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This special ICTP Math Colloquium will be given by Prof. Tomasz Mrowka. The talk on "More deformations of the cohomology ring of the moduli space of representations of the fundamental group of a surface coming from instantons" will take place in the Luigi Stasi Lecture Room, on Tuesday 28 June at 16.00 hrs and will be followed by light refreshments.
 

Tomasz Mrowka is Professor of Mathematics. A graduate of MIT, he received the Ph.D. from U.C. Berkeley in 1988 under the direction of Clifford Taubes and Robin Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford and at Caltech (professor 1994-96). Mrowka's research interests focus on problems in differential geometry and gauge theory. In 2007 he received the Veblen Prize in Geometry by the AMS, jointly with Peter Kronheimer, "for their joint contributions to both three- and four- dimensional topology through the development of deep analytical techniques and applications." Their book, Monopoles and Three Manifolds (Cambridge University Press) also garnered the 2011 Joseph Doob Prize of the AMS. He was appointed Singer Professor of Mathematics from 2007 to 2017. In 2018 he gave a plenary address at ICM18 in Rio de Janeiro. He is a Fellow of the American Academy of Arts & Sciences (2007) and Member of the National Academy of Sciences (2015).

Abstract: The space of (conjugacy classes of) representations of the fundamental group of Riemann surface (sometime with punctures) into SU(2) (and other Lie groups) has been the focus of study in mathematics and physics for many years. Besides having a concrete topological description these spaces have other incarnations.

• In algebraic geometry as moduli spaces of certain stable rank 2 holomorphic vector bundles.

• In differential geometry as moduli spaces of flat connections in a principal bundle over the Riemann surface.

• In physics as the critical points of the Yang-Mills functional on connections in a principal bundle over the Riemann surface.

There is a rich story describing the cohomology rings of these spaces in Instanton Floer homology (as does quantum cohomology) provides natural deformations of these ring structures. When the surface has punctures these deformations coming in families that appear at first hard to describe. This talk with review some of the history of this story. Each of the incarnations of the representation space play an important role as evidenced by the large and varied group of mathematicians and physicists that have contributed to the this study over the years. At the end some we’ll give some hints at some of the ideas behind computing these further families of deformations.

Light refreshments will be served. All are welcome to attend.