Assuming a conjecture in non-Archimedean geometry, we show that a metric formulation of the Strominger–Yau–Zaslow (SYZ) conjecture can be proved in large generality.

Chern–Schwartz–MacPherson (CSM) classes generalize to singular varieties the classical total homology Chern class of the tangent bundle of a smooth compact complex algebraic variety. The theory of CSM classes has been extended to the equivariant setting by Ohmoto. We prove that for an arbitrary complex algebraic variety X, the homogenized, torus-equivariant CSM class of a constructible function φ is the restriction of the characteristic cycle of φ via the zero section of the cotangent bundle of X. In the process, we relate the CSM class in question to a Segre operator applied to the characteristic cycle. This extends to the equivariant setting results of Ginzburg and of Sabbah. We specialize X to be a (generalized) flag manifold $G∕B$. In this case, CSM classes are determined by a Demazure–Lusztig (DL) operator. We prove a Hecke orthogonality of CSM classes, determined by the DL operator and its adjoint, and a geometric orthogonality between CSM and Segre–MacPherson classes. This implies a remarkable formula for the CSM class of a Schubert cell in terms of the Segre class of the characteristic cycle of a holonomic Verma ${\mathcal{D}}_X$-module. We deduce a positivity property for CSM classes previously conjectured by Aluffi and Mihalcea, and extending positivity results by Huh in the Grassmann manifold case. As an application, we prove positivity for certain Kazhdan–Lusztig (KL) classes, and for some instances of Mather classes, of Schubert varieties. We also establish an equivalence between CSM classes and stable envelopes; this re-proves results of Rimányi and Varchenko. Finally, we generalize all of this to partial flag manifolds $G∕P$.

We study the quantitative stability of the quadratic optimal transport map between a fixed probability density ρ and a probability measure μ on ${\mathbb{R}}^d$, which we denote $T_{\mathrm{\mu }}$. Assuming that the source density ρ is bounded from above and below on a compact convex set, we prove that the map $\mathrm{\mu }\mapsto T_{\mathrm{\mu }}$ is bi-Hölder continuous on large families of probability measures, such as the set of probability measures whose moment of order $p>d$ is bounded by some constant. These stability estimates show that the linearized optimal transport metric ${\mathrm{W}}_{2,\mathit{\rho }}(\mathrm{\mu },\mathrm{\nu })=\Vert T_{\mathrm{\mu }}-T_{\mathrm{\nu }}{\Vert }_{{\mathrm{L}}^2(\mathit{\rho },{\mathbb{R}}^d)}$ is bi-Hölder equivalent to the 2-Wasserstein distance on such sets, justifying its use in applications.

We introduce a variant of the K-theoretic quantized Coulomb branch constructed by Braverman, Finkelberg, and Nakajima, by application of a new virtual intersection theory. In the abelian case, we define Verma modules for such virtual Coulomb branches, and relate them to the moduli spaces of quasimaps into the corresponding Higgs branches. The descendent vertex functions, defined by K-theoretic quasimap invariants of the Higgs branch, can be realized as the associated Whittaker functions. The quantum q-difference modules and Bethe algebras (analogue of quantum K-theory rings) can then be described in terms of the virtual Coulomb branch. As an application, we prove the wall-crossing result for quantum q-difference modules under the variation of GIT. Nonabelian cases are also treated via abelianization.