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Workshops  Concentration  Seminars 
The Symposium Seminar meets on most Monday afternoons throughout the Symposium. The subjects of the talks cover all things derived and their applications to classical algebraic geometry. The talks are sometimes followed by a social dinner for the participants.
Date:  Mon 27th April, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0517:00  Eduardo Dias (Warwick) 
“Constructing surfaces as covers of the plane”  
17:0018:00  Artan Sheshmani (Ohio State) 
“On modularity of DT invariants of threefolds given by surface fibrations”  
Motivated by Sduality modularity conjectures in string theory, we study the DonaldsonThomas invariants of 2dimensional sheaves inside a nonsingular threefold X. Our main case of study is when X is given by a surface fibration over a curve with the canonical bundle of X pulled back from the base curve. We study the DonaldsonThomas invariants, as defined by Richard Thomas, of the 2dimensional Gieseker stable sheaves in X supported on the fibers. In case where X is a K3 fibration, analogous to the GromovWitten theory formula established by MaulikPandharipande, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the fiber and the NoetherLefschetz numbers of the fibration, and prove that the invariants have modular properties. I may at the end mention the proof of much more generalized version of Sduality conjecture for CY3’s and postpone the detailed discussions about that to a future talk. This talk is based on joint work with Amin Gholampour, Yukinobu Toda and Richard Thomas. 
No further meetings have been scheduled at this time.
Date:  Mon 20th April, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:1017:00  Alicia Cuzzucoli (Roma I, La Sapienza) 
“Almost complex structures on quaternionKaehler manifolds of positive type”  
17:0018:00  Gavin Brown (Warwick) 
“CalabiYau 3folds, double covers and cancelling index 3 points” 
Date:  Mon 2nd March, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00  Sergey Arkhipov (Aarhus) 
“Monoidal action of the affine Hecke category on equivariant matrix factorizations”  
Let G be a reductive algebraic group with the standard Borel subgroup B and let X be a Gvariety. Examples show that while the affine braid group acts naturally on the Bequivariant Kgroup of X, the affine Hecke category does not act in general on Coh^B(X). Nor does it act on the category of Coh^B(T^*X). Instead it acts on a certain category of Gequivariant matrix factorizations on the product of T^* X with the Grothendieck variety of G. We introduce the derived category of equivariant matrix factorizations on a Gvariety Z with a global Ginvariant function called a potential. We recall the formalism of inverse and direct images for matrix factorizations. We define a natural monoidal action of the affine Hecke category on certain derived categories of equivariant matrix factorizations, with the potential given by the moment map. It follows that the latter category possesses a categorical action of the affine braid group. This talk is a part of a Symposium minicourse detailed here. 
Date:  Mon 23rd February, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00  Marti Lahoz (Paris 7) 
TBA 
Date:  Mon 9th February, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00  Kohei Iwaki (Tokyo) 
“Introduction to exact WKB analysis”  
In this talk I explain the following main objects in exact WKB analysis: WKB solutions, the Borel resummation method, Stokes graphs and Voros symbols. 

17:0018:00  Arend Bayer (Edinburgh) 
“Wallcrossing and the minimal model program for moduli spaces”  
TBA 
Date:  Mon 2nd February, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00:  Leticia BrambilaPaz (CIMAT, Guanajuato, Mexico) 
“Chow stability and Butler's conjecture”  
17:3018:30:  Tom Ducat (Warwick) 
“Divisorial extractions from a singular curve in a type A elephant” 
Date:  Mon 26th January, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00:  Matteo Tommasini (Luxembourg) 
“Wallcrossings for moduli spaces of coherent systems of type (2,d,2)”  
17:3018:30:  Margherita LelliChiesa (Pisa) 
“Severi varieties and BrillNoether theory of curves on abelian surfaces” 
Date:  Mon 19th January, 2015 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00:  Ivona Grzegorczyk (California State University) 
“On coherent systems with fixed determinant”  
17:3018:30:  Ciaran Meachan (Edinburgh) 
“Moduli spaces of torsion sheaves on K3 surfaces and symmetries of the derived category” 
Date:  Mon 1st December, 2014 
Place:  Room B3.02 @ Warwick Mathematical Institute 
14:0015:00:  Dario Beraldo (Oxford) 
“Towards the compatibility of geometric Langlands with the Whittaker model”  
Let G be a connected reductive group and X a smooth complete curve, both defined over an algebraically closed field of characteristic zero. Let denote the stack of Gbundles on X. In analogy with the classical theory of Whittaker coefficients for automorphic functions, we construct a “Fourier transform” functor, called , from the DG category of Dmodules on to a certain DG category , called the extended Whittaker category. Combined with work in progress by other mathematicians and the speaker, this construction allows to formulate the compatibility of the Langlands duality functor with the Whittaker model. For G= and G=, we prove that is fully faithful. This result guarantees that, for those groups, is unique (if it exists) and necessarily fully faithful. The proof relies on the theory of Drinfeld's quasimaps and on the contractibility of the space of rational maps . 

