Let $\Delta $ denote a nondegenerate k-simplex in $\mathbb {R}^k$. The set $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices in $\mathbb {R}^k$ similar to $\Delta $ is diffeomorphic to $\operatorname {O}(k)\times [0,\infty )\times \mathbb {R}^k$, where the factor in $\operatorname {O}(k)$ is a matrix called the pose. Among $(k-1)$-spheres smoothly embedded in $\mathbb {R}^k$ and isotopic to the identity, there is a dense family of spheres, for which the subset of $\operatorname {\mathrm {Sim}}(\Delta )$ of simplices inscribed in each embedded sphere contains a similar simplex of every pose $U\in \operatorname {O}(k)$. Further, the intersection of $\operatorname {\mathrm {Sim}}(\Delta )$ with the configuration space of $k+1$ distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in $\operatorname {O}(k)$ via the pose map. This gives a high-dimensional generalisation of classical results on inscribing families of triangles in plane curves. We use techniques established in our previous paper on the square-peg problem where we viewed inscribed simplices in spheres as transverse intersections of submanifolds of compactified configuration spaces.

We prove that the hitting measure is singular with respect to the Lebesgue measure for random walks driven by finitely supported measures on cocompact, hyperelliptic Fuchsian groups. Moreover, the Hausdorff dimension of the hitting measure is strictly less than one. Equivalently, the inequality between entropy and drift is strict. A similar statement is proven for Coxeter groups.

In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand to optimise accuracy or computational complexity. In numerical polynomial algebra, a single norm (attributed to Weyl) dominates the literature. This article initiates the use of $L_p$ norms for numerical algebraic geometry, with an emphasis on $L_{\infty }$ . This classical idea yields strong improvements in the analysis of the number of steps performed by numerous iterative algorithms. In particular, we exhibit three algorithms where, despite the complexity of computing $L_{\infty }$ -norm, the use of $L_p$ -norms substantially reduces computational complexity: a subdivision-based algorithm in real algebraic geometry for computing the homology of semialgebraic sets, a well-known meshing algorithm in computational geometry and the computation of zeros of systems of complex quadratic polynomials (a particular case of Smale’s 17th problem).

A recent paper of Balogh, Li and Treglown [3] initiated the study of Dirac-type problems for ordered graphs. In this paper, we prove a number of results in this area. In particular, we determine asymptotically the minimum degree threshold for forcing

1. (i) a perfect H-tiling in an ordered graph, for any fixed ordered graph H of interval chromatic number at least $3$ ;

2. (ii) an H-tiling in an ordered graph G covering a fixed proportion of the vertices of G (for any fixed ordered graph H);

3. (iii) an H-cover in an ordered graph (for any fixed ordered graph H).

The first two of these results resolve questions of Balogh, Li and Treglown, whilst (iii) resolves a question of Falgas-Ravry. Note that (i) combined with a result from [3] completely determines the asymptotic minimum degree threshold for forcing a perfect H-tiling. Additionally, we prove a result that, combined with a theorem of Balogh, Li and Treglown, asymptotically determines the minimum degree threshold for forcing an almost perfect H-tiling in an ordered graph (for any fixed ordered graph H). Our work therefore provides ordered graph analogues of the seminal tiling theorems of Kühn and Osthus [Combinatorica 2009] and of Komlós [Combinatorica 2000]. Each of our results exhibits some curious, and perhaps unexpected, behaviour. Our solution to (i) makes use of a novel absorbing argument.

The superspace ring $\Omega _n$ is a rank n polynomial ring tensored with a rank n exterior algebra. Using an extension of the Vandermonde determinant to $\Omega _n$, the authors previously defined a family of doubly graded quotients ${\mathbb {W}}_{n,k}$ of $\Omega _n$, which carry an action of the symmetric group ${\mathfrak {S}}_n$ and satisfy a bigraded version of Poincaré Duality. In this paper, we examine the duality modules ${\mathbb {W}}_{n,k}$ in greater detail. We describe a monomial basis of ${\mathbb {W}}_{n,k}$ and give combinatorial formulas for its bigraded Hilbert and Frobenius series. These formulas involve new combinatorial objects called ordered set superpartitions. These are ordered set partitions $(B_1 \mid \cdots \mid B_k)$ of $\{1,\dots ,n\}$ in which the nonminimal elements of any block $B_i$ may be barred or unbarred.

