# GiC: Geometry in Cardiff seminar

The GiC (IPA: /ɡiːk/) seminar is an algebraic geometry seminar for those in and around Cardiff area and, more generally, in the southwest. All are welcome. The seminar meets on Wednesday afternoons 16:00 - 18:00 UK Time, normally in M/2.06. For the moment, owing to the pandemic, we meet online via Zoom.

The subjects of the talks reflect the interests of its current organisers, ranging from the classical algebraic geometry to all things derived, categorical and homotopical. The talks are sometimes followed by a social dinner for the participants.

Please address any questions or queries to one of the seminar organisers:

• Dr. Timothy Logvinenko  LogvinenkoT at cardiff ac uk
• Mr. Christopher Seaman  SeamanCM at cardiff ac uk

In particular, contact Dr. Timothy Logvinenko if you wish to be added to the GiC Seminar mailing list.

### Next Meeting:

•  Date: Wed 21st Jul, 2021 Place: Zoom @ online 16:00-17:30 Timothy Logvinenko (Cardiff) “The Heisenberg category of a category, III” In this series of three talks we discuss how to associate a Heisenberg category to any smooth and proper dg category. In this final talk, we will present the DG categorical version of our construction, comparing it to the additive case construction discussed in the previous talk. The main challenge here is a lack of the genuine Serre functor, and thus the necessity of working with a homotopy one. We will discuss the unique features of our construction introduced to overcome this and the other challenges we encountered. We will also discuss the applications, as well as the reasons for working in the DG setting in the first place. This is joint work with Ádám Gyenge and Clemens Koppensteiner. 17:30-18:15 Oscar Finegan (Cardiff) “Scheme Theory XXXVII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we finish our discussion of sheaves of modules on a ringed space by considering the exercises provided by Hartshorne on this topic.

### Future Meetings:

No further meetings have been scheduled at this time.

### Past Meetings:

•  Date: Wed 14th Jul, 2021 Place: Zoom @ online 16:00-17:30 Ádám Gyenge (Alfréd Rényi, Budapest) “The Heisenberg category of a category, II” Talk slides (PDF) Khovanov introduced recently a categorification of the infinite Heisenberg algebra associated with the free boson or, equivalently, a rank 1 lattice, using a graphical construction involving planar diagrams. We extend Khovanov’s graphical construction to derived categories of smooth and projective varieties or, more generally, to categories having a Serre functor. In our case the underlying lattice will be the (numerical) Grothendieck group of the category. We also obtain a 2-representation of our Heisenberg category on a categorical analogue of the Fock space. Joint work with Clemens Koppensteiner and Timothy Logvinenko. 17:30-18:15 Wing Kei Poon (Bath) “Scheme Theory XXXVI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss sheaves of modules on a ringed space, beginning with the statement that coherent sheaves on a projective scheme can be written as a quotient of a direct sum of twisted structure sheaves O(n).
•  Date: Wed 7th Jul, 2021 Place: Zoom @ online 16:00-17:30 Clemens Koppensteiner (Oxford) “The Heisenberg category of a category, I” In this series of three talks we will discuss how to associate a Heisenberg category to any smooth and proper dg category. In this first introductory talk, we will consider the geometric motivation for the construction, review the theory of Heisenberg algebras, and look at some categorifications already in the literature. This is joint work with Ádám Gyenge and Timothy Logvinenko. 17:30-18:15 Calla Tschanz (Bath) “Scheme Theory XXXV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss sheaves of modules on a ringed space, beginning with sheaves generated by global sections.
•  Date: Wed 30th Jun, 2021 Place: Zoom @ online 16:00-17:30 Timothy Logvinenko (Cardiff) “Scheme Theory XXXIV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space, starting with very ample sheaves.
•  Date: Wed 23rd Jun, 2021 Place: Zoom @ online 16:00-17:30 Oscar Finegan (Cardiff) “Scheme Theory XXXIII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space, starting with the statement that a scheme is projective if and only if it is isomorphic to the Proj of a graded ring finitely generated in degree 1.
