From March 8 to April 19, the Center of Mathematical Sciences and Applications will be hosting a special lecture series on DonaldsonThomas and GromovWitten Theories. Artan Sheshmani (QGM Aarhus and CMSA Harvard) will give eight talks on the topic on Wednesdays and Fridays from 9:0010:30 am, which will be recorded and promptly available on CMSA’s Youtube Channel.
The topic of the talks are as follows:
Date  Title  Abstract 
332017  Modularity of DT invariants on smooth K3 fibrations I  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 smooth 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 whereXis aK3 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. We are intending to go through these lectures with a relatively slow speed, that is: TheoremProof style, (so we will be less handwavy!) and we intend to establish some of the required background, e.g NoetherLefschetz theory, vector valuedmodular forms etc. 
382017  Modularity of DT invariants on smooth K3 fibrations II  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 smooth 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 whereXis aK3 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. We are intending to go through these lectures with a relatively slow speed, that is: TheoremProof style, (so we will be less handwavy!) and we intend to establish some of the required background, e.g NoetherLefschetz theory, vector valued modular forms etc. 
3102017  Conifold Transitions and modularity of DT invariants on Nodal fibrations 
Following lectures I and II we continue the discussion on the moduli space of shaves with two dimensional support on K3fibered threefolds, which can admit finitely many nodal (rational double point singularity at worst) fibers. We will use the conifold transitions and degeneration techniques in this case to relate the geometry of our moduli space and its enumerative invariants to the ones studied in lectures I, II over smooth K3fibrations.

452017  Stable pair PT invariants on smooth fibrations I  We study PandharipandeThomas’s stable pair theory on smooth K3 fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of KawaiYoshioka’s formula for the Euler characteristics of moduli spaces of stable pairs on K3 surfaces and NoetherLefschetz numbers of the fibration. 
472017  Stable pair PT invariants on smooth fibrations II

We study PandharipandeThomas’s stable pair theory on smooth K3 fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of KawaiYoshioka’s formula for the Euler characteristics of moduli spaces of stable pairs on K3 surfaces and NoetherLefschetz numbers of the fibration. 
4122017  Stable pair PT invariants on nodal fibrations: perverse sheaves, Wallcrossings, and an analog of fiberwise Tduality  Following lecture 4, we continue the study of stable pair invariants of K3fibered threefolds., We investigate the relation of these invariants with the perverse (noncommutative) stable pair invariants of the K3fibration. In the case that the fibration is a projective CalabiYau threefold, by means of wallcrossing techniques, we write the stable pair invariants in terms of the generalized DonaldsonThomas invariants of 2dimensional Gieseker semistable sheaves supported on the fibers. 
4142017  DT versus MNOP invariants and S_duality conjecture on general complete intersections  Motivated by Sduality modularity conjectures in string theory, we define new invariants counting a restricted class of twodimensional torsion sheaves, enumerating pairs Z⊂H in a Calabi–Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a onedimensional subscheme of it. The associated sheaf is the ideal sheaf of Z⊂H, pushed forward to X and considered as a certain Joyce–Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X. 
4192017  Proof of Sduality conjecture on quintic threefold I  I will talk about an algebraicgeometric proof of the Sduality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic CalabiYau threefold. We use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity. More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasiprojective surface is a modular form. This is a generalization of the result of OkounkovCarlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of NoetherLefschetz numbers as I will explain, will provide the ingredients to achieve an algebraicgeometric proof of Sduality modularity conjecture. 
4282017  Proof of Sduality conjecture on Quintic threefold II  I will talk about an algebraicgeometric proof of the Sduality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic CalabiYau threefold. We use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity. More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasiprojective surface is a modular form. This is a generalization of the result of OkounkovCarlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of NoetherLefschetz numbers as I will explain, will provide the ingredients to achieve an algebraicgeometric proof of Sduality modularity conjecture. 