## [10/31] The 5th WPI-AIMR Joint Seminar in FY2014

### The 5th WPI-AIMR Joint Seminar in FY2014

#### Title

**Gröbner bases of toric ideals and their application**

**Speaker**

**Prof. Hidefumi Ohsugi**

(Department of Mathematical Sciences, Kwansei Gakuin University)

**Date**

**October 31 (Fri.), 2014 16:00-17:00**

**Venue**

Seminar Room, 2nd floor, WPI-AIMR Main Bldg.

**Abstract**

Gröbner bases has a lot of application in many research areas, and is implemented in various mathematical software. The most basic application is an elimination of variables from a system of polynomial equations. See, e.g., [4]. In this talk, we discuss basic and recent developments in the theory of Gröbner bases of toric ideals. In 1990's, several breakthroughs on toric ideals were done:

- Conti--Traverso algorithm for integer programming using Gröbner bases of toric ideals (see [1]);
- Correspondence between regular triangulations [3] of integral convex polytopes and Gröbner bases of toric ideals (see [5]);
- Diaconis--Sturmfels algorithm for Markov chain Monte Carlo method in the examination of a statistical model using a set of generators of toric ideals (see [2]).

In this talk, starting with introduction to Gröbner bases and toric ideals, we study some topics related with breakthroughs above.

**References**

- P. Conti and C. Traverso, Buchberger algorithm and integer programming,
*In*Proceedings of AAECC-9 (New Orleans), pp.130--139. Springer LNCS 539, 1991. - P. Diaconis and B. Sturmfels, Algebraic algorithms for sampling from conditional distributions,
*The Annals of Statistics*, 26 (1998) 363--397. - I.M. Gel'fand, A.V. Zelevinskii, and M.M. Kapranov, Hypergeometric functions and toral manifolds,
*Functional Analysis and Its Applications*, 23 (1989) 94--106. - T. Hibi (ed.), ``Gröbner Bases: Statistics and Software Systems,'' Springer, 2013.
- B. Sturmfels, Gröbner bases of toric varieties,
*Tohoku Math*. J., 43 (1991) 249--261.

### Contact

General Affairs Section, AIMR Administrative Office

TEL : | +81-22-217-5922 |
---|---|

E-MAIL : | wpi-soumu@wpi-aimr.tohoku.ac.jp |