Seminars
Algebraic Geometry Seminar, 2025
Semester 1, 2025
Organizers: Jack Hall and Lance Gurney
Time: Wednesday, 10-11am, Peter Hall 162
May 28 Chris Hone (Sydney) Geometric extensions
The six functor formalism is a powerful tool for understanding the topology of algebraic varieties, and is especially suited to dealing with singular spaces. One such construction arising from this formalism is intersection cohomology, widely used throughout algebraic geometry and representation theory. In this talk I’ll introduce geometric extensions, formally defined objects in any suitable six functor formalism on algebraic varieties. These may be viewed as a generalisation of intersection cohomology to the setting of any suitable six functor formalism (eg, p-completed K theory), recovering the usual notion for constructible Q sheaves, and parity sheaves for finite coefficients on Schubert-type varieties. This work forms part of my PhD thesis, and is joint with Geordie Williamson (and the main theorem was also noted by Peter McNamara).
May 21 Gufang Zhao (Melbourne) Line Operators from a 3-Calabi–Yau Category
Starting from a symmetric 3-Calabi–Yau category equipped with additional structure, we construct a triangulated monoidal category with a weak braiding. When the 3-Calabi–Yau category arises from the root datum of a simple algebraic group, we compare the resulting category with the equivariant derived category of coherent sheaves on the affine Grassmannian—namely, coherent sheaves on the Hecke stack.
We further examine monoidal categories associated to an algebraic group together with a representation, known as categorified Coulomb branches, or equivalently, categories of line operators in 4d \mathcal{N}=2 gauge theories. Finally, we discuss the abelian heart of the perverse t-structure on coherent sheaves and its cluster structure as studied by Cautis and Williams.
This talk is based on ongoing joint work with Fujita, Yang, and Soibelman.
May 14 Will Troiani (Melbourne) Linear logic and the Hilbert scheme
In the recent pure maths seminar I presented a model of linear logic in a category of schemes where the interpretation of the exponential connective ‘!’ involved the Hilbert scheme. This work extended previous work by Murfet and Troiani which determined a commuting diagram where one of the arrows is given by the dynamics of linear logic, cut-reduction, and another arrow involving the Buchberger algorithm. This algorithm can be used to calculate images of closed maps, and so is of interest to algebraic geometers. It seems as though the extended model involving the Hilbert scheme also admits connections to Elimination Theory, and so in this talk I will present a particularly compelling example and compare it to the original theorem for the restricted logic.
May 7 Dougal Davis (Melbourne) Archimedean zeta functions, singularities, and Hodge theory
Given a holomorphic function f on a complex manifold, the Archimedean zeta function is a meromorphic distribution defined by analytic continuation of integrals of powers of f. Although difficult to compute, its poles are natural singularity invariants for the hypersurface {f = 0}; for example, the largest pole is the negative of the log canonical threshold. In this talk, I will explain joint work with András Lörincz and Ruijie Yang, in which we give tight relationships between the poles of the zeta function and several other singularity invariants, such as the minimal exponent, the Hodge ideals of Mustaţă-Popa, and the cohomology of Milnor fibres. Our main idea is to relate residues at these poles to polarisations in Sabbah and Schnell’s theory of complex Hodge modules (a complex-coefficient recapitulation of Saito’s mixed Hodge modules). This leads to simple proofs of more general results than previously available.
April 30 Scott Mullane (Melbourne) Residues and rigidity in moduli space
Teichmüller dynamics has been a fruitful source for defining algebraic cycles in the moduli space of pointed curves with interesting rigidity properties. These cycles usually come from projections of strata of differentials. In this talk, I will discuss work in progress that aims to extend this idea to the genus zero case where no such cycles exist, via linear manifolds in the strata defined by residue conditions.
April 23 NO SEMINAR
April 16 Bregje Pauwels (Macquarie) From local to global, via approximability
Given a module over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. In other words, every object in the derived category of a ring has a sequence of ‘simpler’ objects converging to it.
In general, a triangulated category is called approximable if this type of ‘approximation by simpler objects’ is possible. The notion of metrics and approximability in triangulated categories (due to Amnon Neeman) is fairly new and not very well understood yet. But the early evidence is that the new techniques, while at first sight purely categorical, are very powerful.
In this talk, I will discuss some applications of this technique. In particular, I will explain how approximability shows certain properties are local, meaning they can be glued along (co)covers in the sense of Rouquier.
April 9 Oliver Li (Melbourne) Weighted blowups and derived categories
Weighted (stacky) blowups have been in the spotlight in recent years, primarily because they yield a more efficient algorithm for resolutions of singularities in characteristic zero compared to the classical one due to Hironaka. In this talk, I will describe what weighted blowups are and report on recent results for their derived category.
