Communications in Contemporary Mathematics

A $W^{1,p}$-metric on an n-dimensional closed Riemannian manifold naturally induces a distance function, provided p is sufficiently close to n. If a sequence of metrics $g_k$ converges in $W^{1,p}$ to a limit metric $g$, then the corresponding distance functions $d_{g_k}$ subconverge to a limit distance function d, which satisfies $d\le d_g$.

As an application, we show that the above convergence result applies to a sequence of conformal metrics with $L^{n/2}$-bounded scalar curvatures, under certain geometric assumptions. In particular, in this special setting, the limit distance function d actually coincides with $d_g$.

In this paper, we study the following mean field type equation:

In this paper, we establish a global $C^2$ estimates for a Hessian type equation with homogeneous Dirichlet boundary. By the method of sub and sup solution, we get an existence and uniqueness result for the eigenvalue problem of a Hessian type operator.

In [M. De Renzi, A. Gainutdinov, N. Geer, B. Patureau-Mirand and I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, preprint (2019), arXiv:1912.02063[math.GT]], we constructed 3-dimensional topological quantum field theories (TQFTs) using not necessarily semisimple modular categories. Here, we study projective representations of mapping class groups of surfaces defined by these TQFTs, and we express the action of a set of generators through the algebraic data of the underlying modular category $\mathcal{𝒞}$. This allows us to prove that the projective representations induced from the non-semisimple TQFTs of the above reference are equivalent to those obtained by Lyubashenko via generators and relations in [V. Lyubashenko, Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity, Comm. Math. Phys. 172(3) (1995) 467–516, arXiv:hep-th/9405167]. Finally, we show that, when $\mathcal{𝒞}$ is the category of finite-dimensional representations of the small quantum group of ${𝔰𝔩}_2$, the action of all Dehn twists for surfaces without marked points has infinite order.

We define and study multiplicative connections in the tangent bundle of a Lie groupoid. Multiplicative connections are linear connections satisfying an appropriate compatibility with the groupoid structure. Our definition is natural in the sense that a linear connection on a Lie groupoid is multiplicative if and only if its torsion is a multiplicative tensor in the sense of Bursztyn–Drummond [Lie theory of multiplicative tensors, Mat. Ann. 375 (2019) 1489–1554, arXiv:1705.08579] and its geodesic spray is a multiplicative vector field. We identify the obstruction to the existence of a multiplicative connection. We also discuss the infinitesimal version of multiplicative connections in the tangent bundle, that we call infinitesimally multiplicative (IM) connections and we prove an integration theorem for IM connections. Finally, we present a few toy examples.

We study the derived critical locus of a function $f:[X/G]\to {𝔸}_{𝕂}^1$ on the quotient stack of a smooth affine scheme $X$ by the action of a smooth affine group scheme G. It is shown that $\mathrm{\text{dCrit}}(f)\simeq [Z/G]$ is a derived quotient stack for a derived affine scheme Z, whose dg-algebra of functions is described explicitly. Our results generalize the classical BV formalism in finite dimensions from Lie algebra to group actions.

We develop a theory which unifies the universal (co)acting bi/Hopf algebras as studied by Sweedler, Manin and Tambara with the recently introduced [A. L. Agore, A. S. Gordienko and J. Vercruysse, On equivalences of (co)module algebra structures over Hopf algebras, J. Noncommut. Geom., doi: 10.4171/JNCG/428.] bi/Hopf-algebras that are universal among all support equivalent (co)acting bi/Hopf algebras. Our approach uses vector spaces endowed with a family of linear maps between tensor powers of A, called $\mathrm{\Omega }$-algebras. This allows us to treat algebras, coalgebras, braided vector spaces and many other structures in a unified way. We study V-universal measuring coalgebras and V-universal comeasuring algebras between $\mathrm{\Omega }$-algebras A and B, relative to a fixed subspace V of $Vect(A,B)$. By considering the case $A=B$, we derive the notion of a V-universal (co)acting bialgebra (and Hopf algebra) for a given algebra A. In particular, this leads to a refinement of the existence conditions for the Manin–Tambara universal coacting bi/Hopf algebras. We establish an isomorphism between the V-universal acting bi/Hopf algebra and the finite dual of the V-universal coacting bi/Hopf algebra under certain conditions on V in terms of the finite topology on $En{d}_F(A)$.

Let G be a semisimple algebraic group over $ℂ$. For a reduced word $i$ of the longest element in the Weyl group of G and a dominant integral weight $\lambda$, one can construct the string polytope ${\mathrm{\Delta }}_i(\lambda )$, whose lattice points encode the character of the irreducible representation $V_{\lambda }$. The string polytope ${\mathrm{\Delta }}_i(\lambda )$ is singular in general and combinatorics of string polytopes heavily depends on the choice of $i$. In this paper, we study combinatorics of string polytopes when $G=\text{S}{\text{L}}_{n+1}(ℂ)$, and present a sufficient condition on $i$ such that the toric variety $X_{{\mathrm{\Delta }}_i(\lambda )}$ of the string polytope ${\mathrm{\Delta }}_i(\lambda )$ has a small toric resolution. Indeed, when $i$  has small indices and $\lambda$ is regular, we explicitly construct a small toric resolution of the toric variety $X_{{\mathrm{\Delta }}_i(\lambda )}$ using a Bott manifold. Our main theorem implies that a toric variety of any string polytope admits a small toric resolution when $n<4$. As a byproduct, we show that if $i$ has small indices then ${\mathrm{\Delta }}_i(\lambda )$ is integral for any dominant integral weight $\lambda$, which in particular implies that the anticanonical limit toric variety $X_{{\mathrm{\Delta }}_i({\lambda }_P)}$ of a partial flag variety $G/P$ is Gorenstein Fano. Furthermore, we apply our result to symplectic topology of the full flag manifold $G/B$ and obtain a formula of the disk potential of the Lagrangian torus fibration on $G/B$ obtained from a flat toric degeneration of $G/B$ to the toric variety $X_{{\mathrm{\Delta }}_i(\lambda )}$.