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1. Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective

   https://doi.org/10.21468/SciPostPhys.6.5.052  

   Closset, Cyril; Del Zotto, Michele; Saxena, Vivek (January 2019, SciPost Physics)

   We revisit the correspondence between Calabi-Yau (CY) threefoldisolated singularities \mathbf{X} 𝐗 and five-dimensional superconformal field theories (SCFTs), which ariseat low energy in M-theory on the space-time transverse to \mathbf{X} 𝐗 .Focussing on the case of toric CY singularities, we analyze the“gauge-theory phases” of the SCFT by exploiting fiberwise M-theory/typeIIA duality. In this setup, the low-energy gauge group simply arises onstacks of coincident D6-branes wrapping 2-cycles in some ALE space oftype A_{M-1} A M − 1 fibered over a real line, and the map between the Kähler parameters of \mathbf{X} 𝐗 and the Coulomb branch parameters of the field theory (masses and VEVs)can be read off systematically. Different type IIA “reductions” giverise to different gauge theory phases, whose existence depends on theparticular (partial) resolutions of the isolated singularity \mathbf{X} 𝐗 .We also comment on the case of non-isolated toric singularities.Incidentally, we propose a slightly modified expression for theCoulomb-branch prepotential of 5d \mathcal{N}=1 𝒩 = 1 gauge theories. 

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2. A Combinatorial Cut-Toggling Algorithm for Solving Laplacian Linear Systems

   https://doi.org/10.1007/s00453-023-01154-8  

   Henzinger, Monika; Jin, Billy; Peng, Richard; Williamson, David P (December 2023, Algorithmica)

   Over the last two decades, a significant line of work in theoretical algorithms has made progress in solving linear systems of the form $$\mathbf{L}\mathbf{x} = \mathbf{b}$$, where $$\mathbf{L}$$ is the Laplacian matrix of a weighted graph with weights $w(i,j)>0$ on the edges. The solution $$\mathbf{x}$$ of the linear system can be interpreted as the potentials of an electrical flow in which the resistance on edge $(i,j)$ is $1/w(i,j)$$. Kelner, Orrechia, Sidford, and Zhu \cite{KOSZ13} give a combinatorial, near-linear time algorithm that maintains the Kirchoff Current Law, and gradually enforces the Kirchoff Potential Law by updating flows around cycles ({\it cycle toggling}). In this paper, we consider a dual version of the algorithm that maintains the Kirchoff Potential Law, and gradually enforces the Kirchoff Current Law by {\it cut toggling}: each iteration updates all potentials on one side of a fundamental cut of a spanning tree by the same amount. We prove that this dual algorithm also runs in a near-linear number of iterations. We show, however, that if we abstract cut toggling as a natural data structure problem, this problem can be reduced to the online vector-matrix-vector problem (OMv), which has been conjectured to be difficult for dynamic algorithms \cite{HKNS15}. The conjecture implies that the data structure does not have an $$O(n^{1-\epsilon})$ time algorithm for any $$\epsilon > 0$$, and thus a straightforward implementation of the cut-toggling algorithm requires essentially linear time per iteration. To circumvent the lower bound, we batch update steps, and perform them simultaneously instead of sequentially. An appropriate choice of batching leads to an $$\widetilde{O}(m^{1.5})$$ time cut-toggling algorithm for solving Laplacian systems. Furthermore, we show that if we sparsify the graph and call our algorithm recursively on the Laplacian system implied by batching and sparsifying, we can reduce the running time to $$O(m^{1 + \epsilon})$$ for any $$\epsilon > 0$$. Thus, the dual cut-toggling algorithm can achieve (almost) the same running time as its primal cycle-toggling counterpart. 

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3. New lower bounds for matrix multiplication and

   https://doi.org/10.1017/fmp.2023.14  

   Conner, Austin; Harper, Alicia; Landsberg, J.M. (January 2023, Forum of Mathematics, Pi)

   Abstract Let$$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$$denote the matrix multiplication tensor (and write$$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$$), and let$$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$$denote the determinant polynomial considered as a tensor. For a tensorT, let$$\underline {\mathbf {R}}(T)$$denote its border rank. We (i) give the first hand-checkable algebraic proof that$$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$$, (ii) prove$$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$$and$$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$$, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$$M_{\langle 2\rangle }$$, (iii) prove$$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$$, (iv) prove$$\underline {\mathbf {R}}(\operatorname {det}_3)=17$$, improving the previous lower bound of$$12$$, (v) prove$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$$for all$$\mathbf {n}\geq 25$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$$was known, as well as lower bounds for$$4\leq \mathbf {n}\leq 25$$, and (vi) prove$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$$for all$$\mathbf {n} \ge 18$$, where previously only$$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$$was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors. The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, calledborder apolaritydeveloped by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorTand an integerr, in a finite number of steps, either outputs that there is no border rankrdecomposition forTor produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenThas a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 

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4. Brief communication: A nonlinear self-similar solution to barotropic flow over varying topography

   https://doi.org/10.5194/npg-25-201-2018  

   Ibanez, Ruy; Kuehl, Joseph; Shrestha, Kalyan; Anderson, William (January 2018, Nonlinear Processes in Geophysics)

   Beginning from the shallow water equations (SWEs), a nonlinear self-similar analytic solution is derived for barotropic flow over varying topography. We study conditions relevant to the ocean slope where the flow is dominated by Earth's rotation and topography. The solution is found to extend the topographic β-plume solution of Kuehl (2014) in two ways. (1) The solution is valid for intensifying jets. (2) The influence of nonlinear advection is included. The SWEs are scaled to the case of a topographically controlled jet, and then solved by introducing a similarity variable, η = cx^n_xy^n_y. The nonlinear solution, valid for topographies h = h₀ − αxy³, takes the form of the Lambert W-function for pseudo velocity. The linear solution, valid for topographies h = h₀ − αxy^−γ, takes the form of the error function for transport. Kuehl's results considered the case −1 ≤ γ < 1 which admits expanding jets, while the new result considers the case γ < −1 which admits intensifying jets and a nonlinear case with γ = −3. 

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5. An ergodic system is dominant exactly when it has positive entropy

   https://doi.org/10.1017/etds.2022.69  

   AUSTIN, TIM; GLASNER, ELI; THOUVENOT, JEAN-PAUL; WEISS, BENJAMIN (January 2022, Ergodic Theory and Dynamical Systems)

   An ergodic dynamical system $$\mathbf{X}$$ is called dominant if it is isomorphic to a generic extension of itself. It was shown by Glasner et al [On some generic classes of ergodic measure preserving transformations. Trans. Moscow Math. Soc. 82 (1) (2021), 15–36] that Bernoulli systems with finite entropy are dominant. In this work, we show first that every ergodic system with positive entropy is dominant, and then that if $$\mathbf{X}$$ has zero entropy, then it is not dominant. 

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