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# Hodge theory at ICTP

School and conference at ICTP introduces young mathematicians to an essential field

Photograph of W.V.D. Hodge by Paul Halmos. Excerpted from P R Halmos, I have a photographic memory (Providence, 1987)

The thriving field of Hodge theory was developed 80 years ago by the Scottish mathematician W.V.D. Hodge. It applies methods from analysis to study the topology and geometry of complex algebraic varieties, and is an indispensable technique for studying singular spaces. The related Hodge conjecture is a Millennium Prize Problem of the Clay Mathematics Institute in Cambridge, United States.

In physics, Hodge theory allows string theorists to compute the properties of higher-dimensional space and has connections with the concept of mirror symmetry. Soon after the idea was established in physics, a similar hypothesis named 'Homological Mirror Symmetry' was established in mathematics.

The ICTP school and conference brought beginners into the fold with three days of lectures on manifolds, algebraic geometry and the basics of Hodge Theory. The remainder of the school introduced the participants to modern Hodge theory and its applications in algebraic geometry, leading up to the front lines of research. Finally, new results were presented during the conference.

Among the organizers was Phillip Griffiths, a major developer of the technique. He presented his work on reconciling Hodge theory, representation theory and areas of arithmetic. “Some of the most fruitful work has been work that connects different areas of mathematics,” Griffiths said. “One field can give insights into another field, can allow us to look with fresh eyes.”

The majority of the attendees were young mathematicians working in the field. Le Dang Thi Nguyen, a Ph.D. student at the Universität Duisburg-Essen, said he learned from the lectures, had fruitful discussions, and gained good contacts. He remembered a particularly helpful conversation with Luc Illusie, mathematician at the Universite Paris-Sud, Orsay and the University of Tokyo. Nguyen discussed his work and Illusie made a profound suggestion.

During the conference, Griffiths gave a general colloquium on "Hodge Theory and Representation Theory." He congratulated ICTP on "doing an absolutely fantastic job hosting the marvelous summer school."

The event was cosponsored by ICTP, the Clay Mathematics Institute, and the Unites States National Science Foundation.

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