•  Date: Wed 16th Jun, 2021 Place: Zoom @ online 16:00-17:30 Wing Kei Poon (Bath) “Scheme Theory XXXII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space, starting with the graded modules associated to quasi-coherent sheaves on a Proj of a graded ring.
•  Date: Wed 9th Jun, 2021 Place: Zoom @ online 16:00-17:30 Calla Tschanz (Bath) “Scheme Theory XXXI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space, starting with quasi-coherent sheaves on a Proj of a graded ring.
•  Date: Wed 2nd Jun, 2021 Place: Zoom @ online 16:00-17:30 Timothy Logvinenko (Cardiff) “Scheme Theory XXX” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space, starting with the properties of ideal sheaves.
•  Date: Wed 26th May, 2021 Place: Zoom @ online 16:00-17:30 Oscar Finegan (Cardiff) “Scheme Theory XXIX” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space, starting with the exactness of the global sections functor on quasi-coherent sheaves on an affine scheme.
•  Date: Wed 19th May, 2021 Place: Zoom @ online 16:00-17:30 Wing Kei Poon (Bath) “Scheme Theory XXVIII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space starting with quasi-coherent sheaves.
•  Date: Wed 12th May, 2021 Place: Zoom @ online 16:00-17:30 Calla Tschanz (Bath) “Scheme Theory XXVII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discussing sheaves of modules on a ringed space starting with free and locally free sheaves.
•  Date: Wed 5th May, 2021 Place: Zoom @ online 16:00-17:30 Timothy Logvinenko (Cardiff) “Scheme Theory XXVI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we start on the notion of a sheaf of modules on a ringed space.
•  Date: Wed 28th Apr, 2021 Place: Zoom @ online 16:00-17:30 Oscar Finegan (Cardiff) “Scheme Theory XXV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, beginning with the notion of an abstract variety.
•  Date: Wed 21st Apr, 2021 Place: Zoom @ online 16:00-17:30 Miles Reid (Warwick) “ The Tate-Oort group TO_p and moduli of Godeaux surfaces” Video recording (YouTube) The Tate-Oort group scheme TO_p is a group scheme of order p defined in mixed characteristic at p. It contains the cyclic groups ℤ/p and μ_p in characteristic 0, and the three group schemes ℤ/p, μ_p, and α_p in characteristic p. Godeaux surfaces X in characteristic 5 with ℤ/5, μ_5, and α_5 in Pic X were constructed respectively by Lang, Miranda and Liedtke as quotients of quintic surfaces Y_5 in ℙ^3 equivariant under an action of the dual group schemes μ_5, ℤ/5, and α_5. All three of these constructions can be put together in a single deformation family, together with the classical Godeaux surfaces. This is joint work with KIM Soonyoung, based in part on her 2014 Sogang Univ. thesis. See also https://homepages.warwick.ac.uk/~masda/TOp/. 17:30-18:15 Oscar Finegan (Cardiff) “Scheme Theory XXIV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, beginning with basic properties of proper morphisms.
•  Date: Wed 14th Apr, 2021 Place: Zoom @ online 16:00-17:30 Wing Kei Poon (Bath) “Scheme Theory XXIII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, beginning with valuative criterion for properness.
•  Date: Wed 7th Apr, 2021 Place: Zoom @ online 16:00-17:30 Calla Tschanz (Bath) “Scheme Theory XXII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, beginning with basic properties of separated morphisms.
•  Date: Wed 31st Mar, 2021 Place: Zoom @ online 16:00-17:30 Timothy Logvinenko (Cardiff) “Scheme Theory XXI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, beginning with basic properties of separated morphisms.
•  Date: Wed 24th Mar, 2021 Place: Zoom @ online 16:00-17:30 Oscar Finegan (Cardiff) “Scheme Theory XX” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, finishing the proof of the valuatative criterion for separatedness.
