Igor Klep (University of Ljubljana）による数学・数理科学グローバル特別講義２が下記の要領で開催されます。履修されている方は下記のGoogleフォームのから申込をお願いします。

講師： Igor Klep (University of Ljubljana、作用素環論分野)

講義日程：2024年 5月20日 10:00-12:00

5月21日～24日 15:00-17:00

場所: 理学研究科 ３号館 1階110室

タイトル：Noncommutative analysis and real algebraic geometry

アブストラクト：

Ever since Gauss it was known that a positive univariate real polynomial can be written as a sum of two squares of real polynomials. In a similar spirit, positive semidefinite quadratic forms (in any number of variables) are sums of squares of linear forms. During his 1885 PhD thesis defense Minkowski got into an argument with Hilbert about whether an extension to higher degree forms holds true (i.e., is every positive polynomial a sum of squares (sos) of polynomials?), thus providing a common generalization of the above two observations. A few years later Hilbert answered this in the negative; his proof was highly non-constructive, and the first explicit example of a positive polynomial that is not sos was given only 80 years later by Motzkin in 1967. Hilbert also posited that positive bivariate polynomials are sums of squares of rational functions, leading him to include the following problem (as #17) among the famous 23 problems for his address to the 1900 International Congress of Mathematicians:

Is every positive polynomial a sum of squares of rational functions?

A positive solution was presented in 1926 by Artin who developed the theory of formally real fields to solve this problem; we call this the beginning of real algebraic geometry (RAG). Nowadays real algebraic geometry is the branch of algebraic geometry studying real algebraic sets, i.e., real-number solutions to systems of polynomial equations. Pillars of RAG are generalizations of the above mentioned theorem of Artin, the so-called Positivstellensätze (=certificates of positivity):

Given polynomials p and q, is p positive where q is positive?

On the other hand, many problems in quantum physics or linear systems design in control theory have matrices as variables, and the formulas naturally contain noncommutative polynomials in matrices. Analyzing such problems has led to the development of a noncommutative (nc) real algebraic geometry. Often, the qualitative properties of the noncommutative case are much cleaner than those of their scalar counterparts. Indeed, the relaxation of scalar variables by matrix variables in several natural situations results in a beautiful structure.

This series of lectures will start by presenting classical results before moving on to newer modern results and techniques in noncommutative analysis and RAG. A selection of the following topics and their applications will be presented: noncommutative sums of squares (Helton's sum of squares theorem; McCullough's factorization theorem; etc.), convexity (linear matrix inequalities and their connection to complete positivity and dilation theory), noncommutative analytic maps (continuous implies analytic; Ax-Grothendieck theorem, nonlinear completely positive maps).

参加申込用URL：https://forms.gle/nEepdhNPJHn5spsv7

締切日： 5月15日 （水）

※数学・数理科学グローバル講義Ⅰは数学・数理科学イノベーション人材育成強化コースにおける中核科目です。

※数学・数理科学グローバル講義Ⅰを履修するにはKULASISでの履修登録が必要です。

（前期は4月17日、18日が登録期間）

数学・数理科学グローバル講義Ⅰでは6名の講師による特別講義が開講されます。

※聴講（本学学生に限る）も可。