Workshop on

Hamiltonian systems

and Lie groups

Waseda University

Nishi-Waseda Campus, Building 55N, Meeting Room 2 (first floor, near Tully's Coffee)

6 April 2019


Update: There will be an additional seminar talk by Peter Crooks as follows.

5 April 2019 (Fri) 16:30-17:30 "Hessenberg varieties, Slodowy slices, and Kostant-Toda lattices", Peter Crooks (Northeastern University, USA) Room 51-17-08

ABSTRACT: Kostant's Lie-theoretic 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 Kostant-Toda lattice in calculations related to the quantum cohomology of the flag variety. It is in this setting that one compactifies the leaves of the Kostant-Toda lattice, thereby constructing a certain class of Hessenberg varieties. This motivates the possibility of defining the Kostant-Toda 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 Mishchenko-Fomenko polynomials. This represents joint work with Hiraku Abe.


10:00-11:00 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 Mari-Procesi-Shayman 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 Kostant-Toda lattice. In this talk, I will give a brief survey on Hessenberg varieties.


11:15:-12:15 Peter Crooks (Northeastern University), "Invariant-theoretic perspectives on the Kostant-Toda 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 Lie-theoretic realization of the (open) Toda lattice, which one sometimes calls the Kostant-Toda 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 invariant-theoretic aspects of the Kostant-Toda lattice, emphasizing two recent constructions. The first is a partial compactification of the Kostant-Toda lattice by means of Hessenberg varieties, Slodowy slices, and Mishchenko-Fomenko algebras, and it represents joint work with Hiraku Abe. The second construction is a Toda-type integrable system on the so-called universal centralizer, a hyperkähler manifold arising in certain representation-theoretic contexts.


14:30-15:30 Martin Guest (Waseda University), "Hamiltonian aspects of the tt*-Toda equations"

Abstract: The 2D topological-antitopological 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 Nan-Kuo Ho (National Tsing-Hua University), and related work by Ryosuke Odoi (Waseda University).


16:00-17:00 Shinsuke Iwao (Tokai University) "Tropical KP equation and Young tableaux"

Abstract: In the early 2000s, Kirillov and Noumi-Yamada established the ''tropical approach'' to the combinatorial problems on Young tableaux. Later, Mikami and Katayama-Kakei discovered some relation between the Tropical KP equation and Young tableaux, which is independent of the results of Noumi-Yamada. 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 Grant-in-Aid for Scientific Research 18H03668 (Martin Guest) and Grant-in-Aid for Early-Career Scientists 18K13413 (Hiraku Abe).