International Journal of Mathematics

A classical theorem of Titchmarsh relates the $L^2$-Lipschitz functions and decay of the Fourier transform of the functions. In this note, we prove the Titchmarsh theorem for Damek–Ricci space (also known as harmonic $NA$ groups) via moduli of continuity of higher orders. We also prove an analogue of another Titchmarsh theorem which provides integrability properties of the Fourier transform for functions in the Hölder Lipschitz spaces.

This paper studies planar algebras of Jones’ style associated with the Young graph. We first see that, given a positive real valued function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the function is harmonic and related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagrammatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagrammatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups.

For any positive integer numbers $n>r$ and $m$, we construct a complex foliated variety $(X,{ℱ})$ of general type with dim $X=n$ and rank ${ℱ}=r$ such that the pluricanonical system $\left|m{K}_{{ℱ}}\right|$ does not give a birational map. This answers a question of Cascini.

Let $X$ be a smooth complex projective rationally connected threefold with $-K_X$ being nef and not semi-ample. In our previous work, we classified all such threefolds when $|-K_X|$ has no fixed divisor. In this paper, we continue our classification when $|-K_X|$ has a non-zero fixed divisor. We show that the mobile part of $|-K_X|$ can be non-rational, and in this case a general member of the mobile part of $|-K_X|$ has exactly two irreducible components. We give a complete classification in this case and we also describe the cones of curves of the threefolds appearing in the classification.

We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case by using representations of factorizable ribbon Hopf algebras as a case study. We focus on the Cohen–Westreich modular data, which is obtained from the Lyubashenko–Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen–Westreich $S$-matrix diagonalizes the mixed fusion rules and reduces to the usual $S$-matrix for semisimple modular categories. The paper includes detailed studies on small quantum groups $U_{q}sl(2)$ and the Drinfeld doubles of Nichols Hopf algebras, especially the $\mathrm{\text{SL}}(2,\mathbb{Z})$-representation on their centers, Cohen–Westreich modular data, and the congruence kernel theorem’s validity.

An old open question in non-Kähler geometry predicts that any compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler or Chern flat. The conjecture is known to be true in dimension $2$ due to the work by Balas–Gauduchon and Apostolov–Davidov–Muskarov in the 1980s and 1990s, but is still open in dimensions $3$ or higher, except in several special cases. The difficulty in this quest for ‘Hermitian space forms’ is largely due to the algebraic complicity or lack of symmetry for the curvature tensor of a general Hermitian metric. In this paper, we confirm the conjecture for all solvmanifolds with complex commutator, extending earlier result on nilmanifolds by Li and Zheng.