15:30  16:30:  Julian Holstein (Cambridge) 
“Morita cohomology, loop spaces and homotopy locally constant sheaves”  
This talk introduces two categorifications of the cohomology of a topological space X obtained by taking coefficients in the model category of differential graded categories. We consider both derived global sections of a constant presheaf and singular cohomology and find the resulting dgcategories are quasiequivalent. Moreover this categorified cohomology is quasiequivalent to homotopy locally constant sheaves on X and to representations in perfect complexes of chains on the based loop space of X. 

17:00  18:00:  Ed Segal (Imperial) 
“All autoequivalences are spherical twists”  
Seidel and Thomas found a new kind of symmetry of a derived category, called a spherical twist, using the idea of a `spherical’ object. Heuristically, this is the mirror to a Dehn twist around a Lagrangian sphere in a symplectic manifold. Their construction was swiftly generalized to produce spherical twists around ‘relativelyspherical’ objects, and from there to a completely abstract construction of a twist around a `spherical functor’. I will explain why this notion of a twist around a spherical functor is in fact so general that any autoequivalence of a derived category can be described as a spherical twist. The argument is completely formal, but I’ll explain a few examples as well. 
Date:  Mon 24th November, 2014 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00:  Andreas Krugg (Bonn and Warwick) 
“Equivariance of derived categories” 
Date:  Mon 17th November, 2014 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00:  Viacheslav Nikulin (Liverpool) 
“Degeneration of Kaehlerian K3 surfaces with finite symplectic automorphism groups”  
We use our recent results about Kahlerian K3 surfaces and Niemeier lattices to classify degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups. See my recent preprint arXiv:1403.6061 for some details. 

17:00  18:00:  Gavin Brown (Loughborough) 
“Canonical 3folds of Gorenstein index 2” 
Date:  Mon 27th October, 2014 
Place:  Room D1.07 @ Warwick Mathematical Institute 
16:0017:00:  Michael McQuillan (Rome) 
“2 Galois theory a.k.a. Book of Giraud on Champs, but for (actually the 2Group/2type ) instead of ”  
A theorem of Whitehead asserts that the topological 2type of a (connected) space is uniquely characterised by the triple (, , ), where the are the homotopy groups, is the Postnikov class in . Indeed all such triples may be realised. They are synomous with a 2group , i.e. a group `object' in the category of categories which plays the same role for 2types as the fundamental group does for 1types. In particular, there is a 2Galois correspondence between the 2category of champs which are etale fibrations over a space and equivariant groupoids generalising the usual 1Galois correspondence between spaces which are etale fibrations over a given space and equivariant sets. The talk will explain the profinite analogue of this correspondence, which takes place in the 2category of champs (stack is a woefull mistranslation which will be avoided) so, albeit only for the 2type, a much simpler and more generally valid description of the etale homotopy than that of ArtinMazur. 