•  Date: Wed 17th Mar, 2021 Place: Zoom @ online 16:00-17:30 David Favero (Alberta) “Geometric invariant theory through group compactifications, derived categories, and derived algebraic geometry” Given a group G acting on an algebraic variety X, geometric invariant theory tells us how to get a (or several) nice quotient space(s) from this data. Traditionally, this comes from the choice of a G-equivariant line bundle on X. I will discuss an alternative approach via partially compactifying the action groupoid. One benefit of this viewpoint is that it produces a natural correspondence between X and itself. This allows us to embedd the derived category of a given GIT quotient in the derived category of [X/G] and make comparisons (and sometimes deduce equivalences) between derived categories of (the several) GIT quotients. If time permits, I will also discuss how to use this approach in the singular setting through the lens of derived algebraic geometry. 17:30-18:15 Oscar Finegan (Cardiff) “Scheme Theory XIX” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss separated and proper morphisms of schemes, beginning with the valuatative criterion for separatedness.
•  Date: Wed 10th Mar, 2021 Place: Zoom @ online 16:00-17:30 Wing Kei Poon (Bath) “Scheme Theory XVIII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss first properties of schemes beginning with the discussion of base extensions.
•  Date: Wed 3rd Mar, 2021 Place: Zoom @ online 16:00-17:30 Yan Soibelman (Kansas) “Algebra of the infrared and Fukaya-Seidel categories with coefficients in perverse schobers” Talk notes (PDF) Video recording (YouTube) Several years ago physicists Gaitto,Moore and Witten discovered a remarkable algebraic structure underlying all 2d N=(2,2) QFTs. They call it "the algebra of the infrared". Mathematical byproduct of that work was an alternative definition of the Fukaya-Seidel category (= Landau-Ginzburg model) of a Kahler manifold. It is given in terms of the critical points of the superpotential of the LG-model and gradient trajectories between them. In the joint paper with Kapranov and Kontsevich we interpreted the algebraic structure of Gaiotto-Moore-Witten in terms of L-infinity and A-infinity algebras associated with the secondary polytope of the convex hull of the set of critical values of the superpotential. In my talk I will explain how our approach can be generalized to the case of Fukaya-Seidel categories with coefficients which are perverse schobers. The talk is based on the recent work, joint with Kapranov and Soukhanov.
•  Date: Wed 24th Feb, 2021 Place: Zoom @ online 16:00-17:30 Calla Tschanz (Bath) “Scheme Theory XVII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discuss first properties of schemes starting with the proof that fibre products exist.
•  Date: Wed 17th Feb, 2021 Place: Zoom @ online 16:00-17:30 Sergey Arkhipov (Aarhus) “Differential forms with logarithmic singularities and categorical braid group actions” Bezrukavnikov and Riche studied the affine Hecke category - a categorification of the affine braid group. One realization of this category is via equivariant coherent sheaves on the Steinberg variety. Braid group generators are provided by explicit coherent sheaves. However, braid relations are proved in a rather indirect way - either by a case by case analysis or by reduction to prime characteristic. Using linear Koszul duality, we propose another realization of the affine Hecke category via equivariant Omega-modules on the corresponding simple algebraic group G. A study of logarithmic differential forms on Bott-Samelson varieties gives a simple and uniform proof of braid relations. The material of the talk is a joint work in progress with my student Sebastian Orsted. 17:30-18:15 Timothy Logvinenko (Cardiff) “Scheme Theory XVI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discuss first properties of schemes, beginning with fibre products.
•  Date: Wed 10th Feb, 2021 Place: Zoom @ online 16:00-17:30 Oscar Finegan (Cardiff) “Scheme Theory XV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discuss first properties of schemes, continuing with the examples of open and closed immersions.
