The conference entitled Geometry and Topology in Low Dimensions explores several aspects of one of the popular research areas of mathematics.
June 22-26, Erdős Center of Rényi Institute will organize a conference at the Hungarian Academy of Sciences on the topology and geometry of low-dimensional manifolds. Approximately 250 outstanding Hungarian and international researchers have indicated their participation in the event, the detailed programme of which is available HERE and HERE. The speakers come from the world's leading universities, including the Massachusetts Institute of Technology, Harvard University, the University of Cambridge and Princeton University.
The five-day conference focuses on manifolds, that is, mathematical objects that, in simplified terms, may also be called “shapes”. Manifolds include such well-known objects as curves (one-dimensional manifolds) and surfaces (two-dimensional manifolds). The conference focuses primarily on three- and four-dimensional manifolds, which are still considered “low-dimensional” objects in mathematics, as opposed to those of dimension five or higher.
Three- and four-dimensional manifolds exhibit a number of surprising properties that do not appear in higher dimensions, and this is why research in this area remains extremely active today. (A good example of this is that in the 1980s two mathematicians, Simon Donaldson and Michael Freedman, were both awarded the Fields Medal, one of the most prestigious distinctions in mathematics, for their breakthrough results in the understanding of four-dimensional manifolds.)
A remarkably wide variety of mathematical methods is available for the study of three- and four-dimensional manifolds (shapes), and these methods are connected to many other areas of mathematics. The organizers of the conference selected five such topics and dedicated each day of the week to one of them. Accordingly, each day of the conference begins with a panorama talk, in which distinguished mathematicians present a particular area of low-dimensional topology. These lectures provide an overview of the current state of the field, the latest developments, and the open questions that remain. Such lectures will be given by Daniel Ruberman (Brandeis University), Gordana Matić (University of Georgia), Peter Ozsváth (Princeton University), Peter Kronheimer (Harvard), and Maggie Miller (University of Texas at Austin).
Furthermore, each day consists of further lectures within the theme of the panorama talk, discussing new results, in which internationally renowned researchers report on the latest developments. In addition, every day three early-career researchers – PhD students or postdoctoral researchers – will present their own work in short talks related to the topic of the day. The lectures of Lisa Piccirillo (MIT), İnanç Baykur (University of Massachusetts at Amherst) and Maggie Miller will contain particularly spectacular illustrations.
One of the outstanding events of the conference will be the so-called Clay Lecture of Professor John Morgan from Columbia University in the United States, entitled Physics, Loop Groups, and Low-Dimensional Topology. Morgan achieved results of fundamental importance in the study of four-dimensional manifolds through gauge theory, in particular through the application of Seiberg–Witten invariants. In his lecture, he will speak about the interaction between mathematics and physics in the development of three- and four-dimensional topology. He will highlight two particularly remarkable examples: gauge theory and the Jones polynomial.
The conference is supported partly by the Clay Mathematics Institute, and therefore it will include a lecture that is traditionally delivered by an outstanding mathematician of the field. The Clay Lecturer at the conference is American mathematician John Morgan, Professor Emeritus at Columbia University as well as a member and former founding director of the Simons Center for Geometry and Physics at Stony Brook University.
The Clay Mathematics Institute (CMI) is a global organization, a private foundation whose purpose is “to further the beauty, power and universality of mathematical thought.” Its mission is to increase and disseminate mathematical knowledge, inform the scientific community about new mathematical discoveries, encourage gifted young people to pursue careers in mathematics, and recognize outstanding achievements in mathematical research.  The Clay Mathematics Institute is best known for announcing the seven Millennium Prize Problems, that is, the seven mathematical questions regarded as the most important challenges of the third millennium. So far, only one Millennium Problem, the Poincaré Conjecture, has been solved, by Grigori Perelman. John Morgan played a significant role in verifying Perelman's proof and was also co-author of the book that presented the proof in greater detail than the original publications. The Millennium Problems were selected by CMI's Scientific Advisory Board, and by decision of its Board of Directors, CMI is prepared to award one million US dollars for the solution of each problem. |
The conference will also feature two public lectures intended for a broader mathematical audience. One of them is the above-mentioned lecture by John Morgan, while the other is Peter Ozsváth's lecture on Heegaard Floer homology. Peter Ozsváth is an American mathematician of Hungarian descent, known among topologists primarily for the development of Heegaard Floer homology together with Zoltán Szabó, who also has Hungarian roots and is currently based at Princeton. This is an extremely powerful tool that has fundamentally transformed research in low-dimensional topology with respect to the study of three-dimensional manifolds, knots and links.
