In Gromov–Witten theory the virtual localization method is used only when the invariant curves are isolated under a torus action. In this paper, we explore a strategy to apply the localization formula to compute the Gromov–Witten invariants by carefully choosing the related cycles to circumvent the continuous families of invariant curves when there are any. For the example of the two-pointed Hilbert scheme of Hirzebruch surface $F_1$, we manage to compute some Gromov–Witten invariants, and then by combining with the associativity law of (small) quantum cohomology ring, we succeed in computing all 1- and 2-pointed Gromov–Witten invariants of genus 0 of the Hilbert scheme with the help of [13].

We establish an explicit formula for the number $C_n\left(q\right)$ of ideals of codimension (colength) $n$ of the algebra ${\mathbb{F}}_q[x,y,x^{-1},y^{-1}]$ of Laurent polynomials in two variables over a finite field ${\mathbb{F}}_q$ of cardinality $q$. This number is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_n\left(q\right)=(q-1{)}^2{P}_n(q)$, where $P_n\left(q\right)$ is another palindromic polynomial; the latter is a $q$-analogue of the sum of divisors of $n$, which happens to be the number of subgroups of ${\mathbb{Z}}^2$ of index $n$.

Given a web (multifoliation) and a linear system on a projective surface, we construct divisors cutting out the locus where some element of the linear system has abnormal contact with the leaf of the web. We apply these ideas to reobtain a classical result by Salmon on the number of lines on a projective surface. In a different vein, we investigate the numbers of lines and disjoint lines contained in a projective surface and tangent to a contact distribution.

We prove a new dyadic representation theorem with applications to the $T\left(1\right)$ and $A_2$ theorems. In particular, we obtain the nonhomogeneous $T\left(1\right)$ theorem under weaker kernel regularity than in the earlier approaches.

We study wall crossings in Bridgeland stability for the Hilbert scheme of elliptic quartic curves in the three-dimensional projective space. We provide a geometric description of each of the moduli spaces we encounter, including when the second component of this Hilbert scheme appears. Along the way, we prove that the principal component of this Hilbert scheme is a double blowup with smooth centers of a Grassmannian, exhibiting a completely different proof of this known result by Avritzer and Vainsencher. This description allows us to compute the cone of effective divisors of this component.

We exhibit Borel probability measures on the unit sphere in ${\mathbb{C}}^d$ for $d\ge 2$ that are Henkin for the multiplier algebra of the Drury–Arveson space, but not Henkin in the classical sense. This provides a negative answer to a conjecture of Clouâtre and Davidson.

In this paper, we show how to recover the relative $\mathbb{Q}$-grading in Heegaard Floer homology from the noncommutative grading on bordered Floer homology.

We compute many new classes of effective divisors in ${\overline{\mathcal{M}}}_{g,n}$ coming from the strata of Abelian differentials. Our method utilizes maps between moduli spaces and the degeneration of Abelian differentials.