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As a quantum affinization, the quantum toroidal algebra ({U_{q,{{overline{q}}}}(ddot{{mathfrak {gl}}}_n)}) is defined in terms of its “left” and “right” halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the “top” and “bottom” halves instead, starting from the evaluation representation ({U_q({dot{{mathfrak {gl}}}}_n)}curvearrowright {{mathbb {C}}}^n(z)) and its usual R-matrix (R(z) in text {End}({{mathbb {C}}}^n otimes {{mathbb {C}}}^n)(z)) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on ({U_{q,{{overline{q}}}}(ddot{{mathfrak {gl}}}_n)}) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra ({U_q({dot{{mathfrak {gl}}}}_n)}subset {U_{q,{{overline{q}}}}(ddot{{mathfrak {gl}}}_n)}).

Let G be a reductive group, and let X be an algebraic curve over an algebraically closed field k with positive characteristic. We prove a version of nonabelian Hodge correspondence for tame G-local systems over X and logarithmic G-Higgs bundles over the Frobenius twist (X'). To obtain a full description of the correspondence for the noncompact case, we introduce the language of parahoric group schemes to establish the correspondence.

We study infinite groups interpretable in three families of valued fields: V-minimal, power bounded T-convex, and p-adically closed fields. We show that every such group G has unbounded exponent and that if G is dp-minimal then it is abelian-by-finite. Along the way, we associate with any infinite interpretable group an infinite type-definable subgroup which is definably isomorphic to a group in one of four distinguished sorts: the underlying valued field K, its residue field ({textbf {k}}) (when infinite), its value group (Gamma ), or (K/mathcal {O}), where (mathcal {O}) is the valuation ring. Our work uses and extends techniques developed in Halevi et al. (Adv Math 404:108408, 2022) to circumvent elimination of imaginaries.

We prove that a large class of nonamenable almost periodic type III(_1) factors M, including all McDuff factors that tensorially absorb (R_infty ) and all free Araki–Woods factors, satisfy Haagerup–Størmer’s conjecture (1988): any pointwise inner automorphism of M is the composition of an inner and a modular automorphism.

We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the (4times 4) problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size (nge 4), which appear new for (nge 5). By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation.

The cone of nonnegative flows for a directed acyclic graph (DAG) is known to admit regular unimodular triangulations induced by framings of the DAG. These triangulations restrict to triangulations of the flow polytope for strength one flows, which are called DKK triangulations. For a special class of framings called ample framings, these triangulations of the flow cone project to a complete fan. We characterize the DAGs that admit ample framings, and we enumerate the number of ample framings for a fixed DAG. We establish a connection between maximal simplices in DKK triangulations and (tau )-tilting posets for certain gentle algebras, which allows us to impose a poset structure on the dual graph of any DKK triangulation for an amply framed DAG. Using this connection, we are able to prove that for full DAGs, i.e., those DAGs with inner vertices having in-degree and out-degree equal to two, the flow polytopes are Gorenstein and have unimodal Ehrhart (h^*)-polynomials.

We prove that if (Msubset {mathbb {R}}^n) is a bounded subanalytic submanifold of ({mathbb {R}}^n) such that ({textbf{B}}(x_0,varepsilon )cap M) is connected for every (x_0in {{overline{M}}}) and (varepsilon >0) small, then, for (pin [1,infty )) sufficiently large, the space ({mathscr {C}}^infty ( {{overline{M}}})) is dense in the Sobolev space (W^{1,p}(M)). We also show that for p large, if (Asubset {{overline{M}}}setminus M) is subanalytic then the restriction mapping ( {mathscr {C}}^infty ( {{overline{M}}})ni umapsto u_{|A}in L^p(A)) is continuous (if A is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of ( {{overline{M}}}) is dropped.

Let X be an algebraic variety equipped with a dominant rational map (phi :Xdashrightarrow X). A new quantity measuring the interaction of ((X,phi )) with trivial dynamical systems is introduced; the stabilised algebraic dimension of ((X,phi )) captures the maximum number of new algebraically independent invariant rational functions on ((Xtimes Y,phi times psi )), as (psi :Ydashrightarrow Y) ranges over all dominant rational maps on algebraic varieties. It is shown that this birational invariant agrees with the maximum (dim X') where ((X,phi )dashrightarrow (X',phi ')) is a dominant rational equivariant map and (phi ') is part of an algebraic group action on (X'). As a consequence, it is deduced that if some cartesian power of ((X,phi )) admits a nonconstant invariant rational function, then already the second cartesian power does.

Let (Gamma ) be a tropical variety in ({mathbb {R}}^m={mathbb {R}}^{m_1}times cdots times {mathbb {R}}^{m_k}). We show that, under a certain condition, the positivity of the stable intersection of (Gamma ) with certain tropical varieties pulled back from each ({mathbb {R}}^{m_i}) is governed by the dimensions of the images of (Gamma ) under all possible projections from ({mathbb {R}}^m). As an application, we give a tropical proof of the criterion of the positivity of the multidegrees of a closed subscheme of a multi-projective space, carried out in the paper Castillo et al. (Adv Math 374:107382, 2020).