Abstract We consider the problem of finding the minimum of inhomogeneous Gaussian lattice sums: Given a lattice $L \subseteq \mathbb {R}^n$ and a positive constant $\alpha $ , the goal is to find the minimizers of $\sum _{x \in L} e^{-\alpha \|x - z\|^2}$ over all $z \in \mathbb {R}^n$ . By a result of Bétermin and Petrache from 2017 it is known that for steep potential energy functions—when $\alpha $ tends to infinity—the minimizers in the limit are found at deep holes of the lattice. In this paper, we consider minimizers which already stabilize for all $\alpha \geq \alpha _0$ for some finite $\alpha _0$ ; we call these minimizers stable cold spots. Generic lattices do not have stable cold spots. For several important lattices, like the root lattices, the Coxeter-Todd lattice, and the Barnes-Wall lattice, we show how to apply the linear programming bound for spherical designs to prove that the deep holes are stable cold spots. We also show, somewhat unexpectedly, that the Leech lattice does not have stable cold spots.

Abstract We extend the construction of the p -adic L -function interpolating unitary Friedberg–Jacquet periods in previous work of the author to include the p -adic variation of Maass–Shimura differential operators. In particular, we develop a theory of nearly overconvergent automorphic forms in higher degrees of coherent cohomology for unitary Shimura varieties generalising previous work for modular curves. The construction of this p -adic L -function can be viewed as a higher-dimensional generalisation of the work of Bertolini–Darmon–Prasanna and Castella–Hsieh, and the inclusion of this extra variable arising from the p -adic iteration of differential operators will play a key role in relating values of this p -adic L -function to p -adic regulators of special cycles on unitary Shimura varieties.

Abstract We study the homological stability of spin mapping class groups of surfaces and of quadratic symplectic groups using cellular $E_2$ -algebras. We get improvements in their stability results, which for the spin mapping class groups we show to be optimal away from the prime $2$ . We also prove that in both cases the $\mathbb {F}_2$ -homology satisfies secondary homological stability. Finally, we give full descriptions of the first homology groups of the spin mapping class groups and of the quadratic symplectic groups.

Abstract For Lebesgue generic $({x}_1,x_2)\in \mathbb {R}^2$ , we investigate the distribution of small values of products $q\cdot \|qx_1\| \cdot \|qx_2\|$ with $q\in \mathbb {N}$ , where $\|\cdot \|$ denotes the distance to the closest integer. The main result gives an asymptotic formula for the number of $1\le q< T$ such that $$ \begin{align*}a <q\cdot \|qx_1\| \cdot \|qx_2\|\le b\quad \text{and} \quad \|qx_1\|, \|qx_2\|\le c \end{align*} $$ for given parameters $a,b, c$ satisfying certain growth conditions.

Abstract In this paper, we give a complete list of strongly tempered anomaly-free hyperspherical Hamiltonian spaces-those that are dual to symplectic vector spaces under the relative Langlands duality. We show that the period integrals attached to the list contain many previously studied Rankin-Selberg integrals and period integrals, thus giving a new conceptual understanding of these integrals. The list also proposes many new interesting period integrals to study.

Abstract Given a simply connected manifold M , we completely determine which rational monomial Pontryagin numbers are attained by fiber homotopy trivial M -bundles over the k -sphere, provided that k is small compared to the dimension and the connectivity of M . Furthermore, we study the vector space of rational cobordism classes represented by such bundles. We give upper and lower bounds on its dimension, and we construct manifolds for which the lower bound is attained. Our proofs are based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory, and we make use of ideas developed by Krannich–Kupers–Randal-Williams. As an application, we show the existence of elements of infinite order in the homotopy groups of the spaces of positive Ricci and positive sectional curvature, provided that M is $\operatorname {Spin}$ , has a nontrivial rational Pontryagin class and admits such a metric. This is done by constructing M -bundles over spheres with nonvanishing ${\hat {\mathcal {A}}}$ -genus. Furthermore, we give a vanishing theorem for generalized Morita–Miller–Mumford classes for fiber homotopy trivial bundles over spheres. In the appendix coauthored by Jens Reinhold, we investigate which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.

Abstract We introduce a framework to prove integral rigidity results for the Seiberg–Witten invariants of a closed $4$ -manifold X containing a nonseparating hypersurface Y satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if X has the homology of a four-torus, and it contains a nonseparating three-torus, then the sum of all Seiberg–Witten invariants of X is determined in purely cohomological terms. Our results can be interpreted as $(3+1)$ -dimensional versions of Donaldson’s TQFT approach to the formula of Meng–Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg–Witten equations on X and reducible ones on Y and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline {\mathit {HM}}_*$ induced by a negative-definite cobordism between three-manifolds, which might be of independent interest.