HODGE THEORY
Hodge theory is a central area of mathematics that explores the relationship between algebraic geometry and differential geometry through the study of Hodge structures. It involves decomposing the complex structure of geometric shapes (varieties) into simpler, understandable parts. This "Hodge decomposition" uses cohomology, a mathematical tool that provides a way to study the properties of spaces by examining functions and differential forms on them. This process helps in understanding the geometric properties of these shapes, solving problems in number theory, and connecting with areas like string theory in physics.
Partially generated by ChuckGPT, a custom GPT trained on my papers.
Partially generated by ChuckGPT, a custom GPT trained on my papers.
- Degenerations, Fibrations, and Higher Rank Landau-Ginzburg Models. Charles Doran, Jordan Kostiuk, and Fenglong You, arXiv:2112.12891v1 [math.AG] 24 Dec 2021: 41 pages
- The Mirror Clemens-Schmid Sequence. Charles Doran and Alan Thompson, arXiv:2109.04849v2 [math.AG] 13 May 2022: 28 pages
- The Motivic Geometry of Two-Loop Feynman Integrals. Charles Doran, Andrew Harder, Pierre Vanhove (with an appendix by Eric Pichon-Pharabod), arXiv:2302.14840 [math.AG] 28 Feb 2023: 67 pages
- Modularity of Landau-Ginzburg Models. Charles Doran, Andrew Harder, Ludmil Katzarkov, Mikhail Ovcharenko, Victor Przyjalkowski, arXiv:2307.15607 [math.AG] 28 Jul 2023: 252 pages
- Normal Forms and Tyurin Degenerations of K3 Surfaces Polarized by a Rank 18 Lattice. Charles Doran, Joseph Prebble, Alan Thompson, arXiv:2311.10394 [math.AG] 17 Nov 2023: 24 pages
- K2 and Quantum Curves. Charles Doran, Matt Kerr, Soumya Sinha Babu. To appear in Advances in Theoretical and Mathematical Physics (2023): 51 pages
- Geometric Variations of Local Systems and Elliptic Surfaces. Charles Doran and Jordan Kostiuk. Israel Journal of Mathematics (2023): 79 pages
- The Doran-Harder-Thompson Conjecture for Toric Complete Intersections. Charles Doran, Jordan Kostiuk, Fenglong You. Advances in Mathematics, Volume 415 (2023), 108893 : 47 pages
- Hypergeometric Decomposition of Symmetric K3 Quartic Pencils. Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, John Voight, Ursula Whitcher. Research in the Mathematical Sciences 7, Article number: 7 (2020): 81 pages
- Calabi-Yau Threefolds Fibred by High Rank Lattice Polarized K3 Surfaces. Charles Doran, Andrew Harder, Andrey Novoseltsev, Alan Thompson. Mathematische Zeitschrift (2020) 294: 783-815
- Calabi-Yau Manifolds Realizing Symplectically Rigid Monodromy Tuples. Charles Doran and Andreas Malmendier. Advances in Theoretical and Mathematical Physics, Volume 23 (2019) Issue 5, 1271-1359
- Specialization of Cycles and the K-Theory Elevator. Pedro Luis del Angel, Charles Doran, Matt Kerr, James Lewis, Jaya Iyer, Stefan Müller-Stach, Deepam Patel. Communications in Number Theory and Physics, Volume 13 (2019) Number 2, 299-349
- Mirror Symmetry for Lattice Polarized Del Pezzo Surfaces. Charles Doran, Alan Thompson. Communications in Number Theory and Physics, Volume 12, Number 3 (2018), 543-580
- Zeta Functions of Alternate Mirror Calabi-Yau Families. Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, John Voight, Ursula Whitcher. Israel Journal of Mathematics, October 2018, Volume 228, Issue 2, 665-705
- Picard-Fuchs Uniformization of Modular Subvarieties. Brent Doran, Charles Doran, Andrew Harder; 2018; In Uniformization, Riemann-Hilbert Correspondence, Calabi- Yau Manifolds, and Picard-Fuchs Equations. Eds. Lizhen Ji and Shing-Tung Yau. International Press/Higher Education Press. Advanced Lectures in Mathematics, Volume 42, 21-54
- Hodge Numbers from Picard-Fuchs Equations. Charles Doran, Andrew Harder, Alan Thompson; 2017; SIGMA 13 (2017), 045, 23 pages
- Vertical D4-D2-D0 Bound States on K3 Fibrations and Modularity. Vincent Bouchard, Thomas Creutzig, Duiliu-Emanuel Diaconescu, Charles Doran, Callum Quigley, Artan Sheshmani; 2017; Communications in Mathematical Physics 350, 1069-1121 (2017)
- Mirror Symmetry, Tyurin Degenerations, and Fibrations on Calabi-Yau Manifolds. Charles Doran, Andrew Harder, Alan Thompson; 2018; In String-Math 2015, American Mathematical Society, Proceedings of Symposia in Pure Mathematics, 96, 93-132
- Special Function Identities from Superelliptic Kummer Varieties. Adrian Clingher, Charles Doran, Andreas Malmendier; 2017; Asian Journal of Mathematics, Volume 21 (2017) Number 5, 909-952
- Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces. Charles Doran, Andrew Harder, Andrey Novoseltsev, Alan Thompson; 2016; Advances in Mathematics, Volume 298, 6 August 2016, 369-392
