Fibrations, Mirror Symmetry and CalabiYau Geometry
Org:
Charles Doran (Alberta),
Andreas Malmendier (Utah State) and
Alan Thompson (Waterloo)
[
PDF]
 LARA ANDERSON, Virginia Tech
Heterotic and Ftheory Compactifications and Geometry [PDF]

We systematically analyze a broad class of dual heterotic and Ftheory models that give rise to six and fourdimensional supergravity theories, and compare geometric constraints on the two sides of the duality. In this talk we will demonstrate that both theories together give new insight into the space of possible effective theories arising from string theory. We will describe explicit new results regarding elliptic and K3fibration structures and string dualities.
 MBOYO ESOLE, Harvard University
Geometric Engineering of SU(3) SU(2) through Elliptic Fibrations: Resolutions and Flops. [PDF]

I will review the different ways to engineer a SU(2)SU(3) gauge theory using elliptic fibrations and discuss the resulting geometry. In particular, I will present many new nonKodaira fibers that naturally occur in this setting. I will also discuss the network of flops between different nonsingular elliptic fibrations sharing the same Jacobian (Weierstrass model).
 ANTONELLA GRASSI, University of Pennsylavania
Birational Geometry of Higher dimensional elliptic fibrations [PDF]

I wil present comparison and differences between Calabi Yau threefolds and fourfolds which are elliptically fibered.
 ANDREW HARDER, University of Alberta
Tyurin degenerations, K3 fibrations and the BatyrevBorisov construction [PDF]

If a CalabiYau threefold degenerates to the normal crossings union of a pair of quasiFano threefolds, then mirror symmetry predicts that there is a corresponding K3 fibration on the mirror CalabiYau threefold. I will show that this prediction is compatible with the BatyrevBorisov mirror construction, and I will discuss how the resulting K3 fibration reflects the mirrordual degeneration.
 ATSUSHI KANAZAWA, Harvard CMSA
Holomorphic symplectic geometry of the space of Bridgeland stability conditions [PDF]

For a CalabiYau triangulated category D of odd dimension, the space Stab(D) of Bridgeland stability conditions on D is naturally a holomorphic symplectic manifold. I will discuss geometry of Lagrangian submanifolds of Stab(D) in comparison with the DonagiMarkman integrable system. I will also explain our attempt to identify the gauged Kahler moduli space inside Stab(D) when D comes from CalabiYau geometry. This is a joint work with Y.W. Fan and S.T. Yau.
 ALEX MOLNAR, Queen's University
On CalabiYau threefolds of CMtype [PDF]

The first known examples of elliptic curves satisfying the Birch and SwinnertonDyer conjecture were elliptic curves with complex multiplication. With this in mind, one may try to study the BeilinsonBloch conjecture for (rigid) CalabiYau threefolds by first examining (rigid) CalabiYau threefolds of CMtype.
We will discuss an elementary construction of such threefolds (over the complex numbers) towards a classification up to birational equivalence, as well as the limits of this approach and an expected classification. We will also briefly mention how, specifically, the CMtype allows us to achieve arithmetic results.
 SIMON ROSE, Copenhagen University
Quasimodularity of generalized sumofdivisors functions [PDF]

In this talk I will present a generalization of the sumofdivisors introduce by P. A. MacMahon, and show how they are quasimodular forms. I will finish with some speculation as to the geometric significance of these functions.
 EMANUEL SCHEIDEGGER, Freiburg
 EGOR SHELUKHIN, Institute for Advanced Study
Nontrivial Hamiltonian fibrations via Ktheory quantization [PDF]

We show how quantization of families with values in Ktheory can detect nontrivial Hamiltonian fibrations, yielding examples that are not detected by previous methods (the characteristic classes of Reznikov for example). We also upgrade a theorem of Spacil on the cohomologysurjectivity of a natural map of classifying spaces by providing it with an "almost" weak retraction. Joint work with Yasha Savelyev.
 WASHINGTON TAYLOR, MIT
Classification and enumeration of elliptic CalabiYau threefolds and fourfolds [PDF]

Recent work motivated by physics has led to progress in understanding elliptic CalabiYau threefolds and fourfolds, using new mathematical and computational tools for analyzing the geometry of the bases that support such fibrations. This talk will give an introduction to some aspects of this research program, including: the identification (work with D. Morrison) of irreducible geometric structures in the base geometry that facilitate the classification of allowed bases, connections between codimension two singularities and representation theory, MordellWeil groups, a systematic approach to enumerating elliptic CalabiYau threefolds with large $h_{2,1}$, and a Monte Carlo study of $\sim 10^{50}$ distinct toric threefold bases that support elliptic CalabiYau fourfolds. A brief description will also be given of applications to physics including hints at how the observed standard model of particle physics may emerge from "typical" features of CalabiYau fourfolds.
 URSULA WHITCHER, University of WisconsinEau Claire
Arithmetic Mirror Symmetry and Isogenies [PDF]

Arithmetic mirror symmetry is a relationship between the number of points on appropriately chosen mirror pairs of CalabiYau varieties over finite fields. We investigate whether arithmetic mirror relationships observed for diagonal pencils in weighted projective spaces can be extended to mirror families obtained via the BatyrevBorisov construction. Our results show that arithmetic mirror symmetry is controlled by an isogeny structure. This talk describes joint work with Christopher Magyar.
© Canadian Mathematical Society