BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Jinbo Ren (University of Virginia) DTSTART:20210521T020000Z DTEND:20210521T030000Z DTSTAMP:20260423T040710Z UID:POINTS/20 DESCRIPTION:Title: Some applications of Diophantine Approximation to Group theory\nby Jin bo Ren (University of Virginia) as part of POINTS - Peking Online Internat ional Number Theory Seminar\n\n\nAbstract\nTranscendental Number Theory te lls us an essential difference between transcendental numbers and algebrai c numbers is that the former can be approximated by rational numbers ``ver y well’’ but not the latter. More specifically\, one has the following Fields Medal work by Roth. Given a real algebraic number $a$ of degree $\ \geq 3$ and any $\\delta>0$\, there is a constant $c=c(a\,\\delta)>0$ such that for any rational number $\\eta$\, we have $|\\eta-a|>c H(\\eta)^{-\\ delta}$\, where $H(\\eta)$ is the height of $\\eta$. Moreover\, we have Sc hmidt’s Subspace theorem\, a non-trivial generalization of Roth’s theo rem.\n \nOn the other hand\, we have the notion of Bounded Generation in G roup Theory. An abstract group $\\Gamma$ is called Boundedly Generated if there exist $g_1\,g_2\,\\cdots\, g_r\\in \\Gamma$ such that $\\Gamma=\\lan gle g_1\\rangle \\cdots \\langle g_r\\rangle$ where $\\langle g\\rangle$ i s the cyclic group generated by $g$. While being a purely combinatorial pr operty of groups\, bounded generation has a number of interesting conseque nces and applications in different areas. For example\, bounded generation has close relation with Serre’s Congruence Subgroup Problem and Marguli s-Zimmer conjecture.\n \nIn my recent joint work with Corvaja\, Rapinchuk and Zannier\, we applied an ``algebraic geometric’’ version of Roth an d Schmidt’s theorems\, i.e. Laurent’s theorem\, to prove a series of r esults about when a group is boundedly generated. In particular\, we have shown that a finitely generated anisotropic linear group over a field of c haracteristic zero has bounded generation if and only if it is virtually a belian\, i.e. contains an abelian subgroup of finite index.\n \nIn my talk \, I will explain the idea of this proof and give certain open questions.\ n\nZoom ID: 854 7383 4027\n\nPassword: 562471\n LOCATION:https://researchseminars.org/talk/POINTS/20/ END:VEVENT END:VCALENDAR