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1. An Algorithm for Tambara-Yamagami Quantum Invariants of 3-Manifolds, Parameterized by the First Betti Number

   https://doi.org/10.4230/lipics.socg.2025.38  

   Delaney, Colleen; Maria, Clément; Samperton, Eric (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Aichholzer, Oswin; Wang, Haitao (Ed.)

   Quantum topology provides various frameworks for defining and computing invariants of manifolds inspired by quantum theory. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we analyze the computational complexity of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. While these categories are the simplest source of state sum invariants beyond finite abelian groups (whose invariants can be computed in polynomial time) their computational complexities are yet to be fully understood. We first establish that the invariants arising from even the smallest Tambara-Yamagami categories are #P-hard to compute, so that one expects the same to be true of the whole family. Our main result is then the existence of a fixed parameter tractable algorithm to compute these 3-manifold invariants, where the parameter is the first Betti number of the 3-manifold with ℤ/2ℤ coefficients. Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is computable in polynomial time. Thus while one generally expects quantum invariants to be hard to compute classically, our results suggest that the hardness of computing state sum invariants from Tambara-Yamagami categories arises from classical 3-manifold topology rather than the quantum nature of the algebraic input. 

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2. Congruences with Eisenstein series and -invariants

   https://doi.org/10.1112/S0010437X19007127  

   Bellaïche, Joël; Pollack, Robert (May 2019, Compositio Mathematica)

   We study the variation of $$\unicode[STIX]{x1D707}$$ -invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the $$p$$ -adic zeta function. This lower bound forces these $$\unicode[STIX]{x1D707}$$ -invariants to be unbounded along the family, and we conjecture that this lower bound is an equality. When $$U_{p}-1$$ generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the $$p$$ -adic $$L$$ -function is simply a power of $$p$$ up to a unit (i.e.  $$\unicode[STIX]{x1D706}=0$$ ). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms. 

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3. Closure Certificates

   Murali, V; Trivedi, A; Zamani, M (May 2024, In 27th ACM International Conference on Hybrid Systems: Computation and Control (HSCC ’24),)

   A barrier certificate, defined over the states of a dynamical system, is a real-valued function whose zero level set characterizes an in- ductively verifiable state invariant separating reachable states from unsafe ones. When combined with powerful decision procedures— such as sum-of-squares programming (SOS) or satisfiability-modulo- theory solvers (SMT)—barrier certificates enable an automated de- ductive verification approach to safety. The barrier certificate ap- proach has been extended to refute LTL and l -regular specifications by separating consecutive transitions of corresponding l -automata in the hope of denying all accepting runs. Unsurprisingly, such tactics are bound to be conservative as refutation of recurrence properties requires reasoning about the well-foundedness of the transitive closure of the transition relation. This paper introduces the notion of closure certificates as a natural extension of barrier certificates from state invariants to transition invariants. We aug- ment these definitions with SOS and SMT based characterization for automating the search of closure certificates and demonstrate their effectiveness over some case studies. 

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4. The size of wild Kloosterman sums in number fields and function fields

   https://doi.org/10.1007/s11854-023-0325-9  

   Sawin, Will (December 2023, Journal d'Analyse Mathématique)

   We study p-adic hyper-Kloosterman sums, a generalization of the Kloosterman sum with a parameter k that recovers the classical Kloosterman sum when k = 2, over general p-adic rings and even equal characteristic local rings. These can be evaluated by a simple stationary phase estimate when k is not divisible by p, giving an essentially sharp bound for their size. We give a more complicated stationary phase estimate to evaluate them in the case when k is divisible by p. This gives both an upper bound and a lower bound showing the upper bound is essentially sharp. This generalizes previously known bounds in the case of ℤ/p. The lower bounds in the equal characteristic case have two applications to function field number theory, showing that certain short interval sums and certain moments of Dirichlet L-functions do not, as one might hope, admit square-root cancellation. 

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5. Hydrodynamics of Collisions and Close Encounters between Stellar Black Holes and Main-sequence Stars

   https://doi.org/10.3847/1538-4357/ac714f  

   Kremer, Kyle; Lombardi, James C.; Lu, Wenbin; Piro, Anthony L.; Rasio, Frederic A. (July 2022, The Astrophysical Journal)

   Abstract Recent analyses have shown that close encounters between stars and stellar black holes occur frequently in dense star clusters. Depending upon the distance at closest approach, these interactions can lead to dissipating encounters such as tidal captures and disruptions, or direct physical collisions, all of which may be accompanied by bright electromagnetic transients. In this study, we perform a wide range of hydrodynamic simulations of close encounters between black holes and main-sequence stars that collectively cover the parameter space of interest, and we identify and classify the various possible outcomes. In the case of nearly head-on collisions, the star is completely disrupted with roughly half of the stellar material becoming bound to the black hole. For more distant encounters near the classical tidal-disruption radius, the star is only partially disrupted on the first pericenter passage. Depending upon the interaction details, the partially disrupted stellar remnant may be tidally captured by the black hole or become unbound (in some cases, receiving a sufficiently large impulsive kick from asymmetric mass loss to be ejected from its host cluster). In the former case, the star will undergo additional pericenter passages before ultimately being disrupted fully. Based on the properties of the material bound to the black hole at the end of our simulations (in particular, the total bound mass and angular momentum), we comment upon the expected accretion process and associated electromagnetic signatures that are likely to result. 

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