In this paper, we study the cohomology of the unitary unramified PEL Rapoport-Zink space of signature $(1,n-1)$ at hyperspecial level. Our method revolves around the spectral sequence associated to the open cover by the analytical tubes of the closed Bruhat-Tits strata in the special fiber, which were constructed by Vollaard and Wedhorn. The cohomology of these strata, which are isomorphic to generalized Deligne-Lusztig varieties, has been computed in an earlier work. This spectral sequence allows us to prove the semisimplicity of the Frobenius action and the non-admissibility of the cohomology in general. Via p-adic uniformization, we relate the cohomology of the Rapoport-Zink space to the cohomology of the supersingular locus of a Shimura variety with no level at p. In the case $n=3$ or $4$, we give a complete description of the cohomology of the supersingular locus in terms of automorphic representations.

Let $S_g$ denote the genus g closed orientable surface. A coherent filling pair of simple closed curves, $(\alpha,\beta)$ in $S_g$, is a filling pair that has its geometric intersection number equal to the absolute value of its algebraic intersection number. A minimally intersecting filling pair, $(\alpha,\beta)$ in $S_g$, is one whose intersection number is the minimal among all filling pairs of $S_g$. In this paper, we give a simple geometric procedure for constructing minimally intersecting coherent filling pairs on $S_g, \ g \geq 3,$ from the starting point of a coherent filling pair of curves on a torus. Coherent filling pairs have a natural correspondence to square-tiled surfaces, or origamis, and we discuss the origami obtained from the construction.

We introduce the $\ell ^1$-ideal intersection property for crossed product ${\mathrm {C}}^*$-algebras. It is implied by ${\mathrm {C}}^*$-simplicity as well as ${\mathrm {C}}^*$-uniqueness. We show that topological dynamical systems of arbitrary lattices in connected Lie groups, arbitrary linear groups over the integers in a number field and arbitrary virtually polycyclic groups have the $\ell ^1$-ideal intersection property. On the way, we extend previous results on ${\mathrm {C}}^*$-uniqueness of -groupoid algebras to the general twisted setting.

In this paper, we study the existence of $k$-$11$-representations of graphs. Inspired by work on permutation patterns, these representations are ways of representing graphs by words where adjacencies between vertices are captured by patterns in the corresponding letters. Our main result is that all graphs are $1$-$11$-representable, answering a question originally raised by Cheon et al. in 2018 and repeated in several follow-up papers – including a very recent paper, where it was shown that all graphs on at most $8$ vertices are $1$-$11$-representable. Moreover, we prove that all graphs are permutationally $1$-$11$-representable – that is representable as the concatenation of permutations of the vertices – answering the existence question in extremely strong fashion. Our construction leads to nearly optimal bounds on the length of the words, as well. It can, moreover, be adapted to represent all acyclic orientations of graphs; this generalizes the fact that word-representations capture semi-transitive orientations of graphs. Our construction also adapts easily to other $k \geq 2$ as well, giving representations using a linear number of permutations when the best known previous bounds used a quadratic number. Finally, we also consider the (non-)existence of ‘even–odd’-representations of graphs. This answers a question raised by Wanless after a conference talk in 2018.

The Generalised Baker–Schmidt Problem (1970) concerns the Hausdorff measure of the set of $\psi$-approximable points on a non-degenerate manifold. Beresnevich-Dickinson-Velani (in 2006, for the homogeneous setting) and Badziahin-Beresnevich-Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary non-degenerate manifolds. The divergence part has also been resolved for the $p$-adic setting by Datta-Ghosh in 2022, for the inhomogeneous setting. The corresponding convergence counterpart represents a challenging open question. In this paper, we prove the homogeneous $p$-adic convergence result for hypersurfaces of dimension at least three with some mild regularity condition, as well as for some other classes of manifolds satisfying certain conditions. We provide similar, slightly weaker results for the inhomogeneous setting. We do not restrict to monotonic approximation functions.

In this paper, we are concerned with the following Dirichlet problems for nonlinear equations involving the fractional $p$-Laplacian:

\begin{equation*}\begin{cases}(-\Delta)_p^\alpha u=f(x,u,\nabla u),\ \ u \gt 0,\ \ \text{in}\ \ E,\\\ \ \ \ \ \ u\equiv0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ \ \mathbb{R}^{n}\setminus E,\end{cases}\end{equation*}

where $E \subseteq \mathbb{R}^{n}$ is a coercive epigraph, i.e., there exists a continuous function $\phi: \, \mathbb{R}^{n-1} \rightarrow \mathbb{R}$ satisfying

\begin{equation*}\lim_{|x'|\rightarrow+\infty}\phi(x')=+\infty,\end{equation*}

such that $E:=\{x=(x',x_{n}) \in \mathbb{R}^{n}|\,x_{n} \gt \phi(x')\}$, where $x':= (x_{1},...,x_{n-1}) \in \mathbb{R}^{n-1}$. Under some mild assumptions on the nonlinearity $f(x,u,\nabla u)$, we prove strict monotonicity of positive solutions to the above Dirichlet problems involving fractional $p$-Laplacian in coercive epigraph $E$.

