Joint meeting with the Calf Seminar -- Contributed Talks By PhD Students
Schedule (8 September, Tuesday)
14.00 - 14.15: Adrian Cook (Edinburgh)
14.15 - 14:30 Andreas Gieringer (Mainz)
14.30 - 14.45 Nikhil Ken (Florence)
14.45 - 15.00 Sae Koyama (Cambridge)
15.00 - 15.15 Raphael Picovschi (Paris-Saclay)
15.15 - 15.30 Break
15.30 - 15.45 Otto Schmidt (Leipzig)
15.45 - 16.00 Nicolas Seroux (Paris-Saclay)
16.00 - 16.15 Daniel Green Tripp (Bristol)
16.15 - 16.30 Chunkai Xu (Warwick)
16.30 - 16.45 Jiangfan Yuan (Paris-Saclay)
Abstracts:
Let $X$ be a Gorenstein-Fano variety over $\mathbb{C}$. An important invariant of $X$ is the slope-stability of its tangent bundle, $T_X$, as this property is related to the existence of K\"ahler-Einstein metrics, K-stability, and Mori contractions of the variety in question. One can construct a new Gorenstein-Fano variety from $X$ by taking its projective cone. We show that varieties constructed this way cannot have stable tangent sheaves with respect to their anticanonical polarization. As an application of this result, we show that toric Gorenstein-Fano varieties with Fano index in a given range do not have semi-stable tangent sheaves with respect to their anticanonical polarization.
Algebraic K-theory is a cohomological invariant of schemes with numerous interesting qualities: Tied to the classification of vector bundles by its definition, K-theory also exhibits connections to several other properties. E. g. for many classes of rings R it is true that R is regular if and only if the algebraic K-theory of R and that of its polynomial ring R[T] agree.
The analogous statement for schemes and their affine line, however, is false. Instead, the relevant geometric objects are non-trivial line bundles associated to ample invertible sheaves (rather than the affine line).
The talk will provide a short introduction to algebraic K-theory and an overview of its relation to regularity of rings and schemes. Time permitting, we conclude by highlighting the central ingredient to prove the last statement for arbitrary equicharacteristic schemes, which is a new result for schemes not of finite type over a field of characteristic zero.
The Euclidean distance degree counts complex critical points of the squared distance to an algebraic variety. For complex varieties equipped with a Hermitian metric, the critical equations involve conjugation, leading to an actual chamber-dependent count and, after replacing conjugate variables by independent ones, a virtual algebraic count.
I will introduce these Hermitian distance invariants and discuss two structured examples. For unitary-invariant matrix varieties, Hermitian critical points reduce to real Euclidean critical points on the singular-value variety. For the Veronese variety, the doubled equations become a fixed-point problem on projective space, yielding a closed form for the virtual algebraic count.
Following work joint with Alessio-Cela on “well-spaced” invariants in toric varieties, we compare these invariants to the associated logarithmic Gromov-Witten invariants in toric three-folds. We discuss how these invariants may be calculated in practice.
Using theorems of Hennecart-Davison-Schlegel-Mejia, as well as methods of Joyce, I prove that certain tautological classes in the COHA of a 2CY category are primitive in the sense of a recent paper of Davison-Kinjo-Schiffmann-Vasserot. This allows me to generalize the methods of the paper « P=W via H2 » of Hausel-Schiffmann-Mellit-Minets, with two applications : a streamlining of their proof (skipping the final step), and the proof of a chi independance statement of relations between tautological classes in the COHA.
In this talk we consider tensor train varieties. These are varieties of tensors arising in a range of fields, including quantum many-body physics and machine learning. Using methods from integral geometry, we obtain a combinatorial expression for their degrees. We provide the ready-to-use julia package TTVarietyDegree.jl.
Given a three-manifold M and a complex reductive group G, the skein module Sk(M,G) is a topological invariant of M constructed from the structure of the quantum group associated to G, which provides a deformation quantization of the stack of G-local systems on M.
Assuming M is product M=CxS1 of a complex curve and a circle, the study of the skein module is reduced to the computation of cohomological Donaldson-Thomas invariants of the moduli stack of G-Higgs bundles on C. This is done via a process of dimensional reduction, and a version of the non-abelian Hodge correspondence.
This method leads to explicit computations of skein modules, and to non-trivial checks of Langlands duality for these three-manifolds. This is part of ongoing works with Ben Davison, Sarunas Kaubrys, Tasuki Kinjo and Pavel Safronov.
We study the cardinality of the fibre of a general point of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable intersection of their (flipped) Bergman fans. Our first main result is that the cardinality of a generic fibre for the Hadamard product of linear spaces is exactly the flip product of their matroids. We also provide a recursive algorithm for computing the flip product of any pair of matroids. This is a joint work with Oliver Clarke, Sean Dewar, Matteo Gallet, Georg Grasegger and Ben Smith.
Very recently, stability conditions on all smooth projective varieties are constructed by Chunyi Li. It is expected the stability conditions constructed by Li is near the 'large volume limit', which should be connected to the more classical Gieseker stability.
In this talk, I will present some results towards this direction. In particular, we will show that in some direction, the limit of heart indeed consists of coherent sheaves. And Limit Bridgeland stability implies Gieseker stability. If time permits, I will also talk about the difficulty in the direction 'Gieseker implies Limit'. Based on work in progress.
I will briefly recall the geometry of moduli stacks of coherent sheaves over curves/Simpson shapes/quiver varieties, which leads to the construction of cohomological/categorical Hall algebras in various setups. I will then explain some representation theoretical applications of them.
This session is organized by Hülya Argüz and Pierrick Bousseau, jointly with the Calf Team: Ines Chung-Halpern, Marc Truter, and Charlotte Satchwell.
For invited UK-based PhD students who will be speaking at the workshop:
The COW/Calf can cover the cost of travel for UK-based PhD students and postdocs who do not have their own source of travel funding. The COW is currently funded by the Isaac Newton Institute and the Heilbronn Institute for Mathematical Research under the UKRI/EPSRC Additional Funding Programme for Mathematical Sciences (EPSRC EP/V521917/1) and by the London Mathematical Society under a scheme 3 grant. For more information about what can be covered and how to make a claim, please see the COW/CALF webpage.