Abstracts for Seminar on Algebra, Geometry and Physics

The talks describe a derived de Rham cohomology
approach to comparison theorems in p-adic Hodge theory.

In this talk I'll describe what a quantum cluster algebra is, and how quantum cluster coefficients arise in the Donaldson-Thomas theory of a quiver with potential.  In particular, we'll be focusing on the link between purity statements on cohomology of vanishing cycles, and positivity in quantum cluster mutation.  In the case of a graded quiver with nondegenerate graded potential, it turns out that we can prove the general positivity conjecture, and even the stronger "Lefschetz property" considered by A.

The tropical vertex group is generated by certain formal symplectomorphisms of the 2-dimensional algebraic torus. It plays a role in some problems in algebraic geometry and mathematical physics, e.g. wall-crossing. It is known that the group itself can be understood in many ways, for example in terms of "counting" certain rational curves or representations of quivers. This leads to nice correspondences. I will discuss joint work with M. Reineke and T.

Classical deformation theory is based on the Classical Master Equation (CME), a.k.a. the Maurer-Cartan
Equation: dS + 1/2 [S,S] = 0. Physicists have been using a quantized CME, called the Quantum Master Equation
(QME), a.k.a. the Batalin-Vilkovisky (BV) Master Equation: dS + h \DeltaS + 1/2 {S,S} = 0. The CME is defined in a dg
Lie algebra g, whereas the QME is defined in a space V [[h]] of formal power series with values in a differential graded
(dg) BV algebra V. One can anticipate a generalization of classical deformation theory arising from the QME or quantum