14th International Symposium on Natural Sciences

14th International Symposium on Natural Sciences [Homepage, Poster, Timetable]

The 14th international symposium on natural sciences is an annual event of the research institute of basic sciences at Incheon national university. It is held on Oct 6 (Thu) ~ 7 (Fri), 2022 via zoom.  (Registration required for Zoom link or please contact me by ilseungjang@inu.ac.kr)


Invited Speakers:

Jang Soo Kim (Sungkyunkwan University), 

Myungho Kim (Kyung Hee University),

Travis Scrimshaw (Hokkaido University).


Titles and abstracts (* Please note that the timetable is based on UTC (+9)):

The Gordon-Bender-Knuth identities are determinant formulas for the sum of Schur functions of partitions with bounded height, which have interesting combinatorial consequences such as connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this talk we give an affine analog of the Gordon-Bender-Knuth identities, which are determinant formulas for the sum of cylindric Schur functions. We also consider combinatorial aspects of these identities. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and r-noncrossing and s-nonnesting matchings. This is joint work with JiSun Huh, Christian Krattenthaler, and Soichi Okada.


 2. Myungho Kim (Kyung Hee University) 10:50~11:50 Oct 6 (Thu)

The category of finite-dimensional (graded)-modules over the quiver Hecke algebra is a monoidal category whose Grothendieck ring is isomorphic to the (quantum) coordinate ring C[N] of the maximal unipotent subgroup N. If one localizes the ring C[N] at a set of unipotent minors, then one obtains the coordinate ring of the unipotent cell, which is the intersection of N with the maximal Schubert cell.  Thus it is desirable to obtain a localization of the category of modules over quiver Hecke algebra which recovers the above localization of the coordinate ring.  We develop a general localization process for k-linear abelian monoidal categories via family of simple objects.  As an example, we obtain the desired localization for the module category of quiver Hecke algebra. This is joint work with Kashiwara, Oh, and Park.


3. Travis Scrimshaw (Hokkaido University) 13:30~14:30 Oct 6 (Thu)

While surveying canals in 1834, John Scott Russell noticed that the waves in the canal kept its shape as it traveled. From this, he did a number of experiments that constructed waves that did not fit into the theory of waves at the time. This led to the development of the Korteweg-de Vries (KdV) equation, which is a nonlinear partial differential equation that can be solved exactly using a technique known as the inverse scattering transform. In particular, any solution separates out into solitary waves the Russell observed in his experiments, which are known as solitons. In 1990, Takahashi and Satsuma developed an ultradiscrete version of the KdV equation called the box-ball system. In the box-ball system, we have a finite number of balls in an infinite number of boxes in a line that can hold at most one ball with the time evolution given by a simple process. Collections of balls grouped together become solitons and exhibit the same behaviors as the KdV equation solutions. Furthermore, by considering states in the box-ball system as collections of particles with different states, this becomes an example of a Heisenberg spin chain that can be solved by using the quantum inverse scattering transform and leads to many generalizations. In this talk, we will discuss in more detail the history of the box-ball system, the combinatorial meaning of quantum scattering, and some of the generalizations and further connections with quantum mechanics.