Rational Approximation of Systems of Linear Forms by Felipe Ramírez, Wesleyan University

Rational Approximation of Systems of Linear Forms by Felipe Ramírez, Wesleyan University, Friday December 1, 1:00 – 1:50pn, North Science Building 015, Wachenheim

ABSTRACT: A classical theorem of Dirichlet (1840s) tells us that for every real number $x$, we can find infinitely many rational numbers $p/q$ approximating $x$ to within $1/q^2$. Many modern results of Diophantine approximation are direct descendants of Dirichlet’s theorem. For example, Khintchine’s theorem (1924) tells us what happens if, instead of $1/q^2$, we want to approximate to within some other nonincreasing function of the denominator $q$. At this level of generality, it turns out that we can only make “metric” statements that hold for almost all—rather than all—real numbers $x$. I will survey these and other related results, and I will discuss recent work where, instead of real numbers, we are approximating systems of $m$ linear forms in $n$ variables, that is, $n$-by-$m$ matrices.

This talk is for colloquium credit.

We acknowledge support from the Finnerty Fund.