For a positive braid $\beta \in \mathrm {Br}^{+}_{k}$, we consider the braid variety $X(\beta )$. We define a family of open sets $\mathcal {U}_{r, w}$ in $X(\beta )$, where $w \in S_k$ is a permutation and r is a positive integer no greater than the length of $\beta $. For fixed r, the sets $\mathcal {U}_{r, w}$ form an open cover of $X(\beta )$. We conjecture that $\mathcal {U}_{r,w}$ is given by the nonvanishing of some cluster variables in a single cluster for the cluster structure on $\mathbb {C}[X(\beta )]$ constructed in Casals et al. (2025, J. Amer. Math. Soc. 38, 369–479), Galashin et al. (2026, Invent. Math. 243, 1079–1127), and Galashin et al. (2022, Braid variety cluster structures, I: 3D plabic graphs) and that $\mathcal {U}_{r,w}$ admits a cluster structure given by freezing these variables. Moreover, we show that $\mathcal {U}_{r, w}$ is always isomorphic to the product of two braid varieties, and we conjecture that this isomorphism is quasi-cluster. In some important special cases, we are able to prove our conjectures.

Let V be a finite-dimensional complex vector space. Assume that V is a direct sum of subspaces each of which is equipped with a nondegenerate symmetric or skew-symmetric bilinear form. In this article, we introduce a stratification of the Grassmannian $\text {Gr}_k(V)$ related to the action of the appropriate product of orthogonal and symplectic groups, and we study the topology of this stratification. The main results involve sheaves with coefficients in a field of characteristic other than $2$. We prove that there are “enough” parity sheaves, and that the hypercohomology of each parity sheaf also satisfies a parity-vanishing property. This situation arises in the following context: let x be a nilpotent element in the Lie algebra of either $G = \text {Sp}_N(\mathbb {C})$ or $G = \text {SO}_N(\mathbb {C})$, and let $V = \ker x \subset \mathbb {C}^N$. Our stratification of $\text {Gr}_k(V)$ is preserved by the centralizer $G^x$, and we expect our results to have applications in Springer theory for classical groups.

For any power q of the positive ground field characteristic, a smooth q-bic threefold—the Fermat threefold of degree $q+1$, for example—has a smooth surface S of lines which behaves like the Fano surface of a smooth cubic threefold. I develop projective, moduli-theoretic, and degeneration techniques to study the geometry of S. Using, in addition, the modular representation theory of the finite unitary group and the geometric theory of filtrations, I compute the cohomology of the structure sheaf of S when q is prime.

We investigated the symplectic geometry of homogeneous spaces associated with semisimple Lie groups, focusing on cotangent bundles of maximal flag manifolds. Our work provides an explicit description of the canonical symplectic structure on these spaces using connections and curvatures of principal bundles naturally associated with the underlying Lie groups. We extend classical results concerning the exactness of symplectic forms on adjoint orbits, previously known for specific Lie algebras, to arbitrary simple Lie groups. In particular, we identify conditions under which the Kostant–Kirillov–Souriau form on a regular adjoint orbit coincides with the canonical symplectic form of the cotangent bundle, yielding exact symplectic structures. The approach combines differential-geometric techniques with Lie-theoretic constructions, offering a unifying framework that connects the geometry of coadjoint orbits with symplectic structures on homogeneous spaces.

We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogs of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

We show that a divisor in a rational homogenous variety with split normal sequence is the preimage of a hyperplane section in either the projective space or a quadric.

We compute the Čech cohomology ring of a countable product of infinite projective spaces, and that of an infinite flag manifold. The method of our first result in fact computes the cohomology ring of a countably infinite product of paracompact Hausdorff spaces, under some mild assumptions.

We show that a very general hypersurface of degree $d \geq 4$ and dimension $N \leq (d+1)2^{d-4}$ over a field of characteristic $\neq 2$ does not admit a decomposition of the diagonal; hence, it is neither stably nor retract rational, nor $\mathbb {A}^1$-connected. Similar results hold in characteristic $2$ under a slightly weaker degree bound. This improves earlier results in [44] and [33].

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers–Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behaviour of thecoefficients at $q=1$ is Gaussian.

We give a presentation of the torus-equivariant (small) quantum K-ring of flag manifolds of type C as an explicit quotient of a Laurent polynomial ring; our presentation can be thought of as a quantization of the classical Borel presentation of the ordinary K-ring of flag manifolds. Also, we give an explicit Laurent polynomial representative for each special Schubert class in our Borel-type presentation of the quantum K-ring.

For each of the four particle processes given by Dieker and Warren, we show the n-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.

