170 papers
This paper provides an informal introduction to the integrable systems approach to the Schottky problem, detailing how Jacobian theta functions yield solutions to the KP equation and culminating in Krichever's proof of Welters' trisecant conjecture for the most degenerate flex line case.
This paper constructs exotic Hecke correspondences between the special fibers of PEL-type Shimura varieties with potentially bad reduction by utilizing Pappas-Rapoport splitting models, thereby establishing new geometric Jacquet-Langlands correspondences and verifying generic instances of the Tate conjecture.
This paper introduces a determinantal method for computing minimal local generalized additive decompositions (GADs) of homogeneous polynomials by minimizing the rank of symbolic inverse systems, proving that this approach guarantees finding all minimal decompositions without tensor extensions whenever the local GAD-rank does not exceed the form's degree.
This paper extends the study of crystal melting models for toric Calabi-Yau 4-folds by developing an algorithm to construct crystals from periodic quivers, analyzing their behavior and partition functions under triality cascades, and introducing stable variables that reveal stabilization patterns to guide the search for generalized cluster algebras in 2d (0,2) quiver theories.
This paper proposes a novel machine learning framework that combines gradient descent on the Grassmannian manifold with Donaldson's algorithm to efficiently compute Ricci-flat approximations of Calabi-Yau metrics, demonstrating its effectiveness and the emergence of nontrivial local minima on the Dwork family of threefolds.
This paper introduces a new method utilizing syzygies to demonstrate that certain numerical semigroups are not Weierstrass, notably providing the first known examples of non-Weierstrass semigroups with multiplicity 6 and genus 13.
This paper proves the strong monodromy conjecture for defining polynomials of projective hypersurfaces with weighted homogeneous isolated singularities in the specific cases where the hypersurface is a reduced curve or has homogeneous singularities in dimension at least four, demonstrating that an "amazing cancellation" prevents potential counterexamples.
This paper establishes and provides an algorithmic method to verify a new criterion ensuring that a quasi-finite, flat morphism with locally constant geometric fibers between real algebraic varieties induces a covering map on their real points in the Euclidean topology.
This paper constructs a smooth projective complex surface with a representable Chow group of 0-cycles that lacks a universal 0-cycle, thereby providing a two-dimensional analogue to Voisin's counterexample and yielding the first known instance of a smooth projective threefold of Kodaira dimension zero with a non-torsion, non-algebraic Hodge class of degree 4.
This paper constructs a monoidal category of representations for the quantum affine algebra of a simple, simply connected, simply laced algebraic group, whose Grothendieck ring contains a cluster algebra with an initial seed corresponding to the coordinate ring of twisted products of flag varieties, including braid varieties and reduced double Bruhat cells.
This paper establishes a structure theorem for the connected automorphism groups of smooth complete toroidal horospherical varieties by characterizing extendable Demazure roots, thereby providing a criterion for their reductivity and proving the K-unstability of certain -bundles over rational homogeneous spaces.
This paper constructs smooth affine algebras of dimension over an algebraically closed field of characteristic zero that admit nontrivial projective modules of rank with trivial total Chern classes, thereby demonstrating the existence of nontrivial vector bundles whose topological invariants vanish.
This paper establishes a formula for the Euclidean distance degree of curves parameterized by rational functions and applies it to resolve conjectures regarding the ED degree of one-dimensional line multiview varieties in computer vision.
This paper classifies all local connected components of non-discriminant sets near parabolic function singularities, thereby proving and refining previous conjectures, enumerating local Petrovskii lacunas for hyperbolic PDE wavefronts, and demonstrating that certain parabolic singularities possess nontrivial one-dimensional homology groups in their discriminant complements, all achieved through a novel method combining Picard–Lefschetz theory with computer-assisted Morse surgery analysis.
This paper investigates the Chow group of zero-cycles on bielliptic surfaces over arbitrary fields of characteristic not 2 or 3, proving that the kernel of the Albanese map is a torsion group with a specific exponent and providing explicit examples over -adic fields where this kernel contains nontrivial elements arising from abelian surfaces.
This paper demonstrates that adjoining -adic analogs of and to the ring of analytic functions on the Fargues-Fontaine curve eliminates its Galois cohomology in degrees , thereby enabling the formulation of and -type conjectures for the compact support cohomology of -adic analytic varieties.
This paper establishes that a field is large if and only if it has an elementary extension that is the fraction field of a non-field henselian local domain, a result derived from new findings showing that the étale-open topology refines or is refined by the newly introduced finite-closed topology depending on whether the field is perfect or bounded.
This paper classifies projective Legendrian subvarieties that are homogeneous under their stabilizers within the projectivized nilpotent orbits of complex semi-simple Lie algebras, specifically providing a classification of equivariant Legendrian embeddings of rational homogeneous spaces into adjoint varieties.
This paper investigates the asymptotic counting of matrices over finite fields that commute with their Frobenius image (and its iterates), providing explicit results for specific cases such as size-2, diagonalizable, and -defined eigenspace matrices while outlining the approach for the general case.
This paper establishes an explicit formula for the -polynomials of van der Geer--van der Vlugt curves in characteristic 2 by leveraging the structure of Heisenberg groups and Lang torsors, ultimately enabling the construction of examples that attain the Hasse--Weil bound.