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In this paper we resolve a conjecture of Kresch and Tamvakis [Duke Math. J. 110 (2001), pp. 359–376]. Our result is the following. Theorem : For any positive integer D and any integers i,j (0\leq i,j \leq D), \; the absolute value of the following hypergeometric series is at most 1: \begin{equation*} {_4F_3} \left [ \begin {array}{c} -i, \; i+1, \; -j, \; j+1 \\ 1, \; D+2, \; -D \end{array} ; 1 \right ]. \end{equation*} To prove this theorem, we use the Biedenharn-Elliott identity, the theory of Leonard pairs, and the Perron-Frobenius theorem. [ABSTRACT FROM AUTHOR]

By employing a quadratic summation formula due to Gasper and Rahman [ Basic hypergeometric series , 2nd ed, vol. 96, Cambridge University Press, Cambridge, 2004] and the creative microscoping method developed by Guo and Zudilin [Adv. Math. 346 (2019), pp. 329–358], we establish some new parametric q-supercongruences, the corresponding supercongruences of which can be deemed the variants of Van Hamme's (J.2) and (L.2) supercongruences. Moreover, we obtain three families of Ramanujan-type formulas on \pi and propose a challenging conjecture. [ABSTRACT FROM AUTHOR]

In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G is a finite group, p is a prime dividing the group order, and k(G) is the number of conjugacy classes of G, then k(G)\geq 2\sqrt {p-1}, and they proved this conjecture for solvable G and showed that it is sharp for those primes p for which \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p, and we prove it for solvable groups, and when p is large, also for arbitrary groups. [ABSTRACT FROM AUTHOR]

Hovey and Palmieri [ The structure of the Bousfield lattice , Amer. Math. Soc., Providence, RI, 1999] proposed the retract conjecture on the Bousfield lattice of the stable homotopy category. The author, Shimomura and Tatehara [Publ. Res. Inst. Math. Sci. 50 (2014), pp. 497–513] defined the notion of monoidal posets as a generalization of the Bousfield lattice. In this paper, we prove that an analogue of the retract conjecture holds on a sublattice of monoidal posets. [ABSTRACT FROM AUTHOR]

A well known conjecture due to Hassett asserts that a cubic fourfold X whose transcendental cohomology TX cannot be realized as the transcendental cohomology of a K3 surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic 2-cycles on them, it is natural to ask for the most algebraic cubic fourfolds X to which this conjecture is still applicable. In this paper, we show that for an appropriate "algebraicity index" κX ∈ Q+, there exists a unique class of cubics maximizing κX, not having an associated K3 surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett's irrationality conjecture for cubic fourfolds. [ABSTRACT FROM AUTHOR]

We show that the celebrated operad of pre-Lie algebras is very rigid: it has no "non-obvious" degrees of freedom from either of the three points of view: deformations of maps to and from the "three graces of operad theory", homotopy automorphisms, and operadic twisting. Examining the latter, it is possible to answer two questions of Markl from 2005 [Czechoslovak Math. J. 57 (2007), pp. 253–268; J. Lie Theory 17 (2007), pp. 241–261], including a Lie-theoretic version of the Deligne conjecture. [ABSTRACT FROM AUTHOR]

In this paper we prove the containment I(nm) ⊂ M(n-1)mIm, for a radical ideal I of s general points in Pn, where s ≥ 2n. As a corollary we get that the Chudnovsky Conjecture holds for a very general set of at least 2n points in Pn. [ABSTRACT FROM AUTHOR]

We describe an explicit formula of the canonical pairing on the twisted de Rham cohomology associated with the category of local matrix factorizations and by characterizing its relation to Saito's higher residue pairings we reprove the conjecture of Shklyarov [Adv. Math. 292 (2016), pp. 181–209]. [ABSTRACT FROM AUTHOR]

In 1996 Stolz [Math. Ann. 304 (1996), pp. 785–800] conjectured that a string manifold with positive Ricci curvature has vanishing Witten genus. Here we prove this conjecture for toric string Fano manifolds and for string torus manifolds admitting invariant metrics of non-negative sectional curvature. [ABSTRACT FROM AUTHOR]

The notion of strict closedness of rings was given by J. Lipman [Amer. J. Math. 93 (1971), pp. 649–685] in connection with a conjecture of O. Zariski. The present purpose is to give a practical method of construction of strictly closed rings. It is also shown that the Stanley-Reisner rings of simplicial complexes (resp. F-pure rings satisfying the condition (S2) of Serre) are strictly closed (resp. weakly Arf) rings. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new expected dimension formula for certain rank two Brill-Noether loci with fixed special determinant. This answers a question asked by Osserman and also leads to a new and much simpler proof of a theorem in his 2015 work. Our result generalizes the well-known result by Bertram, Feinberg and independently Mukai on expected dimension of rank two Brill-Noether loci with canonical determinant and partially verifies a conjecture (in rank two) of Grzegorczyk and Newstead on coherent systems. [ABSTRACT FROM AUTHOR]

We prove a sharp upper bound for the resurgence of sums of ideals involving disjoint sets of variables, strengthening work of Bisui–Hà–Jayanthan–Thomas [Collect. Math. 72 (2021), pp. 605–614]. Complete solutions are delivered for two conjectures proposed by these authors. For given real numbers a and b, we consider the set Res(a,b) of possible values of the resurgence of I+J where I and J are ideals in disjoint sets of variables having resurgence a and b, respectively. Some questions and partial results about Res(a,b) are discussed. [ABSTRACT FROM AUTHOR]

In this note, we prove some new stability results for plethysm coefficients. As special cases, we verify a conjecture of Wildon, and show the stability of sequences recently predicted by Bessenrodt, Bowman and Paget [Trans. Amer. Math. Soc. 375 (2022), pp. 5151–5194] to be weakly increasing. [ABSTRACT FROM AUTHOR]

A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed in complex dimension 2, by the work of Balas-Gauduchon [Math. Z. 189 (1985), pp. 193–210]. (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov [Trans. Amer. Math. Soc. 348 (1996), pp. 3051–3063] (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to the class of complex nilmanifolds and confirm the conjecture in that case. [ABSTRACT FROM AUTHOR]

Let E be an elliptic curve over a number field K, cv the Tamagawa number of E at v, and let cE = ∏v cv. Lorenzini proved that v13(cE) is positive for all elliptic curves over quadratic fields with a point of order 13. Krumm conjectured, based on extensive computation, that the 13-adic valuation of cE is even for all such elliptic curves. In this note we prove this conjecture and furthermore prove that there is a unique such curve satisfying v13(cE)=2. [ABSTRACT FROM AUTHOR]