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1. Drawing Shortest Paths in Geodetic Graphs

   Cornelsen, S; Pfister, M; Foerster, H; Gronemann, M; Hoffmann, M; Kobourov, S; Schneck, T (October 2020, 28th International Symposium on Graph Drawing and Network Visualization (GD))

   Motivated by the fact that in a space where shortest paths are unique, no two shortest paths meet twice, we study a question posed by Greg Bodwin: Given a geodetic graph G, i.e., an unweighted graph in which the shortest path between any pair of vertices is unique, is there a philogeodetic drawing of G, i.e., a drawing of G in which the curves of any two shortest paths meet at most once? We answer this question in the negative by showing the existence of geodetic graphs that require some pair of shortest paths to cross at least four times. The bound on the number of crossings is tight for the class of graphs we construct. Furthermore, we exhibit geodetic graphs of diameter two that do not admit a philogeodetic drawing. 

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2. Quantum representations and monodromies of fibered links.

   Renaud Detcherry, Efstratia Kalfagianni (July 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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3. Cosets of monodromies and quantum representations

   Renaud Detcherry, Efstratia Kalfagianni (April 2019, Advances in mathematics)

   Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of |TVr(M)| for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU conjecture. Building on earlier work \cite{DK} we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponential r-growth. As a result, for any g>n⩾2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 

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4. Detailed Asymptotics of the Delay-Reliability Tradeoff of Random Linear Streaming Codes

   https://doi.org/10.1109/ISIT54713.2023.10206973  

   Su, Pin-Wen; Huang, Yu-Chih; Lin, Shih-Chun; Wang, I-Hsiang; Wang, Chih-Chun (June 2023, IEEE)

   Streaming codes eliminate the queueing delay and are an appealing candidate for low latency communications. This work studies the tradeoff between error probability p_e and decoding deadline ∆ of infinite-memory random linear streaming codes (RLSCs) over i.i.d. symbol erasure channels (SECs). The contributions include (i) Proving pe(∆) ∼ ρ∆^{−1.5}e^{−η∆}. The asymptotic power term ∆^{−1.5} of RLSCs is a strict improvement over the ∆^{−0.5} term of random linear block codes; (ii) Deriving a pair of upper and lower bounds on the asymptotic constant ρ, which are tight (i.e., identical) for one specific class of SECs; (iii) For any c > 1 and any decoding deadline ∆, the c-optimal memory length α^*_c (∆) is defined as the minimal memory length α needed for the resulting pe to be within a factor of c of the best possible p^*_e under any α, an important piece of information for practical implementation. This work studies and derives new properties of α^*_c (∆) based on the newly developed asymptotics. 

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5. A reduction principle for Fourier coefficients of automorphic forms

   https://doi.org/10.1007/s00209-021-02784-w  

   Gourevitch, Dmitry; Gustafsson, Henrik P.; Kleinschmidt, Axel; Persson, Daniel; Sahi, Siddhartha (March 2022, Mathematische Zeitschrift)

   Abstract We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group $$\mathbf {G}(\mathbb {A}_\mathbb {K})$$ G ( A K ) , which we refer to as Fourier coefficients associated to the data of a ‘Whittaker pair’. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are ‘Levi-distinguished’ Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a $${\mathbb K}$$ K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients. 

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