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1. The Hilbert series of the superspace coinvariant ring

   https://doi.org/10.1017/fmp.2024.14  

   Rhoades, Brendon; Wilson, Andrew Timothy (January 2024, Forum of Mathematics, Pi)

   Abstract Let$$\Omega _n$$be the ring of polynomial-valued holomorphic differential forms on complexn-space, referred to in physics as the superspace ring of rankn. The symmetric group$${\mathfrak {S}}_n$$acts diagonally on$$\Omega _n$$by permuting commuting and anticommuting generators simultaneously. We let$$SI_n \subseteq \Omega _n$$be the ideal generated by$${\mathfrak {S}}_n$$-invariants with vanishing constant term and study the quotient$$SR_n = \Omega _n / SI_n$$of superspace by this ideal. We calculate the doubly-graded Hilbert series of$$SR_n$$and prove an ‘operator theorem’, which characterizes the harmonic space$$SH_n \subseteq \Omega _n$$attached to$$SR_n$$in terms of the Vandermonde determinant and certain differential operators. Our methods employ commutative algebra results that were used in the study of Hessenberg varieties. Our results prove conjectures of N. Bergeron, Colmenarejo, Li, Machacek, Sulzgruber, Swanson, Wallach and Zabrocki. 

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2. 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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3. Modular curves and Néron models of generalized Jacobians

   https://doi.org/10.1112/S0010437X23007662  

   Jordan, Bruce W; Ribet, Kenneth A; Scholl, Anthony J (May 2024, Compositio Mathematica)

   Let$$X$$be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring$$R$$, and$$\mathfrak {m}$$a modulus on$$X$$, given by a closed subscheme of$$X$$which is geometrically reduced. The generalized Jacobian$$J_\mathfrak {m}$$of$$X$$with respect to$$\mathfrak {m}$$is then an extension of the Jacobian of$$X$$by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of$$X$$over$$R$$. This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves$$X_0(N)$$with moduli supported on the cusps. 

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4. THE PAUCITY PROBLEM FOR CERTAIN SYMMETRIC DIOPHANTINE EQUATIONS

   https://doi.org/10.1017/S000497272200096X  

   WOOLEY, TREVOR D. (August 2023, Bulletin of the Australian Mathematical Society)

   Abstract Let$$\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$$be integral linear combinations of elementary symmetric polynomials with$$\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$$, where$$1\le k_1<\cdots . Subject to the condition$$k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$$, we show that there is a paucity of nondiagonal solutions to the Diophantine system$$\varphi _j({\mathbf x})=\varphi _j({\mathbf y})\ (1\le j\le r)$$. 

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5. Almost all orbits of the Collatz map attain almost bounded values

   https://doi.org/10.1017/fmp.2022.8  

   Tao, Terence (January 2022, Forum of Mathematics, Pi)

   Abstract Define theCollatz map$${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$$on the positive integers$$\mathbb {N}+1 = \{1,2,3,\dots \}$$by setting$${\operatorname {Col}}(N)$$equal to$$3N+1$$whenNis odd and$$N/2$$whenNis even, and let$${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$$denote the minimal element of the Collatz orbit$$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $$. The infamousCollatz conjectureasserts that$${\operatorname {Col}}_{\min }(N)=1$$for all$$N \in \mathbb {N}+1$$. Previously, it was shown by Korec that for any$$\theta> \frac {\log 3}{\log 4} \approx 0.7924$$, one has$${\operatorname {Col}}_{\min }(N) \leq N^\theta $$for almost all$$N \in \mathbb {N}+1$$(in the sense of natural density). In this paper, we show that foranyfunction$$f \colon \mathbb {N}+1 \to \mathbb {R}$$with$$\lim _{N \to \infty } f(N)=+\infty $$, one has$${\operatorname {Col}}_{\min }(N) \leq f(N)$$for almost all$$N \in \mathbb {N}+1$$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$$3$$-adic cyclic group$$\mathbb {Z}/3^n\mathbb {Z}$$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency. 

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