### Zero divisors in the Grothendieck ring of varieties via K3 surfaces

Thu, Nov 3, 2016 Speaker: Evgeny Shinder (Sheffield)Location: F38

Time: 16:00

Location: F38

Time: 16:00

*Abstract:*

The geometry of contracting rational curves in three folds can be studied via noncommutative algebra, and I will recap how you can pass from the geometry to the algebra and back again via moduli space constructions. As an example of the advantages of the noncommutative algebra viewpoint I plan to discuss a noncommutative interpretation of the Gopakumar-Vafa invariants attached to the curve realised by Toda.

Speaker: Joe Karmazyn (Sheffield)Location: J11

Time: 16:00

*Abstract:*

I will start by introducing BPS structures and their variations. The point of this is to axiomatise the behaviour of Donaldson-Thomas invariants under changes in stability parameters, but it is not necessary to understand anything about that to understand the definitions. I will then introduce a natural Riemann-Hilbert problem associated to such structures, which involves holomorphic maps on the complex plane with prescribed discontinuities along a collection of rays. I will also explain how to solve this problem in the simplest example using the gamma function.

Speaker: Tom Bridgeland (Sheffield)Location: J11

Time: 16:00

*Abstract:*

We decided that it would be nice if everyone in the algebraic geometry group gave a talk or two to give a flavour of their research interests. I volunteered to go first and immediately realised that I would need to give (at least) a lecture of background first. So this will be fairly basic: I will remind you about Calabi-Yau manifolds, the Kahler cone and periods, and give a rough statement of mirror symmetry as formulated by Candelas et al: coefficients of series expansions of periods give enumerative invariants of the mirror.

Speaker: Tom Bridgeland (Sheffield)Location: J11

Time: 16:00

*Abstract:*

Let G be a split semisimple linear algebraic group over a field k, let E be a G-torsor over k. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel (e.g.Chow ring or an algebraic cobordism). Consider a twisted form E/B of the variety of Borel subgroups G/B. Following the motivic Galois group approach and the Kostant-Kumar results on equivariant cohomology of flag varieties we establish an equivalence between the h-motivic subcategory generated by E/B and the category of projective modules of certain Hecke-type algebra H which depends on the root system of G, its isogeny class, on E, and on the formal group law of the theory h. In particular, taking h to be the Chow groups with finite coefficients F_p and E to be a generic torsor we obtain that all irreducible modules of the affine nil-Hecke algebra H of G with coefficients in F_p are isomorphic and correspond to the generalized Rost-Voevodsky motive for (G,p).

Kirill Zaynullin (University of Ottawa)Location: J11

Time: 15:00

*Abstract:*

Gushel-Mukai varieties are dimensionally transverse intersections of a cone over Grassmannian Gr(2,5) with a quadric. They include all mildly singular Fano varieties with Picard number 1, coindex 3, and degree 10. I will discuss the geometry of this class of varieties (including their birational properties and the structure of their derived categories) and their relation to Eisenbud-Popescu-Walter sextics. This material is covered by joint works with Olivier Debarre and Alex Perry.

Alexander Kuznetsov (Steklov Institute)Location: J11

Time: 16:00

Location: J11

Time: 12:00

*Abstract:*

It is well known that the variety of lines on a cubic 4-fold X is an irreducible holomorphic symplectic (IHS) manifold. More recently Lehn et. al. have constructed another IHS manifold Z which is related to the variety of twisted cubics on X. We will discuss how to describe an open subset of this manifold in terms of moduli spaces of sheaves on X. We will see that the existence of symplectic form on Z is related to the structure of the derived category of X. The talk will be based on a joint work with E. Shinder.

Andrey Soldatenkov (Bonn)Location: J11

Time: 16:00

*Abstract:*

Wall-crossing for stability conditions on derived categories has, over the last few years, led to many new results purely within algebraic geometry. I will illustrate some of the methods behind these applications by reproving an old theorem, namely Lazarsfeld’s Brill-Noether theorem for curves on K3 surfaces.