- Toric Degenerations and Laurent Polynomials Related to Givental's Landau-Ginzburg Models. Charles Doran, Andrew Harder; 2016; Canadian Journal of Mathematics, Volume 68 (2016), 784-815
- Calabi-Yau Threefolds Fibred by Kummer Surfaces Associated to Products of Elliptic Curves. Charles Doran, Andrew Harder, Andrey Novoseltsev, Alan Thompson; 2016; In String-Math 2014, American Mathematical Society, Proceedings of Symposia in Pure Mathematics 93, 278-303
- Humbert Surfaces and the Moduli of Lattice Polarized K3 Surfaces. Charles Doran, Andrew Harder, Hossein Movasati, Ursula Whitcher; 2016; In String-Math 2014, American Mathematical Society, Proceedings of Symposia in Pure Mathematics 93, 124-155
- The 14th Case VHS via K3 Fibrations. Adrian Clingher, Charles Doran, Jacob Lewis, Andrey Novoseltsev, Alan Thompson; 2016; In Recent Advances in Hodge Theory: Period Domains, Algebraic Cycles, and Arithmetic, Cambridge University Press, London Mathematical Society Lecture Note Series 427, 165-227
- Families of Lattice Polarized K3 Surfaces with Monodromy. Charles Doran, Andrew Harder, Andrey Novoseltsev, Alan Thompson; 2015; International Mathematics Research Notices, 2015 (23): 12265-12318
- Algebraic Cycles and Local Quantum Cohomology. Charles Doran, Matt Kerr; 2014; Communications in Number Theory and Physics, Volume 8 (2014), Number 4, pp. 703-727
- Normal Functions, Picard-Fuchs Equations, and Elliptic Fibrations on K3 Surfaces. Xi Chen, Charles Doran, Matt Kerr, James Lewis ; 2014 ; Journal für die reine und angewandte Mathematik (Crelles Journal), DOI: 10.1515/crelle-2014-0085, November 2014
- Lattice Polarized K3 Surfaces and Siegel Modular Forms. Adrian Clingher, Charles Doran; 2012; Advances in Mathematics, Volume 231, Issue 1, 172–212
- Hori-Vafa Mirror Periods, Picard-Fuchs Equations, and Berglund-Hübsch-Krawitz Duality. Charles Doran, Richard Garavuso; 2011; Journal of High Energy Physics, October 2011, 2011:128
- Algebraic K-Theory of Toric Hypersurfaces. Charles Doran, Matthew Kerr; 2011; Commun. Number Theory Phys, Vol 5, No 2, pp. 397-600
- Note on a Geometric Isogeny of K3 Surfaces. Adrian Clingher, Charles Doran; 2011; International Mathematics Research Notices, 2011 (16): 3657-3687
- Closed Form Expressions for Hodge Numbers of Complete Intersection Calabi-Yau Threefolds in Toric Varieties. Charles Doran, Andrey Novoseltsev; 2010; In Mirror Symmetry and Tropical Geometry, Contemporary Mathematics, Vol 527, pp. 1-14
- Normal Forms, K3 Surface Moduli, and Modular Parametrizations. Adrian Clingher, Charles Doran, Jacob Lewis, Ursula Whitcher; 2009; In Groups and Symmetries, CRM Proceedings and Lecture Notes, 47, 81-98
- Families of Quintic Calabi-Yau 3-Folds with Discrete Symmetries. Charles Doran, Brian Greene, Simon Judes; 2008; Communications in Mathematical Physics, Volume 280, Number 2, 675-725
- On Stokes Matrices of Calabi-Yau Hypersurfaces. Charles Doran, Shinobu Hosono; 2007; Advances in Theoretical and Mathematical Physics, Volume 11, Issue 1, 147-174
- Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties. Charles Doran, John Morgan; 2007; Geometry and Topology, 11, 597-642
- Modular Invariants for Lattice Polarized K3 Surfaces. Adrian Clingher, Charles Doran; 2007; Michigan Mathematical Journal, 55, Issue 2, 355-393
- On K3 Surfaces with Large Complex Structure. Adrian Clingher, Charles Doran; 2007; Advances in Mathematics, 215, 504-539
- Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds. Charles Doran, John Morgan; 2006; In Mirror Symmetry V, AMS/IP Studies in Advanced Mathematics, 38, 517-537
- Algebraic and Geometric Isomonodromic Deformations. Charles Doran; 2001; Journal of Differential Geometry, 59, 33-85
- Algebro-geometric Isomonodromic Deformations Linking Hauptmoduls: Variation of the Mirror Map. Charles Doran; 2001; Centre de Recherches Mathematiques: In Proceedings on Moonshine and Related Topics, CRM Proceedings and Lecture Notes, 30, 27-35
- Picard-Fuchs Uniformization and Modularity of the Mirror Map. Charles Doran; 2000; Communications in Mathematical Physics, 212, 625-647
- Picard-Fuchs Uniformization: Modularity of the Mirror Map and Mirror-Moonshine. Charles Doran; 2000; In the Arithmetic and Geometry of Algebraic Cycles, CRM Proceedings and Lecture Notes, 24, 257-281
- String-Math 2014. Vincent Bouchard, Charles Doran, Stefan Méndez-Diez, Callum Quigley, Eds.; 2016; American Mathematical Society, Proceedings of Symposia in Pure Mathematics 93, 396 pages
- Yau's Work on Moduli, Periods, and Mirror Maps for Calabi-Yau Manifolds. Charles Doran; 2010; In “Geometry and Analysis,” Volume I. Pages 93-10
- Modular Forms and String Duality. Noriko Yui, Helena Verrill, Charles Doran, Eds; 2008; Fields Institute Communications, 54, 297 pages