•  Date: Wed 3rd Feb, 2021 Place: Zoom @ online 16:00-17:30 Will Donovan (Tsinghua) “Classification of simple flops and variation of GIT” Though classification of flops remains very challenging in general, progress on classification of simple flops has been made by D. Li and A. Kanemitsu, focusing on their relation with Fano manifolds. Derived equivalence are conjectured for all, and remain open in many cases. I review this, and discuss approaches to proving new equivalences using variation of GIT, in joint work with Weilin Su. 17:30-18:15 Oscar Finegan (Cardiff) “Scheme Theory XIV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discuss first properties of schemes, beginning with open and closed immersions.
•  Date: Wed 27th Jan, 2021 Place: Zoom @ online 16:00-17:30 Wing Kei Poon (Bath) “Scheme Theory XIII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue discuss first properties of schemes, beginning with those of being noetherian or locally noetherian.
•  Date: Wed 20th Jan, 2021 Place: Zoom @ online 16:00-17:30 Calla Tschanz (Bath) “Scheme Theory XII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we discuss first properties of schemes.
•  Date: Wed 13th Jan, 2021 Place: Zoom @ online 16:00-17:30 Harry Rainbird (Bath) “Scheme Theory XI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme beginning by considering the functor from varieties to schemes.
•  Date: Wed 16th Dec, 2020 Place: Zoom @ online 16:00-17:30 Oscar Finegan (Cardiff) “Scheme Theory X” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme beginning with the proof that Proj of a graded ring is a scheme.
•  Date: Wed 9th Dec, 2020 Place: Zoom @ online 16:00-17:30 Alan Thompson (Loughborough) “Mirror symmetry for fibrations and degenerations” In a 2004 paper, Tyurin briefly hinted at a novel relationship between Calabi-Yau mirror symmetry and the Fano-LG correspondence. More specifically, if one can degenerate a Calabi-Yau manifold to a pair of (quasi-)Fanos, then one expects to be able to express the mirror Calabi-Yau in terms of the corresponding Landau-Ginzburg models. Some details of this correspondence were worked out by C. F. Doran, A. Harder, and I in a 2017 paper, but much remains mysterious. In this talk I will describe recent attempts to better understand this picture, and how it hints at a broader mirror symmetric correspondence between degeneration and fibration structures. As an example of this correspondence, I will discuss the question of finding mirrors to certain exact sequences which describe the Hodge theory of degenerations. The material in this talk is joint work in progress with C. F. Doran. 17:30-18:15 Oscar Finegan (Cardiff) “Scheme Theory IX” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme beginning with the Proj construction.
•  Date: Wed 2nd Dec, 2020 Place: Zoom @ online 16:00-17:30 Harkaran Uppal (Bath) “Scheme Theory VIII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme beginning with its formal definition.
•  Date: Wed 25th Nov, 2020 Place: Zoom @ online 16:00-17:30 Agnieszka Bodzenta-Skibinska (Warsaw) “Exact categories and abelian envelopes” Video recording available (YouTube) For exact categories I will develop a theory parallel to the theory well-known for triangulated categories; left and right admissible subcategories, and (semi-orthogonal) decompositions. In particular, I will introduce thin exact categories, i.e. exact categories will full exceptional collections. I will discuss left and right abelian envelopes of an exact category and will show that highest weight categories are precisely the abelian envelopes of thin exact categories. I will also discuss Ringel duality from this point of view. This is a joint work with A. Bondal. 17:30-18:15 Calla Tschanz (Bath) “Scheme Theory VII” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme beginning with the functoriality of the Spec construction.
•  Date: Wed 18th Nov, 2020 Place: Zoom @ online 16:00-17:30 Wing Kei Poon (Bath) “Scheme Theory VI” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme and the basic properties of the structure sheaf of the spectrum of a ring.
