We introduce the characteristic numbers and the chromatic polynomial of a linear subspace of matrices, or equivalently of a tensor. Our approach generalizes and unifies the chromatic polynomial of a graph and of a matroid, characteristic numbers of quadrics in Schubert calculus, Betti numbers of complements of hyperplane arrangements, Euler characteristic of determinantal hypersurfaces, and the maximum likelihood degree for general linear concentration models in algebraic statistics.

Using Lück’s Chern character isomorphism we obtain a general formula in terms of centralisers for the p-adic Farrell–Tate K-theory of any discrete group G with a finite classifying space for proper actions. We apply this formula to $\operatorname {\mathrm {Out}}(F_n)$. The case $n=p+1$ turns out to be especially interesting for the following reason: Up to conjugacy there is exactly one order p element in $\operatorname {\mathrm {Out}}(F_{p+1})$ which does not lift to an order p element in $\operatorname {\mathrm {Aut}}(F_{p+1})$. We compute the rational cohomology of the centraliser of this element and as a consequence obtain a full calculation of the p-adic Farrell–Tate K-theory of $\operatorname {\mathrm {Out}}(F_{p+1})$ for any prime $p \geq 5$. Our arguments provide an infinite family of $\mathbb {Q}_p$ summands in $K^1(B\operatorname {\mathrm {Out}}(F_n)) \otimes _{\mathbb {Z}} \mathbb {Q}$, with no need for computer calculations. The smallest value of p to which this applies is $p=11$. In this case we obtain a $\mathbb {Q}_{11}$ summand in $K^1(B\operatorname {\mathrm {Out}}(F_{12})) \otimes _{\mathbb {Z}} \mathbb {Q}$.

We introduce three families of vectors $|\underline {\lambda }^{so}\rangle $, $|\underline {\lambda }^{sp}\rangle $ and $|\underline {\lambda }^{o}\rangle $ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains [33, Section 7] and Betea [6, Section 3] are determinantal with respect to some explicit correlation kernels.

Furthermore, we establish a number of generalized Littlewood identities summed over certain restricted partitions by computing the inner products with elements indexed by one-column partitions or generalized partitions $(0^m)$ in the complete dual Fock space. In particular, for each positive integer n, we obtain generalized Littlewood identities for $(-n)$-asymmetric partitions. We show that these generalized Littlewood identities contain several well-known Littlewood-type identities as special cases. Consequently we also give a new proof of the generalized Littlewood identity [28, (5.25)] for Lie superalgebras.

The circular $\beta $ ensemble for $\beta =1,2$ and 4 corresponds to circular orthogonal, unitary and symplectic ensemble respectively as introduced by Dyson. The statistical state of the eigenvalues is then a determinantal point process ($\beta = 2$) and Pfaffian point process ($\beta = 1,4$). The explicit functional forms of the correlation kernels then imply that the general n-point correlation functions exhibit an asymptotic expansion in $1/N^2$, which moreover can be lifted to an asymptotic expansion in $1/N^2$ for the spacing distributions and their generating function. We use $\sigma $-Painlevé characterisations to show that the functional form of the first correction is related to the leading term via a second derivative. In the case $\beta = 2$ this finding has immediate consequences in interpreting the empirical Riemann zeros spacing distribution at large height, and that of their thinning. Explicit functional forms are used to show that the spectral form factors for $\beta =1,2$ and 4 also admit an asymptotic expansion in $1/N^2$. Differential relations are identified expressing the first and second correction in terms of the limiting functional form, and evidence is presented that they hold for general $\beta $. For even $\beta $ it is proved that the two-point correlation function permits an asymptotic expansion in $1/N^2$, and moreover that the leading correction relates to the limiting functional form via a second derivative.

We study relations between $(n_4)$ incidence configurations and the classical Poncelet Porism. Poncelet’s result concerns two conics and a sequence of points and lines that inscribes one conic and circumscribes the other. Poncelet’s Porism states that whether this sequence closes up after m steps only depends on the conics and not on the initial point of the sequence. In other words: Poncelet polygons are movable.

