Extensions of Hodge Structure and Completions of Fundamental Groups
Explore extensions of mixed Hodge structures, unipotent and relative completions of fundamental groups, and their geometric interpretations in L-functions and motives theory.
Hodge Theory, Higgs Bundles on Moduli Spaces of Manifolds and Hyperbolicity I
Explores connections between Hodge theory, Higgs bundles, and moduli spaces, focusing on geometric properties of log spaces and variations of Hodge structures in complex manifolds.
Comparing Geometric and Hodge-Theoretic Compactifications II
Explores compactifications of moduli spaces for abelian varieties and K3 surfaces, focusing on geometric interpretations and connections to Hodge theory and polyhedral decompositions.
Characteristic Classes of Foliations in Positive Characteristic
Explore derived foliations on schemes and their application in extending vanishing results for characteristic classes to positive characteristic settings, featuring Hodge filtration on algebraic de Rham cohomology.
Semi-Affineness of Wrapped Invariants on Affine Log Calabi-Yau Varieties
Explore mirror symmetry in affine log Calabi-Yau varieties, examining finiteness properties of Floer theoretic invariants and their implications for categorical crepant resolutions.
Top Weight Cohomology of M_g
Explore the top-graded weight filtration on open moduli spaces, with applications to exponential growth in H^{4g-6}(M_g, Q) dimensions, disproving previous conjectures.
Stratifications in the Moduli Space of Stable Surfaces
Explore stratifications in stable surface moduli spaces, focusing on boundary surfaces and Hodge-theoretic approaches. Illustrated with examples and visuals from collaborative research.
Fixed Loci of Anti-Symplectic Involutions
Explore geometric constructions linking Fano manifolds and hyperkähler manifolds, focusing on reversing the process to recover Fano manifolds from hyperkähler structures.
Compactifications of Moduli — Geometry vs. Hodge Theory
Exploration of geometric and Hodge theoretic approaches to compactifying moduli spaces in algebraic geometry, with focus on classical cases and H and I-surfaces.
On Wormholes in the Moduli Space of Surfaces
Explore wormholing phenomena in KSBA compactification of moduli space for surfaces of general type, focusing on extremal P-resolutions and open questions in the field.
Completions of Period Maps
Explore the extension of period maps to compact Moishezon varieties, examining global structures and applications to moduli spaces. Insights from Hodge theory and complex geometry.
Deformations of Cusp Singularities of Algebraic Surfaces and Mirror Symmetry
Explore cusp singularities in algebraic surfaces, their deformations, and connections to mirror symmetry. Delve into Milnor fibers, log Calabi-Yau surfaces, and Kähler cones.
Semistable Reduction - A Progress Report
Explore recent advancements in resolving singularities of complex varieties, including functorial weak semistable reduction and the proof of the semistable reduction conjecture.
Surfaces with One T-Singularity
Explore stable surfaces with specific properties, focusing on I-surfaces with T-singularities. Gain insights into Gorenstein surfaces and applications to KSBA compactification of Gieseker's moduli space.
Surfaces on Cubic Fourfolds via Stability Conditions
Explore geometric applications of stability conditions on K3 categories, focusing on special surfaces on cubic fourfolds and their relation to rationality problems in algebraic geometry.
On the P=W Conjecture I
Exploration of the P=W conjecture for character varieties, its mirror version in Fano mirror symmetry, and applications to Hodge theory of Landau-Ginzburg models. Discusses recent developments and ongoing research in the field.
On the Fukaya Categories
Explore Fukaya categories in symplectic geometry with Mohammed Abouzaid from Columbia University, delving into advanced mathematical concepts and their applications.
Secondary Fan, Theta Functions, and Moduli of Calabi-Yau Pairs
Explore the conjecture on moduli space of triples, unirational components, and toric variety covers. Discover generalizations of secondary fans and universal family constructions in algebraic geometry.
Birational Calabi-Yau's and Floer Cohomology
Explore the relationship between Gromov-Witten invariants of birational Calabi-Yau manifolds using Hamiltonian Floer theory, including the generalized McKay correspondence and small quantum products.
Equivariant Birational Types
Explore recent developments and constructions in equivariant birational geometry, delving into advanced mathematical concepts and their applications.
Blow Up Formulae and Invariants I - Blow-Up Formula for Quantum Cohomology
Exploration of Gromov-Witten invariants in blown up complex algebraic varieties, proposing a formula and proof approach via equivariant localization. Insights into quantum cohomology and algebraic geometry.