•  Date: Wed 11th Nov, 2020 Place: Zoom @ online 16:00-17:30 Daria Polyakova (Copenhagen) “Weakly monoidal structure for the DG-category of representations up to homotopy” Representations up to homotopy of a group G were introduced by Abad and Crainic. They form a DG-category Rep^h(G) whose objects are A-infinity comodules over the coalgebra of functions on G, and whose morphisms are A-infinity Hom complexes. This category enhances the derived category of ordinary representations. Abad-Crainic-Dherin proved that the homotopy category of Rep^h(G) is monoidal. They posed a question to define an appropriate homotopy-coherent structure on the DG-category itself.I will explain how a family of polytopes controls morphisms of A-infinity (co)modules. Then I will present a new observation that this family is nothing else but freehedra, a family constructed earlier by Saneblidze for entirely different reasons as subdivisions of cubes. Abad-Crainic-Dherin monoidal structure appears to follow from Saneblidze’s diagonal for freehedra. I will extend this diagonal to A-infinity coalgebra structure. This is the first ingredient of a “weakly monoidal” structure that I obtain as a DG-lift of Abad-Crainic-Dherin monoidal structure. 17:30-18:15 Chris Seaman (Cardiff) “Scheme Theory V” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss the notion of a scheme, beginning with basic properties of the structure sheaf of the spectrum of a ring.
•  Date: Wed 4th Nov, 2020 Place: Zoom @ online 16:00-18:00 Oscar Finegan (Cardiff) “Scheme theory IV” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we discuss the notion of a scheme, beginning with the definition of the spectrum of a commutative ring.
•  Date: Wed 28th Oct, 2020 Place: Zoom @ online 16:00-17:30 Alastair Craw (Bath) “Gale duality and the linearisation map for quiver moduli” The goal of the talk is to show you a beautiful matrix and then to explain its geometric significance. This will enable me to explain why two rival geometric interpretations of `Reid's recipe' are equivalent. To begin, I'll set the scene by discussing the classical McKay correspondence in dimension two and I'll go on to discuss how this extends naturally to dimension three. I'll introduce Reid's recipe by studying the cyclic quotient singularity of type 1/19(1,3,15), and this gives me the excuse to introduce the matrix that I've fallen in love with. I'll reveal the geometry that this gorgeous matrix encodes, and as a result, we'll see that two conjectures for consistent dimer model algebras are equivalent. 17:30-18:15 Timothy Logvinenko (Cardiff) “Scheme theory III” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss sheaves of abelian groups on a topological space, beginning with short exact sequences of sheaves.
•  Date: Wed 21st Oct, 2020 Place: Zoom @ online 16:00-18:00 Chris Seaman (Cardiff) “Scheme theory II” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we continue to discuss sheaves of abelian groups on a topological space, beginning with the kernel, image, and cokernel constructions.
•  Date: Wed 14th Oct, 2020 Place: Zoom @ online 16:00-18:00 Oscar Finegan (Cardiff) “Scheme theory I” The wheel of time turns, and academic years come and pass, leaving traditions which become legend. In this series of talks we will work our way through the Chapter 2 of Hartshorne's “Algebraic geometry”, covering the basics of scheme theory. In this talk, we discuss sheaves of abelian groups on a topological space.
•  Date: Wed 12th Feb, 2020 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Chris Seaman (Cardiff) “Combinatorial constructions of derived equivalences II” In the first part of this talk, we stated the main theorem of the paper by Halpern-Leistner and Sam (Theorem 3.2 in arXiv:1601.02030). In the second part, we will briefly restate (the motivation for) this theorem, and go on to show that it constructs window-shift equivalences between the derived categories of some variations of the GIT quotient X^{ss}/G, where X is a quasi-symmetric representation of a reductive group G.
•  Date: Wed 5th Feb, 2020 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Clemens Koppensteiner (Oxford) “What is a holonomic log D-module?” We will discuss how to generalize the theory of D-modules to logarithmic geometry and in particular how to define the notion of holonomicity in this context. We will see that this is not entirely trivial, and how to obtain a satisfactory theory anyways. This is joint work with Mattia Talpo. No knowledge of D-modules or logarithmic geometry will be assumed.
•  Date: Wed 29th Jan, 2020 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Christopher Seaman (Cardiff) “Combinatorial constructions of derived equivalences I” We give a brief introduction to the results of the eponymous paper of Dan Halpern-Leistner and Steven Sam (arXiv:1601.02030) which constructs window-shift equivalences between the derived categories of the variations of a GIT quotient.
