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1. Punctured logarithmic maps

   Abramovich, Dan; Chen, Qile; Gross, Mark; Siebert, Bernd (February 2025, EMS press)

   We introduce a variant of stable logarithmic maps, which we call punctured logarith- mic maps. They allow an extension of logarithmic Gromov–Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with trop- ical geometry providing the underlying combinatorics. Punctured Gromov–Witten invariants also play a pivotal role in the intrinsic con- struction of mirror partners by the last two authors, conjecturally relating to symplec- tic cohomology, and in the logarithmic gauged linear sigma model in work of Qile Chen, Felix Janda and Yongbin Ruan. 

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2. Continuity of logarithmic capacity

   https://doi.org/10.1016/j.jmaa.2021.125585  

   Kalmykov, Sergei; Kovalev, Leonid V. (January 2022, Journal of Mathematical Analysis and Applications)
3. Logarithmic Regret from Sublinear Hints

   Bhaskara, Aditya; Cutkosky, Ashok; Kumar, Ravi; Purohit, Manish (December 2021, Advances in Neural Information Processing Systems 34 (NeurIPS 2021))

   We consider the online linear optimization problem, where at every step the algorithm plays a point x_t in the unit ball, and suffers loss for some cost vector c_t that is then revealed to the algorithm. Recent work showed that if an algorithm receives a "hint" h_t that has non-trivial correlation with c_t before it plays x_t, then it can achieve a logarithmic regret guarantee, improving on the classical sqrt(T) bound. In this work, we study the question of whether an algorithm really requires a hint at every time step. Somewhat surprisingly, we show that an algorithm can obtain logarithmic regret with just O(sqrt(T)) hints under a natural query model. We give two applications of our result, to the well-studied setting of optimistic regret bounds and to the problem of online learning with abstention. 

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4. The logarithmic gauged linear sigma model

   https://doi.org/10.1007/s00222-021-01044-2  

   Chen, Qile; Janda, Felix; Ruan, Yongbin (April 2021, Inventiones mathematicae)

   null (Ed.)
5. Holonomic and perverse logarithmic D-modules

   https://doi.org/10.1016/j.aim.2019.02.016  

   Koppensteiner, Clemens; Talpo, Mattia (April 2019, Advances in Mathematics)