In this paper, we introduce a notion of twisted restricted conformal blocks on totally ramified orbicurves and establish an isomorphism between the space of twisted restricted conformal blocks and the space of twisted conformal blocks. The relationships among twisted (restricted) conformal blocks, $g$-twisted (restricted) correlation functions, and twisted intertwining operators are explored. Furthermore, by introducing a geometric generalization of Zhu’s algebra and its modules, we obtain a description of the space of coinvariants by modules over associative algebras and show it is finite-dimensional under some conditions. In particular, a more conceptual proof of the $g$-twisted fusion rules theorem in vertex operator algebra theory is provided.

In this paper, we introduce a notion of $g$-twisted restricted conformal block on the three-pointed twisted projective line $\mathfrak{x}\colon\overline{C}\to\mathbb{P}^1$ associated with an untwisted module $M^1$ and the bottom levels of two $g$-twisted modules $M^2$ and $M^3$ over a vertex operator algebra $V$. We show that the space of twisted restricted conformal blocks is isomorphic to the space of $g$-twisted (restricted) correlation functions defined by the same datum and to the space of intertwining operators among these twisted modules. As an application, we derive a twisted version of the Fusion Rules Theorem.

We study trace functions associated to intertwining operators as a basis of genus-one conformal blocks, relating their modular transformations to the geometry of conformal-block bundles.

We study the representation-theoretic meaning of the strong identity condition, characterizing it through orthogonal expansions, projectivity of canonical modules, and Morita-type equivalences. Examples include the Weyl algebra and irrational vertex operator algebras.

In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as finite Adams, shedding light on its Picard group. Then, we establish a reduction process through base changes, enabling us to effectively compute the Picard group of the $\mathbb{R}$-motivic mod $2$ Steenrod subalgebra $\mathcal{A}^{\mathbb{R}}(1)$. Our computation shows that $\operatorname{Pic}(\mathcal{A}^{\mathbb{R}}(1))$ is isomorphic to $\mathbb{Z}^4$, where two ranks come from the motivic grading, one from the algebraic loop functor, and the last is generated by the $\mathbb{R}$-motivic joker $J$.

This study delves into simplicial distances on Bruhat-Tits buildings of split classical types (namely, types of $A_n$, $B_n$, $C_n$, $D_n$, and any combination thereof). Simplicial distance serves for measuring the proximity between vertices within the simplicial structure of a building. The purpose of this research is three-fold: (i) to present a concrete characterization of the simplicial distance; (ii) to better understand simplicial balls; and (iii) to derive a formula for the simplicial volume and explore its asymptotic growth. To accomplish these goals, vertices in Bruhat-Tits buildings are carefully analyzed under three frameworks: root systems, norms, and lattices. By leveraging concave functions, we interpret simplicial balls as fixed-point sets of Moy-Prasad subgroups and deduce a formula for the simplicial volume. Additionally, the theory of $q$-exponential polynomials is developed to facilitate the asymptotic study.