For an algebraic group $G$, Anderson introduced the notion of Mirković-Vilonen (MV) polytopes as images of MV-cycles under the moment map of the affine Grassmannian. It was shown by Kamnitzer that MV-polytopes and their corresponding cycles can be described as solutions of the tropical Plücker relations. Another construction of MV-cycles, by Gaussent and Littelmann, can be given by using LS-galleries, a more discrete version of Littelmann's path model.

This article gives a direct combinatorial construction of the MV-polytopes using LS-galleries. This construction is linked to the retractions of the affine building and the Bott-Samelson variety corresponding to $G$, leading to a type-independent definition of MV-polytopes not involving the tropical Plücker relations.

Let $G$ be a unitary group over a totally real field, and let $X$ be a Shimura variety associated to $G$. For certain primes $p$ of good reduction for $X$, we construct cycles $X_{{\tau }_0,i}$ on the characteristic $p$ fiber of $X$. These cycles are defined as the loci on which the Verschiebung map has small rank on particular pieces of the Lie algebra of the universal abelian variety on $X$. The geometry of these cycles turns out to be closely related to Shimura varieties for a different unitary group $G\prime$, which is isomorphic to $G$ at all finite places but not isomorphic to $G$ at archimedean places. More precisely, each cycle $X_{{\tau }_0,i}$ has a natural desingularization ${\tilde{X}}_{{\tau }_0,i}$, which is almost isomorphic to a scheme parameterizing certain subbundles of the Lie algebra of the universal abelian variety over a Shimura variety $X\prime$ associated to $G\prime$. We exploit this relationship to construct an injection of the étale cohomology of $X\prime$ into that of $X$. This yields a geometric construction of Jacquet-Langlands transfers of automorphic representations of $G\prime$ to automorphic representations of $G$.

We give a new construction of monopole Floer homology for ${\text{spin}}^c$ rational homology $3$-spheres. As applications, we define two invariants of certain $4$-manifolds with $b_1=1$ and $b^{+}=0$.

We study a deformation theory of $\mu$-semistable quasi-bundles of rank 2 on a polarized surface $(X,O(1))$ such that $(O(1)·K_X)<0$, and we apply it to strange dualities for some rational surfaces.