Geometry, Combinatorics, and Integrable SystemsAutumn 2017Time: Thursdays 34pmLocation: MA 317 

August 31
Thurs, 3pm  Max Glick
(OSU) 
The limit point of the pentagram map 
September 7
Thurs, 3pm 


September 14
Thurs, 3pm 


September 21
Thurs, 3pm  Yuancheng Xie
(OSU) 
Generalized hypergeometric functions on the Grassmannian and integrable systems of hydrodynamic type 
September 28
Thurs, 3pm  Ed Richmond
(Oklahoma State) 
Pattern avoidance and fiber bundle structures on Schubert varieties 
October 5
Thurs, 3pm  Tair Akhmejanov
(Cornell) 
TBA 
October 12
Thurs, 3pm  (fall break)


October 19
Thurs, 3pm MW 154  Cristian Lenart
(Albany) 
TBA 
October 26
Thurs, 3pm 


November 2
Thurs, 3pm 


November 9
Thurs, 3pm 


November 16
Thurs, 3pm 


November 30
Thurs, 3pm 

(Glick): The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the first paper on the subject, R. Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials.
(Xie): In this talk I will describe a connection between generalized hypergeometric functions on the Grassmannian and integrable systems of hydrodynamic type. This talk consists of the following four parts: (1) Symmetries of hydrodynamictype systems in Riemann invariant form; (2) Generalized hypergeometric functions defined on the Grassmannian and their confluences; (3) Integrable hydrodynamictype systems and confluences of Lauricellatype hypergeometric functions; (4) Some preliminary results on generalizations to the Grassman Gr(r,n) with r > 2. A special case of the compatibility conditions of hydrodynamictype systems in Riemann invariant form leads to EulerPoissonDarboux(EPD) system. This system admits Lauricella type hypergeometric functions as solutions, which are special cases of AomotoGel'fand hypergeometric functions defined on the Grassman Gr(r, n) with r=2. In this way each Lauricella type hypergeometric functions gives rise to a hierarchy of integrable systems of hydrodynamic type in Riemann invariant form. The confluences of the classical Gauss hypergeometric functions can be generalized to AomotoGel'fand hypergeometric functions. The confluences of a generalized Lauricella type hypergeometric function also produce hierarchies of integrable systems of hydrodynamic type which are not necessarily in Riemann invariant form anymore. In this talk I will describe these constructions and connections and also some preliminary results on generalizations to the Grassman Gr(r,n) with r>2.
(Richmond): In this talk, I will discuss joint work with Tim Alland where we give a pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. To do this, we introduce the notion of split pattern avoidance and show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns 312 and 231. Continuing, we also characterize when a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties in terms of usual pattern avoidance.