April 2 Fei Peng (Melbourne) Regularity and t-structures for algebraic stacks
Let $X$ a Noetherian scheme of finite Krull dimension. A conjecture of Antieau, Gepner, and Heller states that $X$ is regular if and only if $\operatorname{Perf}(X)$ admits a bounded t-structure. This conjecture was proven recently by Neeman. In this talk, I will briefly explain how to extend Neeman’s theorem to the case of algebraic stacks with quasifinite and separated diagonal. This is based on joint work with Timothy De Deyn, Pat Lank, and Kabeer Manali Rahul.
March 26 Lance Gurney (Melbourne) Elliptic Witt vectors
The usual Witt vector construction can be viewed as coming from the isogenies of the multiplicative group G_m. This suggests that there might exist analogous constructions which, in the same way, come from isogenies of other commutative algebraic groups. In this talk, I’ll explain how this works for elliptic curves. I’ll also explain what the corresponding elliptic analogues of lambda-rings and delta-rings are. I’ll finish with a conjectural elliptic analogue of Chebotarev plus Kronecker-Weber.
This was inspired by (and no doubt overlaps) the work of Charles Rezk. This is joint with James Borger.
March 19 Christian Haesemeyer (Melbourne) K-theory of singularities, revisited once more
“K-regularity” is a measure of how close to homotopy invariant algebraic K-theory of a ring or algebraic variety is. For example, a regular ring or scheme is K_n – regular for all n. Vorst’s conjecture about how much K-regularity is needed to imply regularity for affine algebras is now a theorem over perfect fields. However, many questions remain open, for example: Do weaker K-regularity conditions imply algebraic properties weaker than regularity, as is the case for K_0 and K_1-regularity? Is the bound given in the conjecture sharp in any dimension bigger than 1?
I will report on recent work with C. Weibel giving one answer to the first question using classical K-theoretic arguments that could have been made 40 years ago. I also will discuss intriguing evidence regarding the second question obtained with Weibel, and even more recent results by Wanshun Chen, both using Hodge-theoretic arguments developed by Mustata, Popa, and other algebraic geometers.
Derived Algebraic Geometry Learning Seminar, 2024
Semester 2, 2024
Organizers: Lance Gurney and Oliver Li
Time: Tuesday, 10-11am, Peter Hall 162
The topic of this seminar is to work through Adeel Khan’s notes.
Moduli and Stacks Learning Seminar, 2023
Semester 2, 2023: Alper–Halpern-Leinster–Heinloth
Organizer: Fei Peng
Time: Wednesdays 2-3pm, Peter Hall 162
The topic of the seminar is to work through Existence of moduli spaces for algebraic stacks.
Semester 1, 2023: The Picard Stack
Organizers: Jack Hall and Fei Peng
Time: Tuesdays, 3:15-4:15pm, Peter Hall 162
The topic of this seminar is to prove that the Picard stack and functor are algebraic.
Arithmetic Geometry Seminar, 2021
If you would like to attend or be added to the mailing list please email lance.gurney@unimelb.edu.au.
Semester 2, 2021: oo-categories
Organiser: Lance Gurney
Time: Fridays 5-6pm.
The topic of the seminar is to work through the foundations of oo-categories.
Semester 1, 2021: Prismatic Cohomology
Organisers: Lance Gurney, Jack Hall, Christian Haesemeyer
Time: Fridays 3:15pm-5:00pm
The topic of the seminar is the recent prismatic cohomology of Bhatt-Scholze and its interpretation via stacks due to Drinfeld here and here. [At some point this paper by Kisin will probably pop-up].
The talks will hopefully have notes posted afterwards. A more detailed reference for some things will be found in the evolving notes on prisms (draft) which also covers the case of DVRs other than Zp.
21 May Jack Hall Lefschetz theorems via localization
I will discuss a new approach to Lefschetz style hyperplane theorems using localizations of triangulated categories. The approach is also applicable to the Fargues–Fontaine curve, an object of recent interest in arithmetic geometry.
14 May Lance Gurney Prismatization Notes
The aim is to define prismatization and (time permitting) prismatic cohomology.
7 May Lance Gurney Some computations Notes
I gave an explicit description of the divisor in Sigma and of fibre products over Sigma.
30 April Lance Gurney What is a prism? (continued) Notes
24 April Lance Gurney What is a prism? Notes
I will begin with some motivational remarks about prismatic cohomology. After this I’ll explain the definition of a prism and of Drinfeld’s stack $\Sigma$. Time permitting some associated analogies from the theory of curves over finite fields (and shtukas).
Previous Seminars