Amitsur Subgroups in Arithmetic and Equivariant Geometry
Abstract: For a variety defined over a field, a line bundle over the algebraic closure may fail to descend to a line bundle on the original variety. Similarly, if there is a group action, that action may fail to lift to certain line bundles. The Amitsur subgroup precisely measures this failure, taking values in the Brauer group of the field (or in an equivariant analog). A key property of the Amitsur subgroup is that it is an (equivariant) birational invariant. When all overfields and subgroups are considered, it is fine enough to completely distinguish between varieties in several interesting classes. I will discuss how the language of Mackey functors can be used to describe and compute the Amitsur subgroup, especially in the cases of Fano varieties, toric varieties, and torsors of tori.
Event Type: Seminar. Research Area: Algebra and Number Theory. Location: MSC W303. Speaker Name: Alexander Duncan. Speaker Institution: University of South Carolina.
Tuesday, September 23, 2025, 4:00 PM – 5:00 PM.
Exponential upper bounds on induced Ramsey numbers
Abstract: The induced Ramsey number $R_{ind}(H;r)$ of a graph $H$ is the minimum number $N$ such that there exists a graph with $N$ vertices for which all $r$-colourings of its edges contain a monochromatic induced copy of $H$. We show the exists an absolute constant $C > 0$ such that, for every graph $H$ on $k$ vertices, these numbers satisfy $R_{ind}(H;r) ≤ r^{Crk}$. When $r = 2$, this resolves a conjecture of Erdős from 1975. For $r > 2$, it answers a question of Conlon, Fox and Sudakov in a strong form.
This is joint work with Lucas Aragão, Gabriel Dahia, Rafael Filipe and João Marciano.
Event Type: Seminar. Research Area: Discrete Math and Combinatorics. Location: MSC W301. Speaker Name: Marcelo Campos. Speaker Institution: IMPA. Speaker Website: https://marceloscampos.github.io.
Thursday, September 25, 2025, 4:00 PM – 5:00 PM.
From Teichmüller to Shoen–Yau: Extremal mappings between Riemann surfaces
Abstract: There are two now classical descriptions of the moduli space of a Riemann surface via the theory of extremal mappings. The first from Teichmu ̈ller in the 1940s (rigorously es- tablished by Ahlfors in 1953) and through the existence of extremal quasiconformal mappings. The second is through Schoen-Yau’s existence theory for unique harmonic diffeomorphisms in the 1970s, and developed into a theory of moduli by many, including Wolf, Tromba and Wolpert many years later. The important ingredient in both is the existence of a holomorphic quadratic differential, from the Beltrami coefficient of an extremal quasiconformal mapping (Teichmu ̈ller) or from the Hopf equation (Harmonic). These quadratic differentials define the cotangent space to the moduli space. Here we show that in fact both of these approaches are manifestations of the same theory (that of existence of diffeomorphic extremal mappings of finite distortion) in limiting regimes. We identify parameterised families of moduli spaces (Beltrami coe…
Event Type: Seminar. Research Area: Analysis and Differential Geometry. Location: MSC E408. Speaker Name: Professor Gaven Martin. Speaker Institution: Massey University (New Zealand).
Wednesday, October 22, 2025, 4:00 PM – 5:00 PM.