Arend Bayer (Edinburgh)Location: J11

Time: 14:00

*Abstract:*

The singularity category is a homological invariant of a scheme that captures the singular information. The singularity category of the type A_n Kleinian surface singularity is equivalent to that of the finite dimensional algebra k[x]/(x^n), and this can be shown via Knörrer periodicity. I will discuss joint work with Martin Kalck that extends a similar phenomenon outside of the Kleinian case to produce an equivalence between the singularity category of a cyclic quotient surface singularity and a certain finite dimensional algebra. This equivalence is constructed geometrically by considering subcategories of the derived category of a surface containing a type A configuration of rational curves , and intriguingly the finite? dimensional algebras produced are noncommutative in general.

Joseph Karmazyn (Bath)Location: J11

Time: 16:00

*Abstract:*

To each triangulated category T one can associate a complex manifold Stab(T) of stability conditions on T. The expectation is that Stab(T) is always contractible, and this has been verified in many examples. In this talk, I will explain how to prove that any `finite-type’ component of Stab(T) is contractible, by considering a cellular stratification of Stab(T) arising from the decomposition into regions in which the heart of the stability condition remains constant. I will use examples from the theory of quivers to illustrate the results, and, I hope, explain all the technical terms above. If there is time I will also discuss why the braid and spherical twist groups of a Dynkin quiver are isomorphic. This is joint work with Yu Qiu.

Jon Woolf (Liverpool)Location: J11

Time: 16:00

Location: J11

Time: 13:00

Location: J11

Time: 16:00

Location: J11

Time: 13:00

*Abstract:*

In 2011, Hosono and Takagi constructed an interesting example of two derived equivalent, non-birational Calabi-Yau 3-folds. This example can be explained by phrasing it in terms of Kuznetsov’s theory of homological projective duality. With this as motivation, we compute the homological projective dual of Sym^2 P^n, and by taking n = 4 we recover Hosono and Takagi’s example. I will explain this result and its proof, which showcases a general technique for approaching HP duality, based on “gauged LG models and variation of GIT stability”.

Speaker: Jørgen Rennemo (Oxford)Location: J11

Time: 13:00

Location: J11

Time: 13:00

*Abstract:*

Igusa’s p-adic zeta function Z(s) attached to a polynomial f in N variables is a meromorphic function on the complex plane that encodes the numbers of solutions of the equation f=0 modulo powers of a prime p. It is expressed as a p-adic integral, and Igusa proved that it is rational in p^{-s} using resolution of singularities and the change of variables formula. From this computation it is immediately clear that the order of a pole of Z(s) is at most N, the number of variables in f. In 1999, Wim Veys conjectured that the only possible pole of order N of the so-called topological zeta function of f is minus the log canonical threshold of f. I will explain a proof of this conjecture, which also applies to the p-adic and motivic zeta functions. The proof is inspired by non-archimedean geometry (Berkovich spaces) but the main technique that is used is the Minimal Model program in birational geometry. This talk is based on joint work with Chenyang Xu.

Speaker: Johannes Nicaise (University of Leuven / ICL)Location: J11

Time: 16:00

*Abstract:*

In this talk i will introduce known results on Beauville conjecture. In particular, i will explain boundary conditions for the second Betti number which follows from Rozansky-Witten invariants and so(4, b_2-2)-action on cohomology. Guan has proved that in dimension four there can exist only finite numbers of hyperkahler manifolds. In dimension six and more some results are obtained by Sawon and Kurnosov.

Speaker: Nikon Kurnosov (Higher School of Economics (Moscow))Location: J11

Time: 16:00

*Abstract:*

In this talk I will introduce the notion of holomorphically symplectic category and discuss some of their basic properties. I will focus in particular on a 4 dimensional modular example whose Hochschild numbers reveal very interesting features.

Speaker: Roland Abuaf (ICL)Location: J11

Time: 16:00

*Abstract:*

Let C be a pre-triangulated category. Its homotopy category H^0© is triangulated; in particular, a commutative square induces a morphism between the cones of its rows. I am going to show how an attempt to lift this morphism into the original pre-triangulated category C reveals a structure which is best described as the data of an A-infinity functor. This is based on a joint work with Timothy Logvinenko.