•  Date: Wed 30th Oct, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “Homogeneous spaces and Bruhat decomposition II” Last week we have introduced algebraic groups, their Borel and parabolic subgroups, and the homogeneous spaces G/P obtained as a factor of an algebraic group G by a parabolic subgroup P. We've also worked explicitly through the example of G = SL_2 to conclude that the corresponding space G/B is the projective space P^1. This week we will define the Bruhat decomposition G = BWB induced by the choice of a Borel subgroup B and a maximal torus T ⊂ B. We will then work though the example of G = SL_3.
•  Date: Wed 23rd Oct, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “Homogeneous spaces and Bruhat decomposition I” I will give a beginner-friendly introduction to the geometry of homogeneous spaces G/P via Bruhat decomposition and Schubert calculus. The main focus is on the case G = SL_n, when these homogeneous spaces are projective spaces, Grassmanians, and flag varieties. There will be explicit worked examples for G = SL_3.
•  Date: Wed 9th Oct, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “The Grothendieck–Springer resolution III” In this talk, we define the nilpotent cone and the Springer resolution of a complex connected Lie group. We then define the homogeneous space G/B, and identify the Springer resolution with the cotangent bundle T^∗ G/B.
•  Date: Fri 5th Oct, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Victor Przyjalkowski (HSE/Steklov) “On the mirror P=W conjecture” We discuss Katzarkov-Kontsevich-Pantev conjectures which relate Hodge numbers of Fano varieties with Hodge-type numbers of their Landau-Ginzburg models. We give their proofs in dimensions 2 and 3. We also discuss the mirror P=W conjecture which claims a deeper relation between mixed Hodge structures of log Calabi-Yau varieties and perverse Leray filtrations of Landau-Ginzburg models.
•  Date: Wed 31st Jul, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Christopher Seaman (Cardiff) “Classification of Finite Semi-Simple Lie Algebras II” Having shown last time that root systems can be classified by their associated Dynkin diagrams, we finish off the classification of finite semi-simple Lie algebras by using a theorem of Serre.
•  Date: Wed 31st Jul, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Christopher Seaman (Cardiff) “Classification of Finite Semi-Simple Lie Algebras I” In this talk we will remind ourselves of the definitions of root systems and the Weyl group, then show that irreducible root systems are classified by their associated Dynkin diagrams. This allows us to classify the finite semi-simple Lie algebras.
•  Date: Wed 24th Jul, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “The Grothendieck–Springer resolution II” In today's talk we will remind the basic definitions of Lie algebras and we will discuss some properties of solvable and semisimple Lie algebras.
•  Date: Wed 3rd Jul, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “The Grothendieck–Springer resolution I” In today's talk we will remind the basic definitions of Lie algebras, we will discuss some properties of nilpotent elements and we will give the contruction of the Springer resolution of the nilpotent cone.
•  Date: Wed 26th Jun, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “P^n-functors V” I will explain the new notion of (non-split) P^n-functors introduced recently in the eponymous preprint (arXiv:1905.05740). I will explain the background, define a P^n-structure on an enhanced functor, construct the induced P-twist, and give an outline of the proof that this P-twist is an autoequivalence of the target category. Then I will explain the strong monad condition. Finally, I will give some examples of non-split P^n-functors.
•  Date: Wed 12th Jun, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “P^n-functors IV” I will explain the new notion of (non-split) P^n-functors introduced recently in the eponymous preprint (arXiv:1905.05740). I will explain the background, define a P^n-structure on an enhanced functor, construct the induced P-twist, and give an outline of the proof that this P-twist is an autoequivalence of the target category. Then I will explain the strong monad condition. Finally, I will give some examples of non-split P^n-functors.
