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.
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.
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.
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.
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.
On the P=W Conjecture II
Exploration of the mirror P=W conjecture, its evidence in low dimensions, and potential verification methods using Lagrangian torus fibrations. Discusses work with Katzarkov-Przyjalkowski and Katzarkov-Pantev.
Differential Equations and Mixed Hodge Structures
Explores asymptotic Hodge theory, focusing on period-theoretic aspects through examples. Discusses motivic Gamma functions and their role in encoding invariants of variations of Hodge structures on P^1.
D-Modules and Toric Schobers
Exploring mirror symmetry using perverse sheaves of categories and schobers, with focus on B-side category in toric homological mirror symmetry and associated D-module characteristic cycles.
Hodge-Theoretic Aspects of Mirror Symmetry
Explores mirror symmetry for Fano varieties through Hodge theory, discussing Hodge diamond rotation, Katzarkov--Kontsevich--Pantev conjectures, and the mirror P=W conjecture in relation to mixed Hodge structures.
Tame Geometry and Hodge Theory II
Explores applications of tame geometry in Hodge theory, covering algebraization results, tameness of period maps, functional transcendence, and atypical intersection conjectures in complex algebraic geometry.
Archimedean Height Pairings for Higher Cycles
Explore archimedean height pairings for higher cycles, their interpretation as mixed Hodge structure invariants, and asymptotic behavior in Hodge-Tate variations.
Basic Cohomology of Moment-Angle Manifolds
Explores basic cohomology of moment-angle manifolds, presenting a Davis-Januszkiewicz type formula for calculating de Rham and Dolbeault cohomology of canonical holomorphic foliations on complex moment-angle manifolds.
Intersection of Quadrics, Moment-Angle Manifolds, Complex and Quaternionic Manifolds and Convex Polytope
Explore LVM manifolds, toric manifolds, and their quaternionic extensions. Gain insights into the intersection of quadrics, moment-angle manifolds, and convex polytopes in complex and quaternionic settings.
Tropical Analog of the Hodge Conjecture for Smooth Algebraic Varieties Over Trivially Valued Fields
Explore tropical geometry's approach to cycle class maps, focusing on a proof of the Hodge conjecture's tropical analog for smooth algebraic varieties over trivially valued fields.
NCHS and Rationality
Explore new approaches to nonrationality through Mirror Symmetry, delving into advanced mathematical concepts and their applications in algebraic geometry.
Donaldson-Thomas and Gromov-Witten Theories - Part 10
Explore moduli spaces of algebraic curves and coherent sheaves, their invariants, and connections to geometry, topology, and string theory. Includes computations of Gromov-Witten and Donaldson-Thomas invariants.
Donaldson-Thomas and Gromov-Witten Theories - Part 9
Explore moduli spaces of algebraic curves and coherent sheaves, their invariants, and connections to geometry, topology, and string theory. Includes computations of Gromov-Witten and Donaldson-Thomas invariants.
Tropical Geometry & Artificial Life
Explore self-organized criticality in artificial life models through tropical geometry, examining its applications and implications for understanding complex systems.