# Mathematics

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The School of Mathematics is an international center of research and postdoctoral training in many diverse aspects of mathematics including pure mathematics, theoretical computer science, mathematical physics and applied mathematics.

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- Some of the metrics are blocked by yourconsent settingsAtypical Hodge loci(2023-06)Phillip GriffithsTalk based on the paper [BKU] and related works given in the references in that work, and on extensive discussions with Mark Green and Colleen Robles.
61 76 - Some of the metrics are blocked by yourconsent settingsComparisons between the Hodge-theoretic and algebro-geometric approaches to limiting mixed Hodge structures
;Griffiths, PhillipGreen, Mark223 165 - Some of the metrics are blocked by yourconsent settingsCompletion of the period mappings and ampleness of the Hodge bundles(2020)
;Griffiths, Phillip ;Green, Mark ;Laza, RaduRobles, ColleenI. Introduction II. Construction of a completion of the image of a period mapping III. Curvature properties of the extended Hodge line bundle (A) IV. Curvature properties of the extended Hodge line bundle (B) V. Proof that the extended Hodge line bundle is ample VI. Curvature properties of the Hodge vector bundle360 136 - Some of the metrics are blocked by yourconsent settings
306 177 - Some of the metrics are blocked by yourconsent settingsCompletions of Period Mappings(2020-10-13)
;Griffiths, Phillip ;Green, MarkRobles, ColleenThis work initiates a global study of period mappings at infinity, that is motivated by challenges arising when considering the questions: (i) What are natural completions of a period mapping? (ii) What geometric applications do they have?334 186 - Some of the metrics are blocked by yourconsent settingsCompletions of Period Mappings: Progress Report(2021)
;Griffiths, Phillip ;Green, MarkRobles, ColleenWe give an informal, expository account of a project to construct completions of period maps.419 314 - Some of the metrics are blocked by yourconsent settingsCurvature properties of Hodge bundles(2018-09-12)Griffiths, PhillipIn algebraic geometry the use of complex analytic methods differential geometry and PDEs | is long standing and far reaching for rather deep reasons the results these methods give (Hodge theory, vanishing theorems) have not been replaced by purely algebraic techniques. Two central areas in the subject are moduli and classification of varieties today we will discuss two results in which the use of analytic methods through the curvature properties of the Hodge bundles plays a central role.
246 312 - Some of the metrics are blocked by yourconsent settingsDifferential of a period mapping at a singularity(2021)
;Griffiths, PhillipGreen, MarkTo Herb Clemens. Lefschetz wrote that he put the harpoon of topology into the whale of algebraic geometry. Herb but the harpoon of topology into the whale of Hodge theory. Period mappings, or equivalently variations of Hodge structure, have been used both to study families of algebraic varieties and as a subject in its own right.242 263 - Some of the metrics are blocked by yourconsent settingsExtended Period MappingsGriffiths, PhillipThis lecture will discuss the global structure of period mappings (variation of Hodge structure) defined over complete, 2-dimensional algebraic varieties. Some applications to moduli of general type algebraic surfaces will also be presented.
158 100 - Some of the metrics are blocked by yourconsent settingsExtended Period MappingsGriffiths, PhillipClay Lecture at the INI, June 2022. The lecture is based on joint work with Mark Green and Colleen Robles. Theme of the lecture is global properties of period mappings with applications to the geometry of completions of moduli spaces.
137 105 - Some of the metrics are blocked by yourconsent settingsGlobal properties of period mappings on the boundary(2020)
;Griffiths, Phillip ;Green, MarkRobles, ColleenOutline I. Introduction II. Extension data for a mixed Hodge structure III. Extension data for limiting mixed Hodge structures IV. Period mappings to extension data (A) V. Period mappings to extension data (B) Appendix to Sections IV and V: Examples VI. Local Torelli conditions VII. Global structure of period mappings from complete surfaces325 139 - Some of the metrics are blocked by yourconsent settingsGlobal properties of period mappings on the boundary
;Griffiths, Phillip ;Green, MarkRobles, ColleenI. Introduction II. Extension data for a mixed Hodge structure III. Extension data for limiting mixed Hodge structures IV. Period mappings to extension data (A) V. Period mappings to extension data (B) Appendix to Sections IV and V: Examples VI. Local Torelli conditions VII. Global structure of period mappings from complete surfaces VIII. Period mappings and the canonical bundle References300 166 - Some of the metrics are blocked by yourconsent settingsHodge Theory and Moduli(2018)Griffiths, PhillipNotes based on the lecture presented at the conference “Geometry at the frontier” held at Pucon, Chile during November 2018.
306 557 - Some of the metrics are blocked by yourconsent settingsHodge Theory and Moduli(2018-05)Griffiths, PhillipThis talk will be at the interface of the two topics - moduli and singularities and Hodge theory and degenerations of Hodge structures.
222 310 - Some of the metrics are blocked by yourconsent settingsHodge Theory and Moduli(2023-04-07)Phillip GriffithsHodge theory provides a major tool for the study of moduli. Conversely, moduli have furnished a significant stimulus for the developement of Hodge theory.
96 129 - Some of the metrics are blocked by yourconsent settingsHodge Theory and Moduli(2020)Griffiths, PhillipClay Lecture, based on joint work with Mark Green, Radu Laza, and Colleen Robles. Some of the lecture draws work of and discussions with Marco Fanciosi, Rita Pardini, and Sonke Rollenske.
304 121 - Some of the metrics are blocked by yourconsent settingsHodge theory and ModuliGriffiths, PhillipThe theory of moduli is an important and active area in algebraic geometry. For varieties of general type the existence of a moduli space $\mathcal{M}$ with a canonical completion $\bar{\mathcal{M}}$ has been proved by Koll\'ar/Shepard-Barron/Alexeev. Aside from the classical case of algebraic curves, very little is known about the structure of $\mathcal{M}$, especially it's boundary $\overline{\mathcal{M}} \diagdown\mathcal{M}$. The period mapping from Hodge theory provides a tool for studying these issues. In this talk, we will discuss some aspects of this topic with emphasis on I-surfaces, which provide one of the first examples where the theory has been worked out in some detail. Particular notice will me made of how the extension data in the limiting mixed Hodge structures that arise from singular surfaces on the boundary of moduli may be used to guide the desingularization of that boundary.
373 310 - Some of the metrics are blocked by yourconsent settingsHodge Theory and Moduli(2019-05)Griffiths, PhillipAlgebraic geometry is frequently seen as a very interesting and beautiful subject, but one that is also very difficult to get into; this is partly due to its breadth, as traditionally algebra, topology, analysis, differential geometry, Lie theory and more recently combinatorics, logic,. . .are used to study it. I have tried to make these notes accessible to a general audience by illustrating topics with elementary examples and informal geometric and heuristic arguments, and with occasional side comments for experts in the subject.
266 283 - Some of the metrics are blocked by yourconsent settings
187 335 - Some of the metrics are blocked by yourconsent settingsIsolated Hypersurface Singularities(2019-04-20)Griffiths, PhillipLectures given at the University of Miami during April, 2019
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