We transfer this motion into a flexibility statement about a large class of $(n_4)$ configurations, in which four (straight) lines pass through each point and four points lie on each line. An instance of such configurations in real geometry had been given by Branko Grünbaum and John Rigby in their classical 1990 paper where they constructed a real geometric realisation of a well known combinatorial $(21_4)$ configuration (which had been studied by Felix Klein), now commonly called the Grünbaum–Rigby configuration.

Since then, there has been an intensive search for movable $(n_4)$ configurations, but it is very surprising that the Grünbaum–Rigby $(21_4)$ configuration admits nontrivial motions. It is well-known that the Grünbaum–Rigby configuration is the smallest example of an infinite class of $(n_4)$ configurations, the trivial celestial configurations. A major result of this paper is that all trivial celestial configurations are movable via Poncelet’s Porism and results about properties of Poncelet grids. Alternative approaches via geometry of billiards, in-circle nets, and pentagram maps that relate the subject to discrete integrable systems are given as well.

We study linearizability properties of finite subgroups of the Cremona group $\operatorname {{\mathrm {Cr}}}_n(k)$ in the case where k is a global field, with the focus on the local-global principle. For every global field k of characteristic different from 2 and every $n \ge 3$, we give an example of a birational involution of $\mathbb P^n_k$ (= an element g of order $2$ in $\operatorname {{\mathrm {Cr}}}_n(k)$) such that g is not k-linearizable but g is $k_v$-linearizable in $\operatorname {{\mathrm {Cr}}}_n(k_v)$ for all places v of k. The main tool is a new birational invariant generalizing those introduced by Manin and Voskresenskiĭ in the arithmetic case and by Bogomolov–Prokhorov in the geometric case. We also apply it to the study of birational involutions in the real plane.

In this paper we introduce the shifted twisted Yangian of type AI, following the work of Lu–Wang–Zhang, and we study their semiclassical limits, a class of Poisson algebras. We demonstrate that they coincide with the Dirac reductions of the semiclassical shifted Yangian for $\mathfrak {gl}_n$. We deduce that these shifted twisted Yangians admit truncations which are isomorphic to Slodowy slices for many nonrectangular nilpotent elements in types B, C, D. As a direct consequence we obtain parabolic presentations of the semiclassical shifted twisted Yangian, analogous to those introduced by Brundan–Kleshchev for the Yangian of type A. Finally we give Poisson presentations of Slodowy slices for all even nilpotent elements in types B, C, D, generalising the recent work of the second author.

In [9], we introduced an approach to the question of extendability of projective varieties via degeneration to ribbons. In this article we build on these methods to give a new proof of optimal results on the extendability of general non-prime $K3$ surfaces, the classification of non-prime Fano threefolds and Mukai varieties and the irreducibility of their Hilbert schemes. The methods in this article also show the non-extendability of prime $K3$ surfaces for infinitely many values of g, for example, when g is of the form $g = 4k+1$, $k \geq 5$. This involves degenerations of $K3$ surfaces to ribbons on embedded Hirzebruch surfaces, called $K3$ carpets. We directly give optimal upper bounds on the cohomology of the twisted normal bundle of the $K3$ carpets instead of computing coranks of Gaussian maps of the canonical curve sections as in [16], [17]. As a result of independent interest, we show that such $K3$ carpets also appear as degenerations of smoothable simple normal crossings of two Hirzebruch surfaces embedded by arbitrary linear series intersecting along an anticanonical elliptic curve. Such type II degenerations constitute a smooth locus of codimension $6$ in the Hilbert scheme of $K3$ surfaces.

We develop the Tannakian theory of (analytic) prismatic F-crystals on a smooth formal scheme $\mathfrak {X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between the $\mathcal {G}$-objects of prismatic F-crystals on $\mathfrak {X}$ and $\mathcal {G}$-objects on a newly defined category of $\mathbb {Z}_p$-local systems on $\mathfrak {X}_\eta $: those of prismatically good reduction. Additionally, we develop a shtuka realization functor for (analytic) prismatic F-crystals on p-adic (formal) schemes and show it satisfies several compatibilities with previous work on the Tannakian theory of shtukas over such objects.

