Spring 2025
The Seminar runs on Tuesdays at STRAND BLDG S4.29. This term, some talks will run from 15:00 to 16:00, and others will be in two parts – an introductory talk from 15:00 to 15:40 and a research talk from 15:45 to 16:30.
29 April, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Alapan Mukhopadhyay (EPFL)
Title: Frobenius and Homological Algebra
Abstract: The Frobenius endomorphism or the p-th power map is crucial in defining singularity classes in characteristic p > 0, especially those appearing in the birational classification of algebraic varieties. On the other hand, the obstruction to smoothness is homological, according to a celebrated theorem of Serre. In this talk, we will show that the Frobenius endomorphism witnesses this homological obstruction to smoothness. This provides an explanation for the effectiveness of Frobenius in detecting singularities, from a homological point of view. The key will be to produce (explicit) generators of the bounded derived category of a variety in characteristic p > 0 from perfect complexes using the Frobenius pushforward functor. Our results recover earlier characterizations of smoothness using Frobenius- such as Kunz’s theorem. Time permitting, we will discuss examples hinting relationship between generators of derived categories and the global geometry of the underlying projective variety. Part of the talk will report a joint work with Matthew Ballard, Patrick Lank, Srikanth Iyengar and Josh Pollitz.
13 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Francesca Tripaldi (University of Leeds)
Title: Extracting subcomplexes in the subRiemannian setting
Abstract: On subRiemannian manifolds, the de Rham complex is not the ideal candidate to use to carry out geometric analysis. However, special subcomplexes have successfully been applied in very specific settings, such as Heisenberg groups and the Cartan group. I will give an overview of different techniques used to obtain such subcomplexes, as well as point out their limitations when used on arbitrary Carnot groups, and a possible way to overcome them.
20 May, 15:00-16:00, STRAND BLDG S4.29
Speaker: Shengwen Wang (Queen Mary University London)
Title: Phase transitions with Allen-Cahn mean curvature bounded in Lp.
Abstract: We consider the varifolds associated to a phase transition problem whose first variation of Allen-Cahn energy is Lp integrable with respect to the energy measure. We can see that the Dirichlet and potential part of the energy are almost equidistributed. After passing to the phase field limit, one can obtain an integer rectifiable varifold with bounded Lp mean curvature.
27 May, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Jef Laga (University of Cambridge)
Title: Torsors and polarisations on abelian varieties
Abstract: Abelian varieties are higher-dimensional generalisations of elliptic curves and are ubiquitous in algebraic geometry and number theory. Central to their theory is the concept of a polarisation. If $A$ is an abelian variety over an algebraically closed field, then every polarisation is represented by an ample line bundle on $A$. However, such a line bundle may not exist if the field is not algebraically closed, or when it is replaced by a more general base scheme; in fact, this failure already occurs for Jacobians.
In 1999, Poonen and Stoll asked: can every polarisation be represented by a line bundle on some torsor under $A$? In this talk, I will expand on the background for this question and I will explain why the answer is often ``yes’’, but not always. Along the way, we will encounter Mumford theta groups, Serre’s notion of negligible group cohomology and moduli spaces of abelian varieties.
3 June, 15:00-16:00 STRAND BLDG S4.29
Speaker: Nattalie Tamam (Imperial College London)
Title: Weighted singular vectors for multiple weights
Abstract: It follows from the Dirichlet theorem that every vector has `good’ rational approximations. Singular vectors are the ones for which the Dirichlet theorem can be infinitely improved. An (obvious) example of singular vectors are the ones lying on rational hyperplanes. We will discuss the existence of totally irrational weighted singular vectors on manifolds, and also ones with high weighted uniform exponent. We will also mention some invariance of weighted uniform exponents in the case of manifolds. The talk is based on a joint work with Shreyasi Datta.
10 June, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Adam Klukowski (University of Oxford)
Title: Recent progress towards the Putman-Wieland conjecture
Abstract: It is a known fact about Mapping Class Groups of surfaces that their abelianisation is (generically) finite. The Ivanov conjecture, open for more than 25 years, asks whether the same also holds for their finite-index subgroups. I will give a gentle introduction to Mapping Class Groups, and review my recent work with Vlad Marković towards this question. Some tools that we use include a reformulation due to Putman-Wieland, Magnus embedding, and spectra of Riemann surfaces.
17 June, Part I: 15:00-15:40, Part II: 15:45-16:30, STRAND BLDG S4.29
Speaker: Owen Patashnick (King’s College London)
Title: On constructing motives via algebraic cycles
Abstract: Grothendieck was inspired to come up with the idea of a motive while trying to generalize the classical relationship between a curve and its Jacobian. Roughly speaking, Grothendieck hoped to show that the linear data associated to a variety could be measured by an associated abelian group. The most geometrically natural abelian structure one can associate to a variety is encapsulated in the construction of algebraic cycles, i.e. is the ring of formal sums of subvarieties, with product structure given by intersection product. Grothendieck himself defined the abelian (sub)category of pure motives via algebraic cycles, and it is reasonable to assume that he thought a generalization to all (mixed) motives could similarly be described in terms of algebraic cycles.
In this talk I will try to give a gentle introduction to a long term project to do just that; to describe the abelian category of mixed motives via algebraic cycles (in the way Grothendieck may have intended them to be understood). In particular, after a quick overview of algebraic cycles and Grothendieck’s construction of the (subcategory) of pure motives, I will attempt to show why it is natural to look for a description of the category of mixed motives via algebraic cycles, and in particular to highlight work in progress, joint with A Logan and A Torzewski, of an explicit and computable construction of the category of mixed abelian motives for a very general abelian variety.