In this note we show that the Riemann moduli spaces $M_{\gamma,n}$ equipped with the Weil–Petersson metric are quantum ergodic for $3 \gamma + n \geq 4$. We also provide other examples of singular spaces with ergodic geodesic flow for which quantum ergodicity holds.

We introduce a natural definition of the renormalized volume of a $4$-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of $4$-dimensional Ricci-flat ALE spaces are Kronheimer’s gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer’s period map.

For an immersed minimal surface in $\mathbb{R}^3$, we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index.

Our new estimate resolves several conjectures made by J. Choe and D. Hoffman concerning the classification of low-index minimal surfaces: we show that there is no complete two-sided immersed minimal surface in $\mathbb{R}^3$ of index two, complete embedded minimal surface with index three, or complete one-sided minimal immersion with index one.

We contribute to study the two dimensional chiral conformal field theories that arise from chiral deformations of free theories such as the $\beta \gamma$‑system, bc‑system and chiral bosons. Our method is based on the effective Batalin–Vilkovisky quantization theory. We establish an exact correspondence between renormalized quantum master equations for chiral deformations and Maurer–Cartan equations for the Lie algebra of modes of chiral vertex algebras. The generating functions on the torus are proven to have modular property with mild holomorphic anomaly. As an application, we construct an exact solution of quantum B‑model (BCOV theory) in complex one dimension that solves the full higher genus mirror symmetry conjecture on elliptic curves.

Given an involution on a rational homology $3$-sphere $Y$ with quotient the $3$-sphere, we prove a formula for the Lefschetz number of the map induced by this involution in the reduced monopole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of $4$-manifolds. A key ingredient is a skein-theoretic argument, making use of an exact triangle in monopole Floer homology, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants associated to spin structures on $Y$. We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and $4$-dimensional topology.