School of Mathematics
https://hdl.handle.net/20.500.12111/14
School of Mathematics2020-02-18T00:14:24ZPositivity and Vanishing Theorems
https://hdl.handle.net/20.500.12111/7881
Positivity and Vanishing Theorems
Griffiths, Phillip
Existence theorems are a central part of algebraic geometry. These results frequently involve linear problems where positivity assumptions are used to prove existence of solutions by establishing the vanishing of obstructions to that existence. Beginning with Riemann (algebraic curves), Picard (algebraic surfaces) and continuing into more recent times (Lefschetz, Hodge, Kodaira-Spencer and many others since this work) it has come to be understood that the vanishing theorems are intimatedly related to the topology of algebraic varieties.
What remains is the case that although the results are about algebraic varieties, analytic tools are needed to establish them. Moreover the property of positivity also appears in other aspects where analytic methods are needed, an example being the proof of the Iitaka conjecture which is central in the classification of algebraic varieties. The purpose of these lectures is to present, sometimes from an historical perspective, some of the principal aspects of the theory.
Lectures given at the University of Miami, Spring 2020
A Tale of Two Mathematicians
https://hdl.handle.net/20.500.12111/7874
A Tale of Two Mathematicians
Griffiths, Phillip
This is the story of some of the mathematical work of two mathematicians, Jean Victor Poncelet and Niels Henrik Abel. They were contemporaries in the early 19th century who never met and who were not even aware of each other's work. However, between them Poncelet and Abel laid the cornerstones of the modern field of algebraic geometry, a field
that is central to current work in geometry, arithmetic and theoretical physics. In this talk I will try to explain what each of them did, Poncelet in geometry and Abel in analysis, and
how the fusion of their work revealed one of the deepest aspects of mathematics. This fusion is captured by an amazing property of playing billiards on a table formed by two ellipses.
Lefschetz Lecture at Cinestav
Using Hodge theory to detect the structure of a compactified moduli space
https://hdl.handle.net/20.500.12111/7873
Using Hodge theory to detect the structure of a compactified moduli space
Griffiths, Phillip
Talk given on November 27, 2019 at the IMSA conference held at IMATE at UNAM, Mexico City.
This talk is based on joint work in progress with Mark Green, Radu Laza and Colleen Robles. The example is drawn from the work of Marco Franciosi, Rita Pardini and Sonke Rollenske and on discussions that we have had with them.
2019-11-27T00:00:00ZModuli and Hodge Theory
https://hdl.handle.net/20.500.12111/7868
Moduli and Hodge Theory
Griffiths, Phillip
Talk 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.
Talk at the University of Illinois at Chicago on April 5, 2019.
2019-04-05T00:00:00Z