•  Date: Wed 5th Jun, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “P^n-functors III” I will explain the new notion of (non-split) P^n-functors introduced recently in the eponymous preprint (arXiv:1905.05740). I will explain the background, define a P^n-structure on an enhanced functor, construct the induced P-twist, and give an outline of the proof that this P-twist is an autoequivalence of the target category. Then I will explain the strong monad condition. Finally, I will give some examples of non-split P^n-functors.
•  Date: Fri 31st May, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “P^n-functors II” I will explain the new notion of (non-split) P^n-functors introduced recently in the eponymous preprint (arXiv:1905.05740). I will explain the background, define a P^n-structure on an enhanced functor, construct the induced P-twist, and give an outline of the proof that this P-twist is an autoequivalence of the target category. Then I will explain the strong monad condition. Finally, I will give some examples of non-split P^n-functors.
•  Date: Wed 22nd May, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Timothy Logvinenko (Cardiff) “P^n-functors” I will explain the new notion of (non-split) P^n-functors introduced recently in the eponymous preprint (arXiv:1905.05740). I will explain the background, define a P^n-structure on an enhanced functor, construct the induced P-twist, and give an outline of the proof that this P-twist is an autoequivalence of the target category. Then I will explain the strong monad condition. Finally, I will give some examples of non-split P^n-functors.
•  Date: Wed 15th May, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Chris Seaman (Cardiff) “Perverse schobers” The group of autoequivalences of a given derived category of coherent sheaves on a variety X, D^b(X), has been the subject of much study. In this talk I will present some well-known results about Aut(D^b(X)), as well as more recent technology in the form of perverse schobers. These should be thought of as categorifications of perverse sheaves, a way of thinking which can be made more precise through results of Bondal, Kapranov and Schechtman.
•  Date: Wed 8th May, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Daria Poliakova (Copenhagen) “Homotopy limits in DG categories and applications to derived algebraic geometry” I will explain a construction of Reedy-fibrant replacements for simplicial DG-categories with respect to Dwyer-Kan model structure. This allows for explicit formulas for homotopy limits (due to Block-Holstein-Wei and Arkhipov-Ørsted). As an application, I will demonstrate a computation of homotopy limit of a cosimplicial DG-algebra associated to a commutative Hopf DG-algebra and compare it to the classical cobar construction.
•  Date: Wed 10th Apr, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “Generalised braid actions” In this talk, after giving some background on autoequivalences of derived categories of smooth projective varieties, I will define the generalised braid category and describe its action on the derived categories of (the cotangent bundles of) full and partial flag varieties. Generalised braids are the braids whose strands are allowed to touch in a certain way. The basic building blocks of their action on flag varieties are spherical and non-split P- functors together with the twist equivalences they induce. I will describe our present progress and future expectations. This is a joint project with Rina Anno and Timothy Logvinenko.
•  Date: Wed 27th Mar, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “A quick tour of deformation theory II” Following the discussion of last'week, we will go through the definition of complex space and through the completeness, versality and universality properties of deformations.
•  Date: Wed 20th Mar, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “A quick tour of deformation theory” After recalling the basic definitions of deformation families, we will discuss the Kodaira-Spencer criterion for completeness and the Kuranishi Theorem on the existence of a versal deformation for every compact complex manifold.
•  Date: Wed 13th Mar, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo De Biase (Cardiff) “Bondal-Orlov criteria for derived equivalences V” We will give a brief introduction to deformation families and Kodaira-Spencer maps.
•  Date: Wed 6th Mar, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Chris Seaman (Cardiff) “Bondal-Orlov criteria for derived equivalences IV” We shall look at a lemma by Bridgeland which states that if we have a flat map S → X and an object E ∈ D(S) such that for every closed point x ∈ X the object ι_x^* E ∈ D(S_x) is a sheaf, then E itself is a sheaf on S flat over X.
•  Date: Wed 27th Feb, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Chris Seaman (Cardiff) “Bondal-Orlov criteria for derived equivalences III” Following on from the overview given two weeks ago, we shall present the spectral sequence argument which ensures that the Fourier-Mukai kernel of LF ∈ D(X × X) is a sheaf supported on Δ ⊂ X × X and flat over the projections to X.
