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 M3//EN
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BEGIN:VEVENT
DTSTAMP:20260410T091318Z
DTSTART:20260505T140000Z
DTEND:20260505T150000Z
SUMMARY:Manchester Number Theory Seminar - Joseph Harrison (Warwick)
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}l1u6-mmpi71
 96-neonz3
DESCRIPTION:Speaker: Joseph Harrison (Warwick)\n\nTitle: Sum-product phen
 omena in algebraic groups\n\nAbstract: The cardinality of sumsets and pr
 oduct sets can be regarded as quantitative indicators of additive or mul
 tiplicative structure. Erd{\\H o}s and Szemer{\\' e}di proved that a set
  of integers cannot have both a small sumset and a small product set. In
  other words\, a set of integers cannot be both additively and multiplic
 atively structured. Bourgain and Chang proved that cardinality of the $k
 $-fold sumset or $k$-fold product set of $A$ exceeds any power of $|A|$\
 , as $k$ grows. Mudgal proved that the same is true with the $k$-fold su
 mset of $A$ replaced by the sumset of polynomial images $\\phi_1(A) + \\
 dots + \\phi_k(A)$.\n\nI will report on joint work with Akshat Mudgal an
 d Harry Schmidt\, which generalises these three results\, replacing the 
 multiplicative group and the additive group with arbitrary commutative a
 lgebraic groups of dimension $1$\, in characteristic zero. The open imme
 rsion $\\mathbb{G}_m \\to \\mathbb{G}_a$\, implicit in the work of Erd{\
 \H o}s--Szemer{\\' e}di and Bourgain--Chang\, and the polynomial morphis
 m $\\mathbb{G}_m^k \\to \\mathbb{G}_a^k$ in the work of Mudgal\, is repl
 aced by an arbitrary algebraic correspondence. The proofs employ the wor
 k of David and Philippon on the uniform Mordell--Lang conjecture in prod
 ucts of elliptic curves\, the work of Evertse\, Schlickewei and Schmidt 
 on the $S$-unit equation\, and the recent work of Gowers\, Green\, Manne
 rs and Tao on the polynomial Freiman--Rusza conjecture.\n\nRoom: Frank A
 dams 1
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 1\, Alan Turing Building\, Manchester
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