Let $M^{({k})}_{d}(n)$ be the manifold of n-tuples $(x_1,\ldots,x_n)\in(\mathbb{R}^d)^n$ having non-k-equal coordinates. We show that, for $d\geq2$, $M^{({3})}_{d}(n)$ is rationally formal if and only if $n\leq6$. This stands in sharp contrast with the fact that all classical configuration spaces $M^{({2})}_d(n)=\text{Conf}(\mathbb{R}^d,n)$ are rationally formal, just as are all complements of arrangements of arbitrary complex subspaces with geometric lattice of intersections. The rational non-formality of $M^{({3})}_{d}(n)$ for $n \gt 6$ is established via detection of non-trivial triple Massey products, which are assessed geometrically through Poincaré duality.

We extend the notion of the J-invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the J-invariant.

In this paper, we prove the integrality conjecture for quotient stacks arising from weakly symmetric representations of reductive groups. Our main result is a decomposition of the cohomology of the stack into finite-dimensional components indexed by some equivalence classes of cocharacters of a maximal torus. This decomposition enables the definition of new enumerative invariants associated with the stack, which we begin to explore.

In this work, we conclude our study of fibred $\infty $-bicategories by providing a Grothendieck construction in this setting. Given a scaled simplicial set S (which need not be fibrant) we construct a 2-categorical version of Lurie’s straightening-unstraightening adjunction, thereby furnishing an equivalence between the $\infty $-bicategory of 2-Cartesian fibrations over S and the $\infty $-bicategory of contravariant functors with values in the $\infty $-bicategory of $\infty $-bicategories. We provide a relative nerve construction in the case where the base is a 2-category, and use this to prove a comparison to existing bicategorical Grothendieck constructions.

We show that Rabinowitz Floer homology and cohomology carry the structure of a graded Frobenius algebra for both closed and open strings. We prove a Poincaré duality theorem between homology and cohomology that preserves this structure. This lifts to a duality theorem between graded open–closed topological quantum field theories (TQFTs). We use in a systematic way the formalism of Tate vector spaces. Specializing to the case of cotangent bundles, we define Rabinowitz loop homology and cohomology and explain from a unified perspective pairs of dual results that have been observed over the years in the context of the search for closed geodesics. These concern critical levels, relations to the based loop space, manifolds all of whose geodesics are closed, Bott index iteration, and level potency. Moreover, the graded Frobenius algebra structure gives meaning and proof to a relation conjectured by Sullivan between the loop product and coproduct.

Consider a quadratic polynomial $Q(\xi_{1},\ldots,\xi_{n})$ of independent Rademacher random variables $\xi_{1},\ldots,\xi_{n}$. To what extent can $Q(\xi_{1},\ldots,\xi_{n})$ concentrate on a single value? This quadratic version of the classical Littlewood–Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if $Q(\xi_{1},\ldots,\xi_{n})$ ‘robustly depends on at least m of the $\xi_{i}$’ in the sense that there is no way to pin down the value of $Q(\xi_{1},\ldots,\xi_{n})$ by fixing values for fewer than m of the variables $\xi_{i}$, then we have $\mathrm{Pr}[Q(\xi_{1},\ldots,\xi_{n})=0]\le O(1/\sqrt{m})$. This also implies a similar result in the case where $\xi_{1},\ldots,\xi_{n}$ have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems to high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility.

For log canonical (lc) algebraically integrable foliations on Kawamata log terminal (klt) varieties, we prove the base-point-freeness theorem, the contraction theorem, and the existence of flips. The first result resolves a conjecture of Cascini and Spicer, while the latter two results strengthen a result of Cascini and Spicer by removing their assumption on the termination of flips. Moreover, we prove the existence of the minimal model program for lc algebraically integrable foliations on klt varieties and the existence of good minimal models or Mori fiber spaces for lc algebraically integrable foliations polarized by ample divisors on klt varieties. As a consequence, we show that $\mathbb{Q}$-factorial klt varieties with lc algebraically integrable Fano foliation structures are Mori dream spaces. We also show the existence of a Shokurov-type polytope for lc algebraically integrable foliations.

Let A be an abelian variety defined over a global function field F and let p be a prime distinct from the characteristic of F. Let $F_\infty $ be a p-adic Lie extension of F that contains the cyclotomic $\mathbb {Z}_p$-extension $F^{\mathrm {cyc}}$ of F. In this paper, we investigate the structure of the p-primary Selmer group $\mathrm {Sel}(A/F_\infty )$ of A over $F_\infty $. We prove the $\mathfrak {M}_H(G)$-conjecture for $A/F_\infty $. Furthermore, we show that both the $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F^{\mathrm {cyc}})$ and the generalized $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F_\infty )$ are zero, thereby proving Mazur’s conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of A over the base field F. Assuming the finiteness of the Tate–Shafarevich group, we establish that this corank equals the order of vanishing of the L-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi—originally proved for elliptic curves without complex multiplication—to abelian varieties over global function fields. This is achieved by adapting the notion of generalized Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalized Euler characteristic of $\mathrm {Sel}(A/F_\infty )$ to the Euler characteristic of $\mathrm {Sel}(A/F^{\mathrm {cyc}})$.

Let G and H be finite-dimensional vector spaces over $\mathbb{F}_p$. A subset $A \subseteq G \times H$ is said to be transverse if all of its rows $\{x \in G \colon (x,y) \in A\}$, $y \in H$, are subspaces of G and all of its columns $\{y \in H \colon (x,y) \in A\}$, $x \in G$, are subspaces of H. As a corollary of a bilinear version of the Bogolyubov argument, Gowers and the author proved that dense transverse sets contain bilinear varieties of bounded codimension. In this paper, we provide a direct combinatorial proof of this fact. In particular, we improve the bounds and evade the use of Fourier analysis and Freiman’s theorem and its variants.