Speaker: Rina Anno (University of Pittsburgh)Location: J11

Time: 16:00

*Abstract:*

Let H be a numerical semigroup minimally generated by a1<dots<ar. We show that if we bound the width wd(H):=ar-a1, then the Betti numbers of the tangent cone grmK[H] are bounded as well. We conjecture what these bounds are in terms of the width and we present evidence to support this. This is joint work with Juergen Herzog.

Speaker: Dumitru Stamate (University of Bucharest)Location: LT11

Time: 15:00

*Abstract:*

We discuss constructions of quasiphantom categories on some fake quadrics in details

Speaker: Kyoung-Seog Lee (KIAS)Location: J11

Time: 16:00

*Abstract:*

Non-commutative Hodge theory is the study of Hodge structures on the cyclic homology groups of dg-categories C. In this talk, we will study the case C= D^bCoh(X/G), where X is a projective-over-affine variety and G is an algebraic group. Using Halpern-Leistner’s theory of derived Kirwan surjectivity, we prove the collapse of nc Hodge-to-de Rham spectral sequence in variety of situations for example when X is smooth, G is reductive and G(X,OX)G is finite dimensional. These results on the degeneration of the spectral sequence also extend to categories of singularities. Using homotopy theoretic methods, there is a Chern character from a topological version of algebraic K-theory to this periodic cyclic homology. We show that this map is an isomorphism, thereby putting an integral structure and ultimately a weight zero Hodge structure on the periodic cyclic homology. Along the way, we identify the periodic cyclic homology with a complexified version of equivariant K-homology in the sense of Atiyah and Segal. This is joint work with Dan Halpern-Leistner.

Speaker: Daniel Pomerleano (Imperial College London)Location: J11

Time: 16:00

*Abstract:*

In a celebrated body of work, Mark Haiman set out to prove a combinatorial conjecture of Macdonald about his eponymous symmetric functions, and to do so wound up proving some geometric theorems about the Hilbert scheme of points in the plane. This will be a high-level and idiosyncratic overview of this work – we won’t assume any knowledge of symmetric functions, but we will assume basic knowledge of the Hilbert scheme of points on the level of my talks last semester.

Speaker: Paul Johnson (Sheffield)Location: J11

Time: 16:00

*Abstract:*

Given a finite group G, a representation of G over the complex numbers and a prime p, one can define a representation of G in characteristic p by a process of modular reduction. A natural question to ask is which irreducible representations of G remain irreducible in characteristic p. I will talk about how this question is answered when G is a symmetric or alternating group, before going on to describe some work in progress on double covers of symmetric groups. I will try to keep the talk at a very introductory level.

Speaker: Matthew Fayers (Queen Mary)Location: J11

Time: 16:00

*Abstract:*

The moduli space of line bundles on a curve can be non-compact even when the worst singularities of the curve are nodes. A natural compactification is obtained by adding stable rank-1 torsion-free sheaves: such compactification depends on the choice of a polarization on the nodal curve. Similarly, when compactifying the universal Jacobian over the moduli space of stable curves, one obtains a family of compact birational moduli spaces that depend on a polarization parameter. In this talk I will present a wall-crossing formula that describes how the theta divisor varies as a function of this parameter. This is a joint work with Jesse Kass (South Carolina).

Speaker: Nicola Pagani (Liverpool)Location: J11

Time: 16:00

*Abstract:*

Tate homology and cohomology originated in the realm of group algebras, and the theories generalize to Iwanaga-Gorenstein rings in a straightforward manner. The cohomological theory has a more far-reaching generalization to the setting of associative rings; it is now called stable cohomology, and it agrees with Tate homology over Iwanaga-Gorenstein rings. On the homological side, the picture is not this clear. In the talk I will discuss recent work - joint with Olgur Celikbas, Li Liang, and Grep Piepmeyer - on that topic.

