“The Emmy Noether Program enables me to attract colleagues to join this beautiful project and to build my own research group,” said Tingxiang Zou, who is delighted to have received the grant. “It also provides a platform to strengthen the connections between model theory and combinatorics.”
The Elekes-Szabó problem is a combinatorial problem with connections to geometry, algebra, model theory, and other areas of mathematics. Tingxiang Zou’s newly formed Emmy Noether group at the Mathematical Institute of the University of Bonn will study higher-dimensional versions of this problem.
A set of numbers cannot be highly structured in both an additive and multiplicative sense at the same time. This phenomenon is known as the “sum-product problem.” Let us consider, for example, a finite set of even positive natural numbers: 2, 4, 6, 8, 10. Apart from these numbers, there are only a few other possible sums of two numbers from this list (even allowing repeated selection): 12, 14, 16, 18, and 20. If we extend the sequence and add more even numbers, the number of possible sums continues to grow only moderately. As an arithmetic progression, the numbers exhibit a strong additive structure: An unusually large number of triples (x, y, z) with x, y, and z from the specified set are solutions to the equation x + y = z. In contrast, in addition to the numbers mentioned, the products 12, 16, 20, 24, 32, 36, 40, 48, 60, 64, 80, and 100 also occur. Thus, this set does not exhibit strong multiplicative structure. The opposite is true when looking at a geometric progression such as 2, 4, 8, 16, 32. Here, there is a strong multiplicative structure, but no significant additive structure.
“The Elekes–Szabó problem examines such phenomena in a more general framework,” explains Tingxiang Zou. “Instead of sums and products, one considers algebraic relations given by polynomial equations over the real or complex numbers.” Elekes and Szabó's central observation is that if such an algebraic equation has an unexpectedly large number of solutions within large finite grids, there must be – apart from certain degenerate cases – an underlying hidden algebraic group structure (such as addition or multiplication) that explains this behaviour. For example, if we consider finite sets A, B, and C of size n and an algebraic equation such as a*a + a*b = c, which is given by the polynomial P(x,y,z) = x*x + x*y - z, then there can be at most n*n = n^2 triples (a,b,c) with a from A, b from B, and c from C that satisfy this equation. In this case, however, it has significantly fewer solutions, and thus P(x,y,z) induces neither an additive nor a multiplicative structure. The situation is different for P(x,y,z) = x*x + y*y - z, where there are actually n^2 solutions. “If the number of solutions is very high, i.e., close to n^2, then this indicates that the polynomial essentially behaves like addition or multiplication,” says Tingxiang Zou. “We now want to investigate higher-dimensional variants of this problem in our research project.” Instead of finite sets of numbers, the scientists will then consider finite sets of tuples lying on higher-dimensional geometric objects. Here, too, the aim is to explain when algebraic equations have an unexpectedly large set of solutions in finite grids.
Scholars from around the world have been collaborating closely with the new Emmy Noether group, including Martin Bays of the University of Oxford, Jan Dobrowolski of Xiamen University Malaysia, and Yifan Jing of the Ohio State University. A host of new collaborations will also be initiated with leading researchers in the field, including Artem Chernikov of the University of Maryland and Ehud Hrushovski of the University of Oxford.
Bio
Tingxiang Zou studied philosophy at Peking University and then completed a master’s degree in logic in Amsterdam. As a doctoral student in mathematics, she conducted research at the Institut Camille Jordan at the University of Lyon from 2015 to 2019, and then worked at the Hebrew University of Jerusalem and the Mathematics Cluster of Excellence in Münster before coming to Bonn in early 2024 as a postdoc at the Mathematical Institute and an associate member of the Hausdorff Center for Mathematics (HCM—one of the currently eight Clusters of Excellence at the University of Bonn). She will lead an Emmy Noether research group starting in September 2026, with initial funding of up to 850,000 euros granted by the German Research Foundation. The initial project term is three years, with a possible three-year extension tied to another 710,000 euros of grant funding, subject to interim evaluation and approval
Emmy Noether Program
The Emmy Noether Program, backed by the German Research Foundation (DFG), is designed to create opportunities for high-caliber early-career researchers to qualify as a full university professor within six years by leading their own a research group.