We obtain asymptotic expansions for Toeplitz determinants corresponding to a family of symbols depending on a parameter $t$. For $t$ positive, the symbols are regular so that the determinants obey Szegő’s strong limit theorem. If $t=0$, the symbol possesses a Fisher-Hartwig singularity. Letting $t\to 0$ we analyze the emergence of a Fisher-Hartwig singularity and a transition between the two different types of asymptotic behavior for Toeplitz determinants. This transition is described by a special Painlevé V transcendent. A particular case of our result complements the classical description of Wu, McCoy, Tracy, and Barouch of the behavior of a 2-spin correlation function for a large distance between spins in the two-dimensional Ising model as the phase transition occurs.

We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to ${\Gamma }_1\left(3\right)$.

Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with ${\Gamma }_1\left(3\right)$-level structure. The morphism is ${\Gamma }_1\left(3\right)$-equivariant and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of $\Pi$-stability and by mirror symmetry.

We propose a general conjecture for the mixed Hodge polynomial of the generic character varieties of representations of the fundamental group of a Riemann surface of genus $g$ to ${\mathrm{GL}}_n\left(\mathbb{C}\right)$ with fixed generic semisimple conjugacy classes at $k$ punctures. This conjecture generalizes the Cauchy identity for Macdonald polynomials and is a common generalization of two formulas that we prove in this paper. The first is a formula for the E-polynomial of these character varieties which we obtain using the character table of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$. We use this formula to compute the Euler characteristic of character varieties. The second formula gives the Poincaré polynomial of certain associated quiver varieties which we obtain using the character table of $\mathfrak{g}{\mathfrak{l}}_n\left({\mathbb{F}}_q\right)$. In the last main result we prove that the Poincaré polynomials of the quiver varieties equal certain multiplicities in the tensor product of irreducible characters of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$. As a consequence we find a curious connection between Kac-Moody algebras associated with comet-shaped, and typically wild, quivers and the representation theory of ${\mathrm{GL}}_n\left({\mathbb{F}}_q\right)$.

We give a geometric interpretation of the Jones-Ocneanu trace on the Hecke algebra using the equivariant cohomology of sheaves on ${\mathrm{SL}}_n$. This construction makes sense for all simple groups, so we obtain a generalization of the Jones-Ocneanu trace to Hecke algebras of other types. We give a geometric expansion of this trace in terms of the irreducible characters of the Hecke algebra and conclude that it agrees with a trace defined independently by Gomi.

Based on our proof, we also prove that certain simple perverse sheaves on a reductive algebraic group $G$ are equivariantly formal for the conjugation action of a Borel $B$, or equivalently, that the Hochschild homology of any Soergel bimodule is free, as the authors had previously conjectured.

This construction is also closely tied to knot homology. This interpretation of the Jones-Ocneanu trace is a more elementary manifestation of the geometric construction of HOMFLYPT homology given by the authors.