•  Date: Wed 20th Feb, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Chris Seaman (Cardiff) “Bondal-Orlov criteria for derived equivalences II” Following on from the overview given two weeks ago, we shall begin the process of fleshing out the full details of the proof of the Bondal-Orlov criterion for a Fourier-Mukai transform to be fully faithful.
•  Date: Fri 15th Feb, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 13:10-14:10 Enrico Fatighenti (Loughborough) “On technical aspects of constructing Fano varieties of K3 type and IHS manifolds” Subvarieties of Grassmannians (and especially Fano varieties) obtained from section of homogeneous vector bundles are far from being classified. A case of particular interest is given by the Fano varieties of K3 type, for their deep links with hyperk\"ahler geometry. This talk will be mainly devoted to the construction of some new examples of such varieties. This is a work in progress with Giovanni Mongardi.
•  Date: Wed 6th Feb, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:10-18:00 Timothy Logvinenko (Cardiff) “Bondal-Orlov criteria for derived equivalences I” I will give a bird's eye overview of the well-known criterion by Bondal and Orlov for the Fourier-Mukai transform defined by a kernel F ∈ D(X × Y) to be fully faithful. We will look at the kernel composition LF ∈ D(X × X) and the successive steps through which it is established to be the structure sheaf of the diagonal Δ ⊂ X × X.
•  Date: Wed 30th Jan, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:10-18:00 Lorenzo de Biase (Cardiff) “Derived Equivalence vs Birationality III” We will continue the proof of Kawamata's result on the implication of K-equivalence from D-equivalence under the hypothesis of maximality of the (anti-canonical) Kodaira dimension.
•  Date: Wed 23rd January, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:10-18:00 Christopher Seaman (Cardiff) “ Derived Equivalence vs Birationality II” Carrying on from our digression last week, we shall remind ourselves of the statement of Kawamata's result and continue the proof. Time permitting, we shall address some corollaries of the result.
•  Date: Wed 16th January, 2019 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:10-18:00 Christopher Seaman (Cardiff) “Derived Equivalence vs Birationality I” We shall state and prove the result by Kawamata that if two smooth projective varieties are derived equivalent and the Kodaira dimension of the (anti-)canonical divisor is maximal, then the varieties are birational.
•  Date: Wed 12th December, 2018 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:10-18:00 Timothy Logvinenko (Cardiff) “Nefness and birationality under derived equivalence” We will state and prove the famous results of Kawamata on the preservation of the nefness of the canonical bundle and the preservation of the birational class under derived equivalences.
•  Date: Wed 28th November, 2018 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo de Biase (Cardiff) “Geometrical aspects of the Fourier-Mukai kernel” What happens if a Fourier-Mukai kernel $K \in D^b(XxY)$ has support not the whole product XxY but just a subvariety Z? In today's seminar we will prove some results about Supp(K): beyond their intrinsic importance, they will be the technical background for proving Kawamata's theorems on the preservation of the nefness of the canonical bundle and of the numerical Kodaira dimension under derived equivalences.
•  Date: Wed 21th November, 2018 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Christopher Seaman (Cardiff) “Hochschild (co)homology and geometrical aspects of the Fourier-Mukai kernel” After reminding ourselves of the definitions, we will study isomorphisms of Hochschild (co)homologies induced by equivalences of derived categories. Time permitting, we will explore some results on the support of the Fourier-Mukai kernel of an equivalence.
•  Date: Wed 31st October, 2018 Place: Room M/2.06 @ School of Mathematics, Cardiff University 16:00-18:00 Lorenzo de Biase (Cardiff) “Kodaira dimension under derived equivalence” We will prove Orlov's result on preservation of Kodaira dimension and canonical ring under derived equivalences for smooth projective varieties. Then, we will extend the result to Hochschild cohomologies and we will recover the Hochschild-Kostant-Rosemberg isomorphism.

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