In this paper, we study the class of polytopes which can be obtained by taking the convex hull of some subset of the points $\{e_i-e_j \ \vert \ i \neq j\} \cup \{\pm e_i\}$ in $\mathbb {R}^n$, where $e_1,\dots ,e_n$ is the standard basis of $\mathbb {R}^n$. Such a polytope can be encoded by a quiver Q with vertices $V \subseteq \{{\upsilon }_1,\dots ,{\upsilon }_n\} \cup \{\star \}$, where each edge ${\upsilon }_j\to {\upsilon }_i$ or $\star \to {\upsilon }_i$ or ${\upsilon }_i\to \star $ gives rise to the point $e_i-e_j$ or $e_i$ or $-e_i$, respectively; we denote the corresponding polytope as $\operatorname {Root}(Q)$. These polytopes have been studied extensively under names such as edge polytope and root polytope. We show that if the quiver Q is strongly-connected, then the root polytope $\operatorname {Root}(Q)$ is reflexive and terminal; we moreover give a combinatorial description of the facets of $\operatorname {Root}(Q)$. We also show that if Q is planar, then $\operatorname {Root}(Q)$ is (integrally equivalent to) the polar dual of the flow polytope of the planar dual quiver $Q^{\vee }$. Finally, we consider the case that Q comes from the Hasse diagram of a finite ranked poset P and show in this case that $\operatorname {Root}(Q)$ is polar dual to (a translation of) a marked order polytope. We then go on to study the toric variety $Y(\mathcal {F}_Q)$ associated to the face fan $\mathcal {F}_Q$ of $\operatorname {Root}(Q)$. If Q comes from a ranked poset P, we give a combinatorial description of the Picard group of $Y(\mathcal {F}_Q)$, in terms of a new canonical ranked extension of P, and we show that $Y(\mathcal {F}_Q)$ is a small partial desingularisation of the Hibi projective toric variety $Y_{\mathcal {O}(P)}$ of the order polytope $\mathcal {O}(P)$. We show that $Y(\mathcal {F}_Q)$ has a small crepant toric resolution of singularities $Y(\widehat {\mathcal {F}}_Q)$ and, as a consequence that the Hibi toric variety $Y_{\mathcal {O}(P)}$ has a small resolution of singularities for any ranked poset P. These results have applications to mirror symmetry [61].

We prove that the initial degenerations of the flag variety admit closed immersions into finite inverse limits of flag matroid strata, where the diagrams are derived from matroidal subdivisions of a suitable flag matroid polytope. As an application, we prove that the initial degenerations of $\mathrm{F}\ell^{\circ}(n)$–the open subvariety of the complete flag variety $\mathrm{F}\ell(n)$ consisting of flags in general position—are smooth and irreducible when $n\leq 4$. We also study the Chow quotient of $\mathrm{F}\ell(n)$ by the diagonal torus of $\textrm{PGL}(n)$ and show that, for $n=4$, this is a log crepant resolution of its log canonical model.

We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split reductive groups. Our new geometric results include Whitney–Tate stratifications of Beilinson–Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of Deligne’s modification of the dual group and a modified form of Vinberg’s monoid over the integers.

Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

Schubert Vanishing is a problem of deciding whether Schubert coefficients are zero. Until this work it was open whether this problem is in the polynomial hierarchy ${{\mathsf {PH}}}$. We prove this problem is in ${{\mathsf {AM}}} \cap {{\mathsf {coAM}}}$ assuming the Generalized Riemann Hypothesis ($\mathrm{GRH}$), that is, relatively low in ${{\mathsf {PH}}}$. Our approach uses Purbhoo’s criterion [57] to construct explicit polynomial systems for the problem. The result follows from a reduction to Parametric Hilbert’s Nullstellensatz, recently analyzed in [2]. We extend our results to all classical types.

We study the moduli space of constant scalar curvature Kähler (cscK) surfaces around toric surfaces. To this end, we introduce the class of foldable surfaces: smooth toric surfaces whose lattice automorphism group contains a non-trivial cyclic subgroup. We classify such surfaces and show that they all admit a cscK metric. We then study the moduli space of polarised cscK surfaces around a point given by a foldable surface, and show that it is locally modelled on a finite quotient of a toric affine variety with terminal singularities.

It was conjectured by McKernan and Shokurov that for any Fano contraction $f:X \to Z$ of relative dimension r with X being $\epsilon $-lc, there is a positive $\delta $ depending only on $r,\epsilon $ such that Z is $\delta $-lc and the multiplicity of the fiber of f over a codimension one point of Z is bounded from above by $1/\delta $. Recently, this conjecture was confirmed by Birkar [9]. In this article, we give an explicit value for $\delta $ in terms of $\epsilon ,r$ in the toric case, which belongs to $O(\epsilon ^{2^r})$ as $\epsilon \rightarrow 0$. The order $O(\epsilon ^{2^r})$ is optimal in some sense.

We prove a ‘Whitney’ presentation, and a ‘Coulomb branch’ presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm {Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation is obtained by quantum deforming the product of the Hirzebruch $\lambda _y$ classes of the tautological bundles. In physics, the $\lambda _y$ classes arise as certain Wilson line operators. The second presentation is obtained from the Coulomb branch equations involving the partial derivatives of a twisted superpotential from supersymmetric gauge theory. This is closest to a presentation obtained by Gorbounov and Korff, utilizing integrable systems techniques. Algebraically, we relate the Coulomb and Whitney presentations utilizing transition matrices from the (equivariant) Grothendieck polynomials to the (equivariant) complete homogeneous symmetric polynomials. Along the way, we calculate K-theoretic Gromov-Witten invariants of wedge powers of the tautological bundles on $\mathrm {Gr}(k;n)$, using the ‘quantum=classical’ statement.

For a split reductive group G we realise identities in the Grothendieck group of $\widehat{G}$-representations in terms of cycle relations between certain closed subschemes inside the affine Grassmannian. These closed subschemes are obtained as a degeneration of e-fold products of flag varieties and, under a bound on the Hodge type, we relate the geometry of these degenerations to that of moduli spaces of G-valued crystalline representations of $\operatorname{Gal}(\overline{K}/K)$ for $K/\mathbb{Q}_p$ a finite extension with ramification degree e. By transferring the aforementioned cycle relations to these moduli spaces we deduce one direction of the Breuil–Mézard conjecture for G-valued crystalline representations with small Hodge type.