Let l and p be distinct primes, let F be a local field with residue field of characteristic p, and let $\mathfrak {X}$ be the irreducible component of the moduli space of Langlands parameters for $GL_3$ over $\mathbb {Z}_l$ corresponding to parameters of Steinberg type. We show that $\mathfrak {X}$ is Cohen–Macaulay and compute explicit equations for it. We also compute the Weil divisor class group of the special fibre of $\mathfrak {X}$, motivated by work of Manning for $GL_2$. Our methods involve the calculation of the cohomology of certain vector bundles on the flag variety, and build on work of Snowden, Vilonen–Xue, and Ngo.

We construct a moduli space that parametrises stable proper holomorphic submersions over a fixed compact Kähler base. Stability is described in terms of the existence of a canonical relatively Kähler metric on the submersion, called an optimal symplectic connection. The construction of the moduli space combines techniques from geometric invariant theory with the study of the geometric PDE defining an optimal symplectic connection. A special case of this moduli space is the moduli space of stable vector bundles over a compact Kähler manifold. We also show that the moduli space is a Hausdorff complex space equipped with a Weil-Petersson type Kähler metric.

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.

We classify the imaginaries in a large class of equicharacteristic zero henselian valued fields that contains all those with bounded inertia group, and more. To do so, we consider a mix of sorts introduced in earlier works of the two authors and prove elimination of imaginaries down to the field, the k-linear imaginaries, and the imaginaries of the value group.

We show that no matter what subset of a normed space is given, a typical 1-Lipschitz mapping into a Banach space is non-differentiable at a typical point of the set in a very strong sense: the set of partial limits of its derivative ratios, which is separable, contains all linear operators of norm at most $1$ from any fixed separable subspace of the operator space.

For subsets of finite-dimensional normed spaces which can be covered by a countable union of closed purely $1$-unrectifiable sets this extreme non-differentiability holds for a typical Lipschitz mapping at every point.

Both results are new even for Lipschitz mappings with a finite-dimensional codomain.

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and nondecorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

We determinewhen an exotic sphere $\Sigma $ of dimension $d\not {\equiv }1\ (\mathrm {mod}\ 4)$ can be detected through the homotopy type of its truncated $\mathscr {D}\mathrm {isc}$-presheaf. The latter records the diagram of framed configuration spaces of bounded cardinality in $\Sigma $ with natural point-forgetting and -splitting maps between them, and it gives rise to the finite stages in Goodwillie–Weiss’ embedding calculus tower. Our proof involves three ingredients that could be of independent interest: a gluing result for $\mathscr {D}\mathrm {isc}$-presheaves of manifolds divided into two codimension zero submanifolds, a version of Atiyah duality in the context of $\mathscr {D}\mathrm {isc}$-presheaves, and a computation of the finite residual of the mapping class group of the connected sums $\sharp ^g(S^{2k+1}\times S^{2k+1})$.

In this paper, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small. As a consequence, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which answers a question posed by Damanik, Fillman, and Gorodetski [24].

This is the fourth in a sequence of four papers, where we prove the arithmetic Siegel–Weil formula in co-rank $1$ for Kudla–Rapoport special cycles on exotic smooth integral models of unitary Shimura varieties of arbitrarily large even arithmetic dimension. Our arithmetic Siegel–Weil formula implies that degrees of Kudla–Rapoport arithmetic special $1$-cycles are encoded in near-central first derivatives of unitary Eisenstein series Fourier coefficients.

In this paper, we pin down precise normalizations for some $U(m,m)$ Siegel Eisenstein series, give local Siegel–Weil special value formulas with explicit constants, and record a geometric Siegel–Weil result for degrees of complex $0$-cycles. Using these, we complete the proof of our arithmetic Siegel–Weil results by patching together the local main theorems from our companion papers.