All the videos will be available at the end of the workshop.
10.00-11.00 Don Zagier video
12.10-13.00 joint (virtual) lunch. You are welcome to stay and share some annedoctes on Lothar.
13.000-14.00 Geir Ellingsrud video
10.00-11.00 Pierrick Bousseau video
11.10-12.10 Fabio Perroni video
12.10-13.00 joint (virtual) lunch. You are welcome to stay and share some annedoctes on Lothar.
13.00-14.00 Melissa Liu video
I will present a proof of the flow tree formula previously conjectured by Alexandrov and Pioline expressing general Donaldson-Thomas invariants of quivers with potentials in terms of the generally much simpler attractor Donaldson-Thomas invariants. The formula takes the form of a sum over trees weighted with Block-Göttsche multiplicities. This is joint work with Hülya Argüz.
The theorem says that if two normal, projective varieties over uncountable, algebraically closed fields K and L of characteristic zero have homeomorphic Zariski topologies, then they are isomorphic as schemes. I shall report on this result and indicate a slightly simplified proof. I shall also report on my work in (slow) progression trying to establish a version valid in positive characteristic.
I will report on joint paper with Y. Dutta in which we try to control the variation of curves in an ample linear system on a K3 surface by two techniques: Degeneration to the Hitchin system and restriction theorems for the tangent bundle.
In her 2019 PhD thesis, Catherine Cannizzo proved a version of homological mirror symmetry relating the derived category of coherent sheaves on a genus two curve to the Fukaya-Siedel category of a Landau-Ginzburg model (Y,W) where Y is a non-compact Calabi-Yau threefold and W is a holomorphic function on Y known as the superpotential. The critical locus of W is a banana configuration of three P^1's whose symplectic areas A_1, A_2, A_3 are Kahler parameters of Y. Cannizzo considered a ray in the Kahler moduli corresponding to A_1=A_2=A_3. In this talk, I will describe a version of global homological mirror symmetry over the complex three dimensional stringy Kahler moduli of Y which can be identified with a covering space of the moduli of complex structures on a genus two curve. Joint work with Haniya Azam, Catherine Cannizzo, and Heather Lee.
I will explain how to think about the Gromov-Witten theory of hypersurfaces in projective space (and more generally complete intersections). The interesting new aspect is the control of the primitive cohomology. Full use of monodromy, degeneration, and a new theory of nodal relative geometry, leads to an inductive solution. A consequence is that all Gromov-Witten cycles for hypersurfaces (and complete intersections) lie in the tautological ring of the moduli space of curves. Joint work with Argüz, Bousseau, and Zvonkine.
In this talk, we will study the moduli stack of twisted stable maps from pointed genus 0 curves to BG, for G a finite group, i.e. the stack of admissible G-covers of such curves. In particular, we will address the problem of determining the Betti numbers of this stack, by first considering its class in the Grothendieck group of stacks. We will show how such "motive" can be recursively reduced to the motive of the corresponding substack of covers of smooth curves. This is an ongoing joint work with Massimo Bagnarol.
Bridgeland studied moduli of stable sheaves on elliptic surfaces by usuing relative Fourier-Mukai transforms. In particular, under suitable conditions on the Chern classes, moduli spaces are birationally equivalent to the Hilbert scheme of points. We shall slightly modifiy his argument to describe the Picard groups of the moduli spaces if there is no multiple fibers.
Sheaves on non-reduced curves can appear in moduli spaces of 1-dimensional semistable sheaves over a surface, and moduli spaces of Higgs bundles as well. We estimate the dimension of the stack M_X(nC, \chi) of pure sheaves supported at the non-reduced curve nC (n ≥ 2) with C an integral curve on X. We prove that the Hilbert-Chow morphism h_{L,\chi} : M_X^H(L, \chi) → |L| sending each semistable 1-dimensional sheaf to its support have all its fibers of the same dimension for X Fano or with trivial canonical line bundle and |L| contains integral curves. The strategy is to firstly deal with the case with C smooth and then do induction on the arithmetic genus of C which once can decrease by a blow-up given C singular. As an application, we generalize the result of Maulik-Shen on the cohomology \chi-independence of M_X^H(L,\chi) to X any del Pezzo surface not necessarily toric.
The theory of mock modular forms was created in the 2002 thesis of Sander Zwegers as a generalization of the famous "mock theta functions" of Ramanujan, of which Ramanujan had given many examples but no intrinsic characterization. One of the approaches given by Zwegers involves the Taylor expansion of meromorphic Jacobi forms, which have a "wall-crossing" property as some contour of integration crosses a pole of the function, and he noticed that the functions defined some years earlier in a paper by Lothar Göttsche and myself on Donaldson invariants of 4-manifolds, which were also based on a wall-crossing property, were in fact examples of mock modular forms before either the notion or the term existed. The same wall-crossing property occurred some years later in mathematical physics in the study of the string theory of black holes, the explanation there too being given by mock modular forms. In the lecture I will try to explain what mock modular forms and how these various properties arise.