In this paper, we prove a Clifford type inequality for the curve $X_{2,2,2,4}$, which is the intersection of a quartic and three general quadratics in $\mathbb {P}^5$. We thus prove a stronger Bogomolov–Gieseker inequality for characters of stable vector bundles and stable objects on Calabi–Yau complete intersection $X_{2,4}$. Applying the scheme proposed by Bayer, Bertram, Macrì, Stellari and Toda, we can construct an open subset of Bridgeland stability conditions on $X_{2,4}$.

The goal of this paper is to generalise, refine and improve results on large intersections from [2, 8]. We show that if G is a countable discrete abelian group and $\varphi , \psi : G \to G$ are homomorphisms, such that at least two of the three subgroups $\varphi (G)$ , $\psi (G)$ and $(\psi -\varphi )(G)$ have finite index in G, then $\{\varphi , \psi \}$ has the large intersections property. That is, for any ergodic measure preserving system $\textbf {X}=(X,\mathcal {X},\mu ,(T_g)_{g\in G})$ , any $A\in \mathcal {X}$ and any $\varepsilon>0$ , the set

$$ \begin{align*} \{g\in G : \mu(A\cap T_{\varphi(g)}^{-1} A \cap T_{\psi(g)}^{-1} A)>\mu(A)^3-\varepsilon\} \end{align*} $$

is syndetic (Theorem 1.11). Moreover, in the special case where $\varphi (g)=ag$ and $\psi (g)=bg$ for $a,b\in \mathbb {Z}$ , we show that we only need one of the groups $aG$ , $bG$ or $(b-a)G$ to be of finite index in G (Theorem 1.13), and we show that the property fails, in general, if all three groups are of infinite index (Theorem 1.14).

One particularly interesting case is where $G=(\mathbb {Q}_{>0},\cdot )$ and $\varphi (g)=g$ , $\psi (g)=g^2$ , which leads to a multiplicative version of the Khintchine-type recurrence result in [8]. We also completely characterise the pairs of homomorphisms $\varphi ,\psi $ that have the large intersections property when $G = {{\mathbb Z}}^2$ .

The proofs of our main results rely on analysis of the structure of the universal characteristic factor for the multiple ergodic averages

$$ \begin{align*} \frac{1}{|\Phi_N|} \sum_{g\in \Phi_N}T_{\varphi(g)}f_1\cdot T_{\psi(g)} f_2. \end{align*} $$

In the case where G is finitely generated, the characteristic factor for such averages is the Kronecker factor. In this paper, we study actions of groups that are not necessarily finitely generated, showing, in particular, that, by passing to an extension of $\textbf {X}$ , one can describe the characteristic factor in terms of the Conze–Lesigne factor and the $\sigma $ -algebras of $\varphi (G)$ and $\psi (G)$ invariant functions (Theorem 4.10).

We establish a Hirzebruch–Riemann–Roch-type theorem and a Grothendieck–Riemann–Roch-type theorem for matrix factorizations on quotient Deligne–Mumford stacks. For this, we first construct a Hochschild–Kostant–Rosenberg-type isomorphism explicit enough to yield a categorical Chern character formula. Then, we find an expression of the canonical pairing of Shklyarov under the isomorphism.

We give an upper bound on the volume $\operatorname {vol}(P^*)$ of a polytope $P^*$ dual to a d-dimensional lattice polytope P with exactly one interior lattice point in each dimension d. This bound, expressed in terms of the Sylvester sequence, is sharp and achieved by the dual to a particular reflexive simplex. Our result implies a sharp upper bound on the volume of a d-dimensional reflexive polytope. Translated into toric geometry, this gives a sharp upper bound on the anti-canonical degree $(-K_X)^d$ of a d-dimensional Fano toric variety X with at worst canonical singularities.

