Abstract We study the space of traces associated with arbitrary full free products of unital, separable $C^*$ -algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the same object: the Poulsen simplex, that is, the unique infinite-dimensional metrizable Choquet simplex whose extreme points are dense. Moreover, we show that whenever such a trace space is the Poulsen simplex, the extreme points are dense in the Wasserstein topology. Concretely for the case of groups, we find that, unless the trivial character is isolated in the space of characters, the space of traces of any free product of non-trivial countable groups is the Poulsen simplex. Our main technical contribution is a new perturbation result for pairs of von Neumann subalgebras $(M_{1},M_{2})$ of a tracial von Neumann algebra M , providing necessary conditions under which $M_{1}$ and a small unitary perturbation of $M_{2}$ generate a II $_{1}$ factor.

Abstract This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R . The focus is on lattices, namely, finitely generated G -modules that are projective as R -modules, and on the full subcategory of all G -modules projective over R generated by the lattices. The stable category of such G -modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G . Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.

Abstract We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney–Tate stratifications of Beilinson–Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne’s modification of the dual group and a modified form of Vinberg’s monoid over the integers.

Abstract We establish a version of Seiberg–Witten Floer K -theory for knots, as well as a version of Seiberg–Witten Floer K -theory for 3-manifolds with involution. The main theorems are 10/8-type inequalities for knots and for involutions. The 10/8-inequality for knots yields numerous applications to knots, such as lower bounds on stabilizing numbers and relative genera. We also give obstructions to extending involutions on 3-manifolds to 4-manifolds, and detect non-smoothable involutions on 4-manifolds with boundary.

Abstract We prove cohomological vanishing criteria for the Ceresa cycle of a curve C embedded in its Jacobian J : (A) if $\mathrm{H}^3(J)^{\mathrm{Aut}(C)} = 0$ , then the Ceresa cycle is torsion modulo rational equivalence; (B) if $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$ , then the Ceresa cycle is torsion modulo algebraic equivalence, with criterion (B) conditional on the Hodge conjecture. We then use these criteria to study the simplest family of curves where (B) holds but (A) does not, namely the family of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$ . Criterion (B) and work of Schoen combine to show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. We furthermore determine exactly when it is torsion in the Chow group. As a byproduct, we deduce that there exist one-parameter families of plane quartic curves with torsion Ceresa Chow class; that the torsion locus in $\mathcal{M}_3$ of the Ceresa Chow class contains infinitely many components; and that the order of a torsion Ceresa Chow class of a Picard curve over a number field K is bounded, with the bound depending only on $[K\colon \mathbb{Q}]$ . Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$ .

Abstract We study functions f on $\mathbb{Q}$ which satisfy a ‘quantum modularity’ relation of the shape $ f(x+1)=f(x)$ and $f(x) - {| {x} |}^{-k} f(-1/x) = h(x)$ , where $h:\mathbb{R}_{\neq 0} \to \mathbb{C}$ is a function satisfying various regularity conditions. We study the case $\operatorname{Re}(k)\neq 0$ . We prove the existence of a limiting function, denoted by $f^\triangleleft$ or $f^\triangleright$ , depending on the sign of $\operatorname{Re}(k)$ , which extends continuously f to $\mathbb{R}$ in some sense. This means, in particular, that in the $\operatorname{Re}(k)\neq0$ case, the quantum modular form itself has to have at least a certain level of regularity. We deduce that the values $\{f(a/q), 1\leqslant a<q, (a, q)=1\}$ , appropriately normalized, tend to equidistribute along the graph of $f^\triangleleft$ or $f^\triangleright$ , and we prove that, under natural hypotheses, the limiting measure is diffuse. We apply these results to obtain limiting distributions of values and continuity results for several arithmetic functions known to satisfy the above quantum modularity: higher weight modular symbols associated to holomorphic cusp forms; Eichler integral associated to Maaß forms; a function of Kontsevich and Zagier related to the Dedekind $\eta$ -function; and generalized cotangent sums.

Abstract A classical argument was introduced by Khintchine in 1926 in order to exhibit the existence of totally irrational singular linear forms in two variables. This argument was subsequently revisited and extended by many authors. For instance, in 1959 Jarník used it to show that for $n \geqslant 2$ and for any non-increasing positive f there are totally irrational matrices $A \in M_{m,n}(\mathbb{R})$ such that for all large enough t there are $\mathbf{p} \in \mathbb{Z}^m, \mathbf{q} \in \mathbb{Z}^n \smallsetminus \{0\}$ with $\|\mathbf{q}\| \leqslant t$ and $\|A \mathbf{q} - \mathbf{p}\| \leqslant f(t)$ . We denote the collection of such matrices by $\operatorname{UA}^*_{m,n}(f)$ . We adapt Khintchine’s argument to show that the sets $\operatorname{UA}^*_{m,n}(f)$ , and their weighted analogues $\operatorname{UA}^*_{m,n}(f, {\boldsymbol{\omega}})$ , intersect many manifolds and fractals, and have strong intersection properties. For example, we show that: (i) when $n \geqslant 2$ , the set $\bigcap_{{\boldsymbol{\omega}}} \operatorname{UA}^*(f, {\boldsymbol{\omega}}) $ , where the intersection is over all weights ${\boldsymbol{\omega}}$ , is non-empty, and moreover intersects many manifolds and fractals; (ii) for $n \geqslant 2$ , there are vectors in $\mathbb{R}^n$ which are simultaneously k -singular for every k , in the sense of Yu; and (iii) when $n \geqslant 3$ , $\operatorname{UA}^*_{1,n}(f) + \operatorname{UA}^*_{1,n}(f) =\mathbb{R}^n$ . We also obtain new bounds on the rate of singularity which can be attained by column vectors in analytic submanifolds of dimension at least 2 in $\mathbb{R}^n$ .

Abstract We give an interpretation of the semi-infinite intersection cohomology sheaf associated with a semisimple simply connected algebraic group in terms of finite-dimensional geometry. Specifically, we describe a procedure for building factorization spaces over moduli spaces of finite subsets of a curve from factorization spaces over moduli spaces of divisors, and show that, under this procedure, the compactified Zastava space is sent to the support of the semi-infinite intersection cohomology (IC) sheaf in the factorizable Grassmannian. We define ‘semi-infinite t-structures’ for a large class of schemes with an action of the multiplicative group, and show that, for the Zastava, the limit of these t-structures recovers the infinite-dimensional version. As an application, we also construct factorizable parabolic semi-infinite IC sheaves and a generalization (of the principal case) to Kac–Moody algebras.