We compute the Hochschild and Gerstenhaber–Schack cohomological dimensions of the universal cosovereign Hopf algebras, when the matrix of parameters is a generic asymmetry. Our main tools are considerations on the cohomologies of free product of Hopf algebras, and on the invariance of the cohomological dimensions under graded twisting by a finite abelian group.

It is shown how the graph category of Borisov and Manin can be constructed from (a variant of) the graph category of Joyal and Kock, essentially by reversing the generic morphisms. More precisely, the morphisms in the Borisov–Manin category are exhibited as cospans of reduced covers and refinement morphisms.

We study Heegner points on elliptic curves, or more generally modular abelian varieties, coming from uniformization by Shimura curves attached to a rather general type of quaternionic orders. We address several questions arising from the Birch and Swinnerton-Dyer (BSD) conjecture in this general context. In particular, under mild technical conditions, we show the existence of non-torsion Heegner points on elliptic curves in all situations in which the BSD conjecture predicts their existence.

We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $\mathcal{C}^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the Dirac system as well as to define traces in a weak sense for functions in the appropriate Sobolev space. Finally, we introduce Calderón projectors associated with the problem and illustrate the method in two special cases: the well-known MIT bag model and an electrostatic $\delta$-shell interaction.

Consider the linear parabolic operator in divergence form $$ \mathcal{H} u :=\partial_t u(X,t)-\operatorname{div}(A(X)\nabla u(X,t)). $$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is well-posed for boundary data in $L^p$, for any elliptic matrix of coefficients $A$ which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation $\partial_t u_\varepsilon(X,t)-\operatorname{div}(A(X/\varepsilon)\nabla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.

Given a Muckenhoupt weight $w$ and a second order divergence form elliptic operator $L$, we consider different versions of the weighted Hardy space $H^1_L(w)$ defined by conical square functions and non-tangential maximal functions associated with the heat and Poisson semigroups generated by $L$. We show that all of them are isomorphic and also that $H^1_L(w)$ admits a molecular characterization. One of the advantages of our methods is that our assumptions extend naturally the unweighted theory developed by S. Hofmann and S. Mayboroda in [19] and we can immediately recover the unweighted case. Some of our tools consist in establishing weighted norm inequalities for the non-tangential maximal functions, as well as comparing them with some conical square functions in weighted Lebesgue spaces.

In this paper, we establish existence, uniqueness, and scale-invariant estimates for fundamental solutions of non-homogeneous second order elliptic systems with bounded measurable coefficients in $\mathbb{R}^n$ and for the corresponding Green functions in arbitrary open sets. We impose certain non-homogeneous versions of de Giorgi–Nash–Moser bounds on the weak solutions and investigate in detail the assumptions on the lower order terms sufficient to guarantee such conditions. Our results, in particular, establish the existence and fundamental estimates for the Green functions associated to the Schrödinger ($-\Delta+V$) and generalized Schrödinger ($-\operatorname{div} A\nabla +V$) operators with real and complex coefficients, on arbitrary domains.

We study regularized determinants of Laplacians acting on the space of Hilbert–Maass forms for the Hilbert modular group of a real quadratic field. We show that these determinants are described by Selberg type zeta functions introduced in [5, 6].

We prove that a finite non-degenerate involutive set-theoretic solution $(X,r)$ of the Yang–Baxter equation is a multipermutation solution if and only if its structure group $G(X,r)$ admits a left ordering or equivalently it is poly-$\mathbb{Z}$.

In this paper, we present a generalisation of a theorem of David and Rob Pollack. In [PP], they give a very general argument for lifting ordinary eigenclasses (with respect to a suitable operator) in the group cohomology of certain arithmetic groups. With slightly tighter conditions, we prove the same result for non-ordinary classes. Pollack and Pollack apply their results to the case of $p$-ordinary classes in the group cohomology of congruence subgroups for $\mathrm{SL}_3$, constructing explicit overconvergent classes in this setting. As an application of our results, we give an extension of their results to the case of non-critical slope classes in the same setting.