In this paper, we prove one divisibility of the Iwasawa–Greenberg main conjecture for the Rankin–Selberg product of a weight two cusp form and an ordinary complex multiplication form of higher weight, using congruences between Klingen Eisenstein series and cusp forms on $\mathrm {GU}(3,1)$, generalizing an earlier result of the third-named author to allow nonordinary cusp forms. The main result is a key input in the third-named author’s proof of Kobayashi’s $\pm $-main conjecture for supersingular elliptic curves. The new ingredient here is developing a semiordinary Hida theory along an appropriate smaller weight space and a study of the semiordinary Eisenstein family.

The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in $\mathbb {P}^n$ and $\mathbb {P}^n/G$ , where G is a finite group that acts on $\mathbb {P}^n$ and preserves the Fermat hypersurface. We generalize this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian $\operatorname {{\mathrm {Gr}}}(n,r)$ and preserves an appropriate Calabi–Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi–Yau hypersurfaces inside $\operatorname {{\mathrm {Gr}}}(n,r)$ and $\operatorname {{\mathrm {Gr}}}(n,r)/G$ . This allows us to describe a compactification of the Eguchi–Hori–Xiong mirror to the Grassmannian, inside a blow-up of the quotient of the Grassmannian by G.

In this work, we prove a version of the Sylvester–Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of $\Sigma ^{[3]}\Pi \Sigma \Pi ^{[2]}$ circuits. Specifically, we prove that, if a finite set of irreducible quadratic polynomials ${\mathcal {Q}}$ satisfies that for every two polynomials $Q_1,Q_2\in {\mathcal {Q}}$ there is a subset ${\mathcal {K}}\subset {\mathcal {Q}}$ such that $Q_1,Q_2 \notin {\mathcal {K}}$ and whenever $Q_1$ and $Q_2$ vanish, then $\prod _{i\in {\mathcal {K}}} Q_i$ vanishes, then the linear span of the polynomials in ${\mathcal {Q}}$ has dimension $O(1)$ . This extends the earlier result [21] that holds for the case $|{\mathcal {K}}| = 1$ .

The purpose of this paper is to quantify the size of the Lebesgue constants $(\boldsymbol {L}_m)_{m=1}^{\infty }$ associated with the thresholding greedy algorithm in terms of a new generation of parameters that modulate accurately some features of a general basis. This fine tuning of constants allows us to provide an answer to the question raised by Temlyakov in 2011 to find a natural sequence of greedy-type parameters for arbitrary bases in Banach (or quasi-Banach) spaces which combined linearly with the sequence of unconditionality parameters $(\boldsymbol {k}_m)_{m=1}^{\infty }$ determines the growth of $(\boldsymbol {L}_m)_{m=1}^{\infty }$ . Multiple theoretical applications and computational examples complement our study.

For $1\le p <\infty $ , we present a reflexive Banach space $\mathfrak {X}^{(p)}_{\text {awi}}$ , with an unconditional basis, that admits $\ell _p$ as a unique asymptotic model and does not contain any Asymptotic $\ell _p$ subspaces. Freeman et al., Trans. AMS. 370 (2018), 6933–6953 have shown that whenever a Banach space not containing $\ell _1$ , in particular a reflexive Banach space, admits $c_0$ as a unique asymptotic model, then it is Asymptotic $c_0$ . These results provide a complete answer to a problem posed by Halbeisen and Odell [Isr. J. Math. 139 (2004), 253–291] and also complete a line of inquiry of the relation between specific asymptotic structures in Banach spaces, initiated in a previous paper by the first and fourth authors. For the definition of $\mathfrak {X}^{(p)}_{\text {awi}}$ , we use saturation with asymptotically weakly incomparable constraints, a new method for defining a norm that remains small on a well-founded tree of vectors which penetrates any infinite dimensional closed subspace.