Mathematical Sciences: Algebraic Combinatorics

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Fomin This award funds Professor Sergey Fomin's exploration of new relations between combinatorics and algebra. A new approach to the theory of Schubert and Grothendieck polynomials based on exponential solutions to the Yang-Baxter equations has suggested new developments in the combinatorics of Coxeter groups. This work will build on this via symmetric functions, local associative algebras and universal algebra. This research falls in the broad category of combinatorics, which is one of the most active fields in today's mathematics. At its roots, combinatorics is the study of systematic counting. Counting can be incredibly difficult when the objects are difficult to list, and combinatorists look for general methods for overcoming these difficulties. Today's combinatorics makes use of a wide variety of the most advanced and modern mathematical techniques. Although its roots go back several centuries, the field has had an explosive development in the past few decades. This growth comes from its importance in communications and information technology and from the success of modern techniques to problems of counting. ***