Manchester Number Theory Seminar - Joseph Harrison (Warwick)
| Dates: | 5 May 2026 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Joseph Harrison (Warwick)
Title: Sum-product phenomena in algebraic groups
Abstract: The cardinality of sumsets and product sets can be regarded as quantitative indicators of additive or multiplicative 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 multiplicatively 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 sumset of $A$ replaced by the sumset of polynomial images $\phi_1(A) + \dots + \phi_k(A)$.
I will report on joint work with Akshat Mudgal and Harry Schmidt, which generalises these three results, replacing the multiplicative group and the additive group with arbitrary commutative algebraic groups of dimension $1$, in characteristic zero. The open immersion $\mathbb{G}_m \to \mathbb{G}_a$, implicit in the work of Erd{\H o}s--Szemer{\' e}di and Bourgain--Chang, and the polynomial morphism $\mathbb{G}_m^k \to \mathbb{G}_a^k$ in the work of Mudgal, is replaced by an arbitrary algebraic correspondence. The proofs employ the work of David and Philippon on the uniform Mordell--Lang conjecture in products of elliptic curves, the work of Evertse, Schlickewei and Schmidt on the $S$-unit equation, and the recent work of Gowers, Green, Manners and Tao on the polynomial Freiman--Rusza conjecture.
Room: Frank Adams 1
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Frank Adams 1
Alan Turing Building
Manchester
