BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Matthew Hase-Liu (MIT) DTSTART:20261020T203000Z DTEND:20261020T213000Z DTSTAMP:20261023T093552Z UID:MITNT/145 DESCRIPTION:Title: A new and improved version of the circle method over function fields\n by Matthew Hase-Liu (MIT) as part of MIT number theory seminar\n\nLecture held in Room 2-449 in the Simons Building (building 2).\n\nAbstract\nBirch 's theorem is a classical result about counting rational points on smooth hypersurfaces with many variables\; its function field analogue counts rat ional curves on them. I'll explain a new variant of the circle method that does the same (and works just as well!) for maps from any projective vari ety $Y$ to a smooth hypersurface $X$. There are many arithmetic and geomet ric applications: Over finite fields\, this gives an higher dimensional an alogue of Birch's theorem for $X$ over function fields. Over the complex n umbers\, this proves that the moduli spaces of maps $Y$ to $X$ are irreduc ible of the expected dimension. The key input is to stratify the "circle" differently by an unusual invariant called the multiplication rank and a d egeneration result from the theory of Gröbner bases. Many parts of this a re joint with Jakob Glas.\n LOCATION:https://researchseminars.org/talk/MITNT/145/ END:VEVENT END:VCALENDAR