17:00  18:00:  Eduardo Dias (Warwick) 
“Constructing surfaces as covers of ” 
The notion of a spherical twist originated as a mirrorsymmetric analogue of a Dehn twist around a Lagrangian sphere on a symplectic manifold. The corresponding notion is an autoequivalence of the derived category D(X) of an algebraic variety X constructed out of a "spherical" object. These were the first examples of genuinely derived autoequivalences and they quickly became indispensible in the study of Aut D(X) and its links with birational geometry of X. It was also observed that spherical twists can give rise to natural categorical group actions on D(X) and thus they became a fundamental tool in geometrical representation theory. A parallel minicourse by Sergey Arkhipov will explore this connection further.
In this introductory lecture course, aimed at graduate students and early postdocs, we will present a cohesive and thorough account of our current understanding of this theory. We will start with the symplectic foundations and connections to mirror symmetry, define the original SeidelThomas objects, present their various generalisations such as spherical functors, Pobjects and Pfunctors, and then proceed to some current workinprogress. Each lecture will be followed by an examples class where we will work together with the students through many of the explicit examples and applications which appear in the literature.
In this lecture, we will unpack the physics terminology and try to explain what mathematicians should think of when they hear a physicist say "the category of branes in a topological twist of the sigma model" as well as other exotic phrases. We will also try to develop some intuition for what the Homological Mirror Symmetry conjecture actually says and then loosely describe the concept of generalised Dehn twists and their link with monodromy maps in the “stringy Kahler moduli space”.
Lecture Notes #1 (.PDF, 10 Mb)
Exercise Sheet #1 (.PDF, 0.169 Mb)
L2 (Tue 10th Feb, 5pm start): Spherical objectsIn this lecture, we introduce a notion of a spherical object in the derived category D(X) of a smooth projective variety. We illustrate this notion with a number of geometrical examples on elliptic curves, K3 surfaces and CY3 threefolds. We then define the notion of a twist of D(X) around an arbitrary object E in it. We conclude by giving a sketch of the proof that the twist around a spherical object is an autoequivalence of D(X).
Lecture Notes #2 (.PDF,6 Mb)
L3 (Tue 17th Feb, 5pm start): Examples of spherical twistsIn this lecture we compute explicitly several examples of wellknown spherical twists. We first show that on an elliptic curve the twist around a point sheaf is simply the functor of tensoring by the corresponding line bundle, while the twist around the structure sheaf is a nontrivial derived autoequivalence which can be identified with the original FourierMukai transform.
We then consider two smooth rational curves intersecting in precisely one point on a K3 surface and show that the corresponding spherical twists satisfy braid relations. This is an example of a more general phenomena
Lecture Notes #3 (.PDF,5.454 Mb)
L4 (Tue 24th Feb): PobjectsIn this lecture, we will introduce the notion of a Pobject and explain why it gives rise to an autoequivalence of the derived category. We will then analyse the relationship between Pobjects and spherical objects when we have a one parameter deformation of the underlying variety. Throughout the talk, I will try to highlight the key steps which will need to be generalised when we come to talk about Pfunctors.
Lecture Notes #4 (.PDF,8.103 Mb)
L5 (Tue 3rd Mar): Spherical functorsBroadly speaking, geometric representation theory is a framework in which symmetries of geometric objects act on invariants of these objects such as cohomology theories and, more generally, derived categories associated to them. Often the representation theoretic results obtained in this way are substantial and beyond the reach of purely algebraic methods.
More specifically, in an algebrogeometric setting we can consider an algebraic group G with a subgroup H. Тhe geometry of the space H\G/H produces a number of interesting algebras and their representations, both classical and categorical. These, in turn, give rise naturally to braid group actions. In this introductory lecture course I will present several examples of this, both wellknown and workinprogress.
It is well known that both conjugacy classes of nilpotent matrices of size n and irreducible representations of the symmetric group in n letters are enumerated by Young diagrams of size n. Springer correspondence both explains and generalises this coincidence. Let G be a reductive algebraic group. We first recall the standard geometric objects related to G: the flag variety Fl_G, the Springer variety T^* Fl_G, the Springer desingularization of the nilpotent cone in the Lie algebra of G, and the Steinberg variety for G. Then we outline KazhdanLusztigGinzburg construction of the group algebra for the Weyl group of G via BorelMoore homology of the Steinberg variety. We conclude the lecture by classification of the irreducible representations of the Weyl group via BorelMoore homology of Springer fibers.
Lecture Notes #1 (.PDF, 0.6 Mb)
Exercise Sheet #1 (.PDF, 2.021 Mb)
L2 (Fri 13th Feb): KazhdanLusztigGinzburg construction and the finite Hecke categoryWe recall the classical theory of algebraic Dmodules on Fl_G for a reductive algebraic group G due to Kashiwara, Beilinson and Bernstein. Its culmination is the famous BeilinsonBernstein localization theorem for representations of the Lie algebra of G. The derived category of Bequivariant Dmodules on Fl_G equipped with a convolution becomes a monoidal category whose Grotehndieck group equals the group algebra of the Weyl group. We call this the finite Hecke category. Characteristic cycle of a holonomic Dmodule on a variety X is a Lagrangian cycle in T^* X. This relates the finite Hecke category to the KazhdanLusztigGinzburg approach to Springer correspondence.
L3 (Fri 27th Feb): BezrukavnikovRiche constructionWe begin by reviewing the Springer variety T^* Fl_G, the Grothendieck variety and the Steinberg variety for a reductive group G. It is discussed in lecture 2 that Steinberg variety is a natural home for the characteristic cycles of Dmodules in the finite Hecke category. A version of characteristic cycle construction of a Dmodule gives rise to an equivariant coherent sheaf on the Steinberg variety. We then take the category of equivariant coherent sheaves on the Steinberg variety and define a convolution monoidal structure on it. The result is the affine Hecke category. Following BezrukavnikovRiche we outline the construction of canonical objects in the affine Hecke category whose convolution satisfies the relations in the affine braid group. We conclude the lecture by constructing various categorical braid group actions from monoidal actions of the affine Hecke category on certain categories of algebrogeometric nature.
L4 (Mon 2nd Mar, Room D1.07, 4pm): Monoidal action of the affine Hecke category on equivariant matrix factorizationsLet G be a reductive algebraic group with the standard Borel subgroup B and let X be a Gvariety. Examples show that while the affine braid group acts naturally on the Bequivariant Kgroup of X, the affine Hecke category does not act in general on Coh^B(X). Nor does it act on the category of Coh^B(T^*X). Instead it acts on a certain category of Gequivariant matrix factorizations on the product of T^* X with the Grothendieck variety of G.
In the lecture, we introduce the derived category of equivariant matrix factorizations on a Gvariety Z with a global Ginvariant function called a potential. We recall the formalism of inverse and direct images for matrix factorizations. We define a natural monoidal action of the affine Hecke category on certain derived categories of equivariant matrix factorizations, with the potential given by the moment map. It follows that the latter category possesses a categorical action of the affine braid group.