
Update: There will be an additional seminar talk by Peter Crooks as follows.
5 April 2019 (Fri) 16:3017:30 "Hessenberg varieties, Slodowy slices, and KostantToda lattices", Peter Crooks (Northeastern University, USA) Room 511708
ABSTRACT: Kostant's Lietheoretic realization of the (open) Toda lattice has given rise to a fascinating hybrid of ideas from symplectic geometry, algebraic geometry, and representation theory. A modern example is the appearance of this KostantToda lattice in calculations related to the quantum cohomology of the flag variety. It is in this setting that one compactifies the leaves of the KostantToda lattice, thereby constructing a certain class of Hessenberg varieties. This motivates the possibility of defining the KostantToda lattice on (the total space of) a family of Hessenberg varieties. I will explain how this is accomplished, defining all relevant objects along the way. I will also emphasize the roles played by Slodowy slices and MishchenkoFomenko polynomials. This represents joint work with Hiraku Abe.
SCHEDULE OF TALKS:
10:0011:00 Hiraku Abe (Osaka Prefectural University), "Hessenberg varieties"
Abstract: Hessenberg varieties are subvarieties of the full flag variety, and they form a relatively new research subject which was introduced by De MariProcesiShayman around 1990. Particular examples are the flag variety itself, the Springer fibres, the Peterson variety, and the permutohedral variety. Similarly to Schubert varieties, it has been found that geometry, combinatorics, and representation theory interact nicely on Hessenberg varieties. Moreover, in joint work with Peter Crooks, we found that a family of certain Hessenberg varieties forms a Poisson variety with an open dense symplectic leaf and it admits a completely integrable system related to the KostantToda lattice. In this talk, I will give a brief survey on Hessenberg varieties.
tea/coffee
11:15:12:15 Peter Crooks (Northeastern University), "Invarianttheoretic perspectives on the KostantToda lattice"
Abstract: Toda lattices play a distinguished role in both the classical and modern theories of completely integrable systems, and they are fruitfully studied at the interface of symplectic geometry and representation theory. One crucial aspect of this study is Kostant's Lietheoretic realization of the (open) Toda lattice, which one sometimes calls the KostantToda lattice. This construction invokes Kostant's prior works on invariant theory, especially his results on regular Slodowy slices and the structure of the adjoint quotient. I will discuss invarianttheoretic aspects of the KostantToda lattice, emphasizing two recent constructions. The first is a partial compactification of the KostantToda lattice by means of Hessenberg varieties, Slodowy slices, and MishchenkoFomenko algebras, and it represents joint work with Hiraku Abe. The second construction is a Todatype integrable system on the socalled universal centralizer, a hyperkähler manifold arising in certain representationtheoretic contexts.
lunch
14:3015:30 Martin Guest (Waseda University), "Hamiltonian aspects of the tt*Toda equations"
Abstract: The 2D topologicalantitopological fusion equations were introduced by Cecotti and Vafa in the 1990's. They are a system of "integrable" nonlinear pde with rich connections to geometry, e.g. Frobenius manifolds and quantum cohomology, but they are difficult to solve. I will review a special case of the 2D tt* equations, the tt*Toda equations, where computations are feasible. Already here, interesting links with Lie theory arise  for example the Coxeter Plane, classical work of Kostant and Steinberg, and the universal centralizer. I will mention joint work with NanKuo Ho (National TsingHua University), and related work by Ryosuke Odoi (Waseda University).
tea/coffee
16:0017:00 Shinsuke Iwao (Tokai University) "Tropical KP equation and Young tableaux"
Abstract: In the early 2000s, Kirillov and NoumiYamada established the ''tropical approach'' to the combinatorial problems on Young tableaux. Later, Mikami and KatayamaKakei discovered some relation between the Tropical KP equation and Young tableaux, which is independent of the results of NoumiYamada. It can be proved that, by using their results nicely, simpler alternative proofs of fundamental theorems about combinatorics (the uniqueness of rectification, Shape Change Theorem, etc) are given. In this talk, I will give an explanation about this method for the case of "the uniqueness of rectification".
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Organizing committee: Hiraku Abe (Osaka Prefectural University), Martin Guest (Waseda University)
Supported by JSPS GrantinAid for Scientific Research 18H03668 (Martin Guest) and GrantinAid for EarlyCareer Scientists 18K13413 (Hiraku Abe).