The main theorem here is the $K$-theoretic analogue of the cohomological ``stable double component formula'' for quiver polynomials in [KMS]. This $K$-theoretic version is still in terms of lacing diagrams, but nonminimal diagrams contribute terms of higher degree. The motivating consequence is a conjecture of Buch [B1] on the sign alternation of the coefficients appearing in his expansion of quiver $K$-polynomials in terms of stable Grothendieck polynomials for partitions.

We describe an alternative approach to the existence results of S.-Y. A. Chang and P. C. Yang for metrics of prescribed scalar curvature on $S^2$ via the prescribed curvature flow. Moreover, we give an example showing that the results of these authors in general cannot be improved upon.

In this paper we give a characterization of locally compact rank one symmetric spaces, which can be seen as an analogue of Ballmann's and Burns and Spatzier's characterizations of nonpositively curved symmetric spaces of higher rank, as well as of Hamenstädt's characterization of negatively curved symmetric spaces. Namely, we show that a complete Riemannian manifold $M$ is locally isometric to a compact, rank one symmetric space if $M$ has sectional curvature at most $1$ and each normal geodesic in $M$ has a conjugate point at $\pi$.

In our previous paper [EV2], to every finite-dimensional representation $V$ of the quantum group $U_q\left(\mathfrak{g}\right)$ we attached the trace function $F^V\left(\lambda ,\mu \right)$ with values in $\mathrm{End}V\left[0\right]$ which was obtained by taking the (weighted) trace in a Verma module of an intertwining operator. We showed that these trace functions satisfy the Macdonald-Ruijsenaars and quantum Knizhnik-Zamolodchikov-Bernard (qKZB) equations, their dual versions, and the symmetry identity. In this paper, we show that the trace functions satisfy the orthogonality relation and the qKZB-heat equation. For $\mathfrak{g}=s{l}_2$, this statement is the trigonometric degeneration of a conjecture from [FV3], proved in [FV3] for the 3-dimensional irreducible $V$. We also establish the orthogonality relation and the qKZB-heat equation for trace functions that were obtained by taking traces in finite-dimensional representations (rather than in Verma modules). If $\mathfrak{g}={}_n$ and $V=S^{kn}{ℂ}^n$, these functions are known to be Macdonald polynomials of type $A$. In this case, the orthogonality relation reduces to the Macdonald inner product identities, and the qKZB-heat equation coincides with the q-Macdonald-Mehta identity that was proved by Cherednik [Ch2].

In this paper we complete the results of our papers [2], [3] and show how to generate from the hypergeometric function $F\left(1/6,1/6;1;\zeta \right)$ fundamental solutions for the classical Tricomi operator relative to any point in the elliptic, parabolic, or hyperbolic region of the operator.

We prove a conjecture of Conrad, Diamond, and Taylor on the size of certain deformation rings parametrizing potentially Barsotti-Tate Galois representations. To achieve this, we extend results of Breuil and Mézard (classifying Galois lattices in semistable representations in terms of "strongly divisible modules") to the potentially crystalline case in Hodge-Tate weights (0, 1). We then use these strongly divisible modules to compute the desired deformation rings. As a corollary, we obtain new results on the modularity of potentially Barsotti-Tate representations.