Junior Algebraic geometry Warwick Seminar (JAWS)
Organisers: Marc Truter and David Hubbard
Reading groups:
Term 1 (GIT) : The reading group this term will be on GIT, following Victoria Hoskins notes. See my webpage for the notes from our talks.
Term 2 (K-Stability) :
Rooms and Times
Term 1: D1.07 12-1pm Thursdays for reading group, D1.07 3-4pm Thursday for talks
Term 2: B1.12 12-1pm Thursdays for reading group, D1.07 3-4pm Thursdays
Term 3: MB0.08 3-4pm Thursdays
Term 2
Week 1 Jan 9
Reading group [Marc Truter, Introduction to K-stability]
Peize Liu (Blowups)
Week 2 Jan 16
Reading group [TBD]
Week 3 Jan 23
Reading group [TBD]
Week 4 Jan 30
Reading group [TBD]
Week 5 Feb 6
Reading group [TBD]
Week 6 Feb 13
Reading group [TBD]
Amy
Week 7 Feb 20
Reading group [TBD]
Week 8 Feb 27
Reading group [TBD]
Week 9 Mar 6
Reading group [TBD]
Week 10 Mar 13
Reading group [TBD]
Joris Köfler - Max Planck (Leipzig)
Term 1
Week 2 Oct 10
Reading group [Marc Truter, Introduction to Moduli problems and GIT] (2-3pm Thursday C1.06)
Week 3 Oct 17
Reading group [Arnaud Vilpert, Moduli problems] (12-1pm Thursday D1.07)
Week 4 Oct 24
Reading group [Chunkai Xu, Groups and actions] (12-1pm Thursday D1.07)
Joe Malbon - Edinburgh (3-4pm Thursday D1.07)
Classification of Algebraic Varieties
Classification - the identification of similar objects and the distinction of different ones, as well as the construction of spaces that parametrise such equivalence classes - is a guiding meta-principle in algebraic geometry. It was realised around the turn of the last century that isomorphisms are too rigid a notion of equivalence for classification to be achievable, and that the more flexible notion of birational equivalence should be used instead. This point of view was successfully applied to algebraic surfaces, whose birational classification was more or less completed by the 1950s.For higher-dimensional varieties, it is often said that all varieties are birationally (and conjecturally) constructed from Fano, Calabi-Yau, and canonically polarised varieties, and thus classification may proceed inductively based on dimension. In this talk, I will explain the birational classification of varieties from the viewpoint of the minimal model program, which is an algorithm that constructs varieties in this inductive way. If time permits, I will explain the application of the minimal model program to the construction of moduli spaces, which parametrise equivalence classes of varieties with similar properties, and illustrate some of my own work on the construction of the moduli space of a particular family of K-polystable Fano threefolds.
Week 5 Oct 31
Reading group [Alvaro Gonzalez Hernandez, Affine GIT](12-1pm Thursday D1.07)
Week 6 Nov 7
Reading group [Tommaso Faustini, Projective GIT] (12-1pm Thursday D1.07)
Thamarai Valli - UCL (LSGNT) (3-4pm Thursday D1.07)
Week 7 Nov 14
Reading group [Marc Truter, Stability] (12-1pm Thursday D1.07)
Week 8 Nov 21
Reading group [Peize Liu, Moduli of vectors bundles I] (12-1pm Thursday D1.07)
Michela Barbieri - Imperial (LSGNT) (3-4pm Thursday D1.07)
Derived Categories and Mirror Symmetry Conjecture
Derived categories were originally developed as a homological algebra tool, but in fact are very interesting from a geometric point of view too. One example of this is Homological Mirror Symmetry (HMS), which comes from a mysterious and poorly understood relationship between complex and symplectic geometry, and originates from physics, specifically string theory. Generally, the HMS conjecture says that given some 'complex geometry' X, there is a mirror 'symplectic geometry' Y such that the derived category of coherent sheaves on X, denoted D^b(Coh X), is equivalent to a Fukaya category of Y, denoted Fuk(Y). In this talk I'll talk about toric HMS from the geometric invariant theory (GIT) perspective - this story is nice because it's well-understood and very combinatorial. No prior knowledge of derived categories, symplectic geometry, GIT, etc will be assumed. One of the main goals of this talk will be to explain what derived categories are informally, with the HMS stuff hopefully giving a motivation for studying them.
Week 9 Nov 28
Reading group [Chunkai Xu, Moduli of vectors bundles II] (12-1pm Thursday D1.07)
Week 10 Dec 5
Reading group [Peize Liu, Hypersurfaces and blowups] (12-1pm Thursday D1.07)
Xuanchun Lu - Cambridge (3-4pm Thursday D1.07)
Double Ramification cycles in the punctured setting
The projective line $\mathbb{P}^1$ is the only algebraic curve on which every degree 0 divisor is principal. For any other algebraic curve, it is thus natural to ask when a 'random' divisor on the said curve becomes principal.
Generalising classical works of Abel--Jacobi, a solution to the above problem may be given as a collection of cycles on the moduli spaces $M_{g, n}$ of genus g, n-marked algebraic curves. These cycles are known as the double ramification (DR) cycles. Since $M_{g, n}$ is not compact, these cycles do not fully capture the geometry of our situation. We thus need to seek an extension of DR cycles to some compactification of $M_{g, n}$.
In this talk, I will explain some of the challenges involved in constructing such an extension, and how they can be resolved using ideas and techniques from logarithmic geometry. I will also highlight some downstream enumerative applications of DR cycles. Time permitting, I will attempt to shed some light on the appearance of the word 'punctured' in the title.