas and scattering on elliptic backgrounds

   https://doi.org/10.1088/1361-6544/accfdf  

   Bertola, M; Jenkins, R; Tovbis, A (May 2023, Nonlinearity)

   Abstract We obtain Fredholm type formulas for partial degenerations of theta functions on (irreducible) nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study of ‘many-soliton’ solutions on an elliptic (cnoidal) background standing wave for the Korteweg–de Vries (KdV) equation starting from a formula that is reminiscent of the classical Kay–Moses formula for N -solitons. In particular, we represent such a solution as a sum of the following two terms: a ‘shifted’ elliptic (cnoidal) background wave and a Kay–Moses type determinant containing Jacobi theta functions for the solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background. The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of Jacobi theta functions. We also show that genus N  + 1 finite gap solutions with random initial phases converge in probability to the deterministic cnoidal wave solution as N bands degenerate to a nodal curve of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV soliton gas on the residual elliptic background. 

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3. Asymptotic expansions, partial theta functions, and radial limit differences of mock modular and modular forms

   https://doi.org/10.1142/S1793042120400126  

   Folsom, Amanda (March 2021, International Journal of Number Theory)

   null (Ed.)

   In 1920, Ramanujan studied the asymptotic differences between his mock theta functions and modular theta functions, as [Formula: see text] tends towards roots of unity singularities radially from within the unit disk. In 2013, the bounded asymptotic differences predicted by Ramanujan with respect to his mock theta function [Formula: see text] were established by Ono, Rhoades, and the author, as a special case of a more general result, in which they were realized as special values of a quantum modular form. Our results here are threefold: we realize these radial limit differences as special values of a partial theta function, provide full asymptotic expansions for the partial theta function as [Formula: see text] tends towards roots of unity radially, and explicitly evaluate the partial theta function at roots of unity as simple finite sums of roots of unity. 

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4. The Elliptic Tail Kernel

   https://doi.org/10.1093/imrn/rnaa038  

   Cuenca, Cesar; Gorin, Vadim; Olshanski, Grigori (March 2020, International Mathematics Research Notices)

   null (Ed.)

   Abstract We introduce and study a new family of $$q$$-translation-invariant determinantal point processes on the two-sided $$q$$-lattice. We prove that these processes are limits of the $$q$$–$zw$ measures, which arise in the $$q$$-deformation of harmonic analysis on $$U(\infty )$$, and express their correlation kernels in terms of Jacobi theta functions. As an application, we show that the $$q$$–$zw$ measures are diffuse. Our results also hint at a link between the two-sided $$q$$-lattice and rows/columns of Young diagrams. 

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5. Asymptotics and Ramanujan's mock theta functions: then and now

   https://doi.org/10.1098/rsta.2018.0448  

   Folsom, Amanda (December 2019, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms—Dyson called this ‘a challenge for the future’ at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later in 2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann–Rolen, Choi–Lim–Rhoades and Griffin–Ono–Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’. 

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