Speaker: Lars Winther Christensen (Texas Tech University)Location: J11

Time: 10:30

*Abstract:*

A central result in the theory of Hilbert schemes of points on surfaces is the identification of their cohomology with the Fock module over a Heisenberg algebra by means of the Nakajima operators. In this talk, I aim to present two constructions on the level of the derived categories of symmetric quotient stacks which are related to these operators.

Speaker: Andreas Krug (Warwick)Location: J11

Time: 16:00

Location: J11

Time: 16:00

*Abstract:*

The aim of these talks is to review the basics on Lyubeznik numbers and to introduce a new set of invariants associated to the linear strands of a minimal free resolution of a graded ideal in the polynomial ring. It turns out that these invariants satisfy some properties analogous to those of Lyubeznik numbers of local rings. For the case of squarefree monomial ideals we have a close relation between both invariants that allows to interpret Lyubeznik numbers of Stanley-Reisner rings as the obstruction to the acyclicity of the linear strands of their associated Alexander dual ideals. Finally, we prove that Lyubeznik numbers of Stanley-Reisner rings are not only an algebraic invariant but also a topological invariant, meaning that they depend on the homeomorphic class of the geometric realization of the associated simplicial complex and the characteristic of the base field. The non-expository parts of these talks are based on joint works with K.Yanagawa and A.Vahidi respectively.

Speaker: Josep Alvarez-Montaner (Universitat Politècnica de Catalunya)Location: F41

Time: 14:00

*Abstract:*

(Relative) singularity categories are triangulated categories associated with (non-commutative resolutions of) singular varieties. I will explain these notions and their mutual relations focusing on the simplest examples - the singularities of type A_1, e.g. k[x|/x^2. For these examples, everything can be understood in a rather elementary way. In particular, familiarity with triangulated categories will NOT be necessary to follow the talk. In the end, I will mention what we know for ADE-singularities in general. This is based on joint work with Dong Yang.

Speaker: Martin Kalck (Edinburgh)Location: J11

Time: 15:00

*Abstract:*

For a smooth projective algebraic variety, according to a conjecture of Dubrovin, the semisimplicity of quantum cohomology is related to the existence of full exceptional collections in the derived category of coherent sheaves. I will start by giving a general introduction to quantum cohomology and, eventually, talk about a recent joint work with S. Galkin and A. Mellit on Dubrovin’s conjecture for the symplectic isotropic Grassmannian IG(2,6). This appears to be the simplest case where one needs to work with the big quantum cohomology to formulate the conjecture.

Speaker: Maxim Smirnov (ICTP)Location: J11

Time: 13:00

*Abstract:*

In this talk I will describe a special class of Laurent polynomials, which we call “maximally-mutable”. These Laurent polynomials arise naturally in the study of Fano manifolds via mirror symmetry. In particular, I will explain why in dimension two, the rigid maximally-mutable Laurent polynomials correspond exactly, under mirror symmetry, with the 10 deformation families of smooth del Pezzo surfaces. A similar result holds in dimension 3, where the rigid maximally-mutable Laurent polynomials supported on a reflexive polytope correspond precisely with the 98 deformation families of smooth Fano 3-folds with very ample -K.

Speaker: Alexander Kasprzyk (ICL)Location: J11

Time: 16:00

*Abstract:*

I will state two conjectures exploring mirror symmetry for surfaces and its implications for classification, and summarise the evidence available so far.

Speaker: Alessio Corti (ICL)Location: J11

Time: 14:30

Location: J11

Time: 15:00

(2:30 - 3:00: Pretalk in J11).

Speaker: Ziyu Zhang (Bath)Location: J11

Time: 15:00

Location: LTA

Time: 14:00

Location: LTA

Time: 13:00

Location: J11

Time: 15:00

*Abstract:*

I will survey the results on automorphisms of Fano threefolds. In particular, I will speak about finiteness of automorphism groups (valid with a couple of explicitly described exceptions) and about the structure of Hilbert schemes of lines and conics that allows one to control the latter groups. Finally, I will describe some consequences for groups of birational automorphisms of rationally connected threefolds.

Speaker: Konstantin Shramov (Steklov Institute of Mathematics)Location: J11

Time: 15:00