We extend the notion of T-duality to manifolds endowed with non-principal torus actions. The singularities of the torus action are controlled by a certain Lie algebroid, called the elliptic tangent bundle. Using this Lie algebroid, we explain how certain invariant generalized complex structures can be transported via T-duality. Along the way, we use the elliptic tangent bundle to define connections for these torus actions, and give new insight into the classification of such actions by Haefliger-Salem.

Joyce vertex algebras are vertex algebra structures defined on the homology of certain $\mathbb {C}$-linear moduli stacks, and are used to express wall-crossing formulae for Joyce’s homological enumerative invariants. This paper studies the generalization of this construction to settings that come from nonlinear enumerative problems. In the special case of orthosymplectic enumerative geometry, we obtain twisted modules for Joyce vertex algebras.

We expect that our construction will be useful for formulating wall-crossing formulae for enumerative invariants for nonlinear moduli stacks. We include several variants of our construction that apply to different flavours of enumerative invariants, including Joyce’s homological invariants, DT4 invariants, and a version of K-theoretic enumerative invariants.

We introduce action-driven flows for causal variational principles, being a class of nonconvex variational problems emanating from applications in fundamental physics. In the compact setting, Hölder continuous curves of measures are constructed by using the method of minimizing movements. As is illustrated in examples, these curves will in general not have a limit point, due to the nonconvexity of the action. This leads us to introducing a novel penalization which ensures the existence of a limit point, giving rise to approximative solutions of the Euler-Lagrange equations. The methods and results are adapted and generalized to the causal action principle in the finite-dimensional case. As an application, we construct a flow of measures for causal fermion systems in the infinite-dimensional situation.

We prove an analogue of Chebyshev’s alternation theorem for linearly independent discrete functions $\Phi _n=\{\varphi _k\}_{k=1}^n$ on the interval $[0,q]_{\scriptscriptstyle \mathbb {Z}}=[0,q]\cap \mathbb {Z}$. In particular, we establish that the polynomial of best uniform approximation of a discrete function admits a Chebyshev alternance set of length $n+1$ if and only if $\Phi _n$ is a Chebyshev $T_{\scriptscriptstyle \mathbb {Z}}$-system. We also obtain a discrete version of Sturm’s oscillation theorem, according to which the number of discrete zeros of the polynomial $\sum _{k=m}^{n}a_k\varphi _k$ is no less than $m-1$ and no more than $n-1$. This implies that $\Phi _n$ is a $T_{\scriptscriptstyle \mathbb {Z}}$-system and a discrete Sturm-Hurwitz spectral gap theorem is valid. As applications, we study the orthogonal polynomials with removed largest zeros. We establish the monotonicity property of coefficients in the Fourier expansions of such polynomials, thereby strengthening the results of H. Cohn and A. Kumar. We apply this to solve a Yudin-type extremal problem for polynomials with spectral gap.

Let G be a finite group and let p be a prime. In this paper, we prove a strengthened version of Brauer’s height zero conjecture for the principal p-block of G that takes the action of a certain group of Galois automorphisms into account. This answers a conjecture recently proposed by Malle, Moretó, Rizo and Schaeffer Fry. We then use this to obtain a structural result which can be seen as a Galois version of the Itô–Michler theorem.

The quantum duality principle (QDP) by Drinfeld predicts a connection between the quantized universal enveloping algebras and the quantized coordinate algebras, where the underlying classical objects are related by the duality in Poisson geometry. The current paper gives an explicit formulization of the QDP for quantum symmetric pairs.

Let $\mathfrak {g}$ be a complex semi-simple Lie algebra, equipped with the standard Lie bialgebra structure. Let $\theta $ be a Lie algebra involution on $\mathfrak {g}$ and denote by $\mathfrak {k}=\mathfrak {g}^\theta $ the fixed point subalgebra. The quantum symmetric pair $(\mathrm {U},\mathrm {U}^\imath )$ is originally defined to be a quantization of the symmetric pair of the universal enveloping algebras $(U(\mathfrak {g}),U(\mathfrak {k}))$. In this paper, we show that an explicit specialization of $(\mathrm {U},\mathrm {U}^\imath )$ gives rise to the pair of the coordinate algebras $(\mathcal {O}(G^*),\mathcal {O}(K^\perp \backslash G^*))$, where $G^*$ is the dual Poisson-Lie group with the Lie algebra $\mathfrak {g}^*$, and $K^\perp \backslash G^*$ is a $G^*$-Poisson homogeneous space. Here $K^\perp $ is the closed subgroup of $G^*$ associated to the complementary dual of $\mathfrak {k}$. Therefore $(\mathrm {U},\mathrm {U}^\imath )$ can be viewed as a pair of quantized coordinate algebras. This generalizes the result of De Concini–Procesi [14] that the quantum group $\mathrm {U}$ provides a quantization of the coordinate algebra of $G^*$.

We systematically study the intersection flatness and Ohm-Rush properties for modules over a commutative ring, drawing inspiration from the work of Ohm and Rush and of Hochster and Jeffries. We establish new structural results for modules that are intersection flat/Ohm-Rush by exhibiting intimate connections between these notions and the seminal work of Raynaud and Gruson on Mittag-Leffler modules. In particular, we develop a theory of Ohm-Rush modules that is parallel to the theory of Mittag-Leffler modules. We also obtain descent and local-to-global results for intersection flat/Ohm-Rush modules. Our investigations reveal a particularly pleasing picture for flat modules over a complete local ring, in which case many otherwise distinct properties coincide.

We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682–706, 2007]) to a nonlinear setting, and from the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1–45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrary-dimensional phase space, and the application of Bourgain’s geometric lemma in [Geom. Funct. Anal., 17(3):682–706, 2007].

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.

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].

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 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 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.

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.

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.

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.