Abstract By applying the theory of the minimal model program for adjoint foliated structures, we establish the Sarkisov program for algebraically integrable foliations on klt varieties: any two Mori fiber spaces of such structure are connected by a sequence of Sarkisov links. Combining with a result of R. Mascharak, we establish the Sarkisov program for foliations in dimension at most $3$ with mild singularities. Log version and adjoint foliated version of the aforementioned Sarkisov programs are also established.

Abstract We investigate the action of Alesker’s Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain–Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre-type differential equation, and thus, is a strictly positive function that is smooth up to one point.

Abstract We prove that if $A_{1}, A_{2}, \dots , A_{n}$ are diffuse abelian tracial von Neumann algebras for $2\leq n \leq \infty $ and $M = A_{1} * \cdots * A_{n}$ is their free product, then any subalgebra $A \subset M$ of the form $A = \sum _{i=1}^{n} u_{i} A_{i} p_{i} u_{i}^{*}$, for some projections $p_{i} \in A_{i}$ and unitaries $u_{i} \in \mathcal{U}(M)$, for $1 \leq i \leq n$, such that $\sum _{i} u_{i} p_{i} u_{i}^{*} = 1$, is freely complemented (FC) in $M$. We also generalize this result in situations where the $A_{i}$ are not assumed abelian but the projections $p_{i}$ are instead assumed central, and subsequently without assuming the $p_{i}$ are central, but with the weaker result of the existence of free Haar unitaries. Moreover, we prove that if $A_{1}, A_{2}, \dots , A_{n}$ are purely non-separable abelian, and $M = A_{1} * \cdots * A_{n}$, then any purely non-separable singular MASA in $M$ is FC. We also show that any of the known maximal amenable MASAs $A\subset L\mathbb{F}_{n}$ (notably the radial MASA) satisfies Popa’s weak FC conjecture, that is, there exist Haar unitaries $u\in L\mathbb{F}_{n}$ that are freely independent from $A$.

Abstract We prove that two natural Markov chains on the set of monotone paths in a strip mix slowly. To do so, we make novel use of the theory of non-positively curved (CAT(0)) cubical complexes to detect small bottlenecks in many graphs of combinatorial interest. Along the way, we give a formula for the number $c_{m}(n)$ of monotone paths of length $n$ in a strip of height $m$. In particular, we compute the exponential growth constant of $c_{m}(n)$ for arbitrary $m$, generalizing results of Williams for $m=2, 3$.

Abstract We construct a probability measure $\mu $ supported on a set of zero $2d/p$-Hausdorff measure such that $\hat{\mu }\in L_{p}(\mathbb{R}^{d})$.

Abstract We study the intersection of the base loci of line bundles that are isomorphic to the canonical bundle in the Néron–Severi group, and show that it also governs the nonexistence of semi-orthogonal decompositions. As an application, we show the bounded derived category of the $i$-th symmetric product of a smooth projective curve $C$ has no nontrivial semi-orthogonal decompositions when the genus $g(C)\geq 2$ and $i\leq g(C)-1$. We show the indecomposability of derived categories of some examples of elliptic surfaces with $p_{g}(X)=0$, and some examples of minimal surfaces of general type. An inequality involving phases of skyscraper sheaves for any Bridgeland stability condition is obtained.

Abstract We show that the algebra $D_\hbar (SL_{n}/U)$ of differential operators on the base affine space of $SL_{n}$ is the quantized Coulomb branch of a certain 3d $\mathcal{N} = 4$ quiver gauge theory. In the semiclassical limit this proves a conjecture of Dancer–Hanany–Kirwan about the universal hyperkähler implosion of $SL_{n}$. We also formulate and prove a generalization identifying the Hamiltonian reduction $T^{*} SL_{n}\ /\!/ _{\psi } U$ as a Coulomb branch for an arbitrary unipotent character $\psi $. As an application of our results, we provide a new interpretation of the Gelfand–Graev symmetric group action on $D_\hbar (SL_{n}/U)$.

Abstract A causal manifold $(M,\gamma )$ is a manifold $M$ endowed with a closed convex proper cone $\gamma $ in the tangent bundle $TM$ satisfying suitable properties. Let $\lambda ={\gamma }^{\circ a}\subset T^{*}M$ be the antipodal of the polar cone of $\gamma $. An open set $U$ of $M$ is called $\gamma $-open if its Whitney normal cone contains $\textrm{Int}(\gamma )$. Similarly, $U$ is called $\lambda $-open if the micro-support of the constant sheaf on $U$ is contained in $\lambda $. We begin by proving that the two notions coincide. Next, we prove that if $(M,\gamma )$ admits a “future time function” (in particular if it is globally hyperbolic) the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $\lambda $ and the derived category of sheaves on $M_\gamma $, the manifold $M$ endowed with the $\gamma $-topology. This generalizes a result of [5], which treated the case of a constant cone in a vector space.

Abstract We consider the problem of approaching real numbers with rational numbers with prime denominator and with a single numerator allowed for each denominator. We obtain basic results, both probabilistic and deterministic, draw connections to twisted diophantine approximation, and present a simple application, related to possible correlations between trace functions and dynamical sequences.

Abstract A vertex algebra with an action of a group $G$ comes with a notion of $g$-twisted modules, forming a $G$-crossed braided tensor category. For a Lie group $G$, one might instead wish for a notion of $(\textrm{d}+A)$-twisted modules for any $\mathfrak{g}$-connection on the formal punctured disk. For connections with a regular singularity, this reduces to $g$-twisted modules, where $g$ is the monodromy around the puncture. The case of an irregular singularity is much richer and involved, and we are not aware that it has appeared in vertex algebra language. The present article is intended to spark such a treatment, by providing a list of expectations and an explicit worked-through example with interesting applications. Concretely, we consider the vertex super algebra of symplectic fermions through its associated Clifford algebra and study its twisted module with respect to irregular $\mathfrak{s}\mathfrak{l}_{2}$-connections. We first determine the category of representations, depending on the formal type of the connection. Then we prove that a Sugawara-type construction gives a Virasoro action and we prove that as Virasoro modules our representations are direct sums of Whittaker modules. Conformal field theory with irregular singularities resp. wild ramification appear in the context of geometric Langlands correspondence, in particular work by Witten [51], and more generally in higher-dimensional context. Our original motivation in [24] comes from semiclassical limits of the generalized quantum Langlands kernel, which fibres over the space of connections (similar to the affine Lie algebra at critical level) and as such gives a family of deformations of the Feigin–Tipunin algebra. Our present article now describes, in the smallest case, the fibres and their representation categories over irregular connections.