# 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 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 settingsLecture Series at the High School of Economics, Moscow(2018-05)Griffiths, PhillipAlgebraic geometry is the study of the geometry of algebraic varieties, defined as the solutions of a system of polynomial equations over a field k. When k = C the earliest deep results in the subject were discovered using analysis, and analytic methods (complex function theory, PDEs and differential geometry) continue to play a central and pioneering role in algebraic geometry. The objective of these talks is to present an informal and illustrative account of some answers to the question in the title.
410 1193 - 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 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and far reaching for rather deep reasons the results these methods give (Hodge theory, vanishing theorems) have not been replaced by purely algebraic techniques.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 settingsPositivity of vector bundles and Hodge theory(2018-10-10)
;Griffiths, PhillipGreen, MarkIt is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge metrics generally have singularities; what can be said about these and their curvatures, Chern forms etc.? (iii) What are some algebro-geometric applications of positivity of Hodge bundles? The purpose of these partly expository notes is fourfold. One is to summarize some of the general measures and types of positivity that have arisen in the literature. A second is to introduce and give some applications of norm positivity. This is a concept that implies the di_erent notions of metric semi-positivity that are present in many of the standard examples and one that has an algebro-geometric interpretation in these examples. A third purpose is to discuss and compare some of the types of metric singularities that arise in algebraic geometry and in Hodge theory. Finally we shall present some applications of the theory from both the classical and recent literature.294 387 - Some of the metrics are blocked by yourconsent settingsModuli and Hodge Theory(2018-11-12)
;Griffiths, Phillip ;Green, Mark ;Laza, RaduRobles, ColleenTalk at the "Geometry at the Frontier" conference, Pucón, Chile (November 12, 2018), and based in part on joint work in progress with Mark Green, Radu Laza and Colleen Robles.237 200 - Some of the metrics are blocked by yourconsent settingsNew Geometric Invariants Arising from Hodge Theory(2018-12)
;Griffiths, Phillip ;Green, Mark ;Laza, RaduRobles, ColleenThe use of the Hodge line bundle, especially its positivity properties, in algebraic geometry is classical. The Hodge line bundle only sees the associated graded to the limiting mixed Hodge structures (LMHS's) that arise from the singular members in a family of algebraic varieties. This talk will introduce a new type of geometric object associated to the extension data in LMHS's (and not present for general MHS's) whose positivity properties play a central role in the application of Hodge theory to moduli. Their construction will be discussed and illustrated.239 121 - Some of the metrics are blocked by yourconsent settingsModuli and Hodge Theory(2019-04-05)Griffiths, PhillipTalk at UIC (April 5, 2019), and based in part on joint work in progress with Mark Green, Radu Laza and Colleen Robles (GLR). Selected references to works quoted in or related to this talk are given at the end.
241 381 - 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
220 152 - Some of the metrics are blocked by yourconsent settings
200 152 - 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 settingsUsing Hodge theory to detect the structure of a compactified moduli space(2019-11-27)Griffiths, PhillipTalk given on November 27, 2019 at the IMSA conference held at IMATE at UNAM, Mexico City.
388 343 - 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 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 settingsPositivity in Hodge theory and algebraic geometry(2020)Griffiths, PhillipLecture give at Cambridge 2020
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337 321 - 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 settingsPeriod Mapping at Infinity(2020-05-06)Griffiths, PhillipHodge theory provides a basic invariant of complex algebraic varieties. For algebraic families of smooth varieties the global study of the Hodge structure on the cohomology of the varieties (period mapping) is a much studied and rich subject. When one completes a family to include singular varieties the local study of how the Hodge structures degenerate to limiting mixed Hodge structures is also much studied and very rich. However, the global study of the period mapping at infinity has not been similarly developed. This has now been at least partially done and will be the topic of this talk. Sample applications include: new global invariants of limiting mixed Hodge structures; a generic local Torelli assumption implies that moduli spaces are log canonical (not just log general type); and extension data and asymptotics of the Ricci curvature; a proposed construction of the toroidal compactification of the image of period mapping. The key point is that the extension data associated to a limiting mixed Hodge structure has a rich geometric structure and this provides a new tool for the study of families of singular varieties in the boundary of families of smooth varieties.
354 240 - Some of the metrics are blocked by yourconsent settingsPeriod Mapping and the properties of the Hodge line bundle(2020-07)
;Griffiths, Phillip ;Green, Mark ;Laza, RaduRobles, Colleen341 280 - 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

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