Geometry and Mathematical Physics Seminar

The Geometry and Mathematical Physics Seminar usually takes place on Tuesday at 16:00 during term time in Watson Building.

Spring 2020

Dimer models, matrix factorizations, and Hochschild cohomology

  • Michael Wong, University College London
  • Tuesday 03 March 2020, 16:00
  • Physics West 115

A dimer model is a type of quiver embedded in a Riemann surface. It gives rise to a noncommutative 3-Calabi-Yau algebra called the Jacobi algebra. In the version of mirror symmetry proved by R. Bocklandt, the wrapped Fukaya category of a punctured surface is equivalent to the category of matrix factorizations of the Jacobi algebra of a dimer, equipped with its canonical potential. With the aim of studying deformations, I will describe the Hochschild cohomologies of the Jacobi algebra and the associated matrix factorization category in terms of dimer combinatorics.

Moduli theory, stability of fibrations and optimal symplectic connections

  • Ruadhaí Dervan, Cambridge
  • Tuesday 18 February 2020, 16:00
  • Physics West 115

I will describe a new notion of stability associated to fibrations in algebraic geometry. The definition generalises, and is analogous to, the notion of slope stability of a vector bundle. Much like Tian-Donaldson’s notion of K-stability, there is an associated notion of a “canonical metric”, in the form of an “optimal symplectic connection”. Our main result shows that the existence of an optimal symplectic connection implies that the fibration is semistable. There is an associated moduli problem for stable fibrations over a fixed base, and I will also explain certain aspects of this. This is joint work with Lars Sektnan.

Title to be confirmed

  • Alfredo Nájera Chávez, UNAM Oaxaca
  • Tuesday 11 February 2020, 16:00
  • Physics West 115

Abstract not available

A primer on Calabi-Yau algebras

  • Dan Kaplan, University of Birmingham
  • Tuesday 04 February 2020, 16:00
  • Physics West 115

Calabi-Yau (CY) algebras are ubiquitous in geometry, arising for example as functions on an affine CY-variety, group algebras of the fundamental group of a manifold, and Jacobi algebras of a quiver with potential. In this talk I’ll provide an example-oriented treatment of their history, properties, and applications. Then I’ll turn to preprojective algebras, which are classically 2-CY in the non-Dynkin case. Finally, I’ll discuss joint work with Travis Schedler where we conjecture the same for a multiplicative version of preprojective algebras—defined by Crawley-Boevey and Shaw—and prove the conjecture for quivers containing a cycle.

Algebraic and combinatorial decompositions of Fuchsian groups

  • Daniel Labardini Fragoso, Universidad Nacional Autónoma de México
  • Friday 24 January 2020, 15:00
  • Lecture Theatre B, Watson Building 

The discrete subgroups of PSL _2(R) are often called ‘Fuchsian groups’. ForFuchsian groups \Gamma whose action on the hyperbolic plane H is free, theorbit space H/\Gamma has a canonical structure of Riemann surface with ahyperbolic metric, whereas if the action of \Gamma is not free, thenH/\Gamma has a structure of ‘orbifold’. In the former case, there is adirect and very clear relation between \Gamma and the fundamental group\pi_1(H/\Gamma,x): a theorem of the theory of covering spaces states thatthey are isomorphic. When the action of \Gamma is not free, the relationbetween \Gamma and \pi_1(H/\Gamma,x) is subtler. A 1968 theorem ofArmstrong states that there is a short exact sequence1->E->\Gamma->\pi_1(H/\Gamma,x)->1, where E is the subgroup of \Gammagenerated by the elliptic elements. For \Gamma finitely generated,non-elementary and with at least one parabolic element, I will presentfull algebraic and combinatorial decompositions of \Gamma in terms of\pi_1(H/\Gamma,x) and a specific finitely generated subgroup of E, thusimproving Armstrong’s theorem.

This talk is based on an ongoing joint project with Sibylle Schroll andYadira Valdivieso-Díaz that aims at describing the bounded derivedcategories of skew-gentle algebras in terms of curves on surfaces withorbifold points of order 2.

Lagrangian tori and cluster charts

  • Marco Castronovo, Rutgers Universit
  • Monday 13 January 2020, 16:30
  • Watson (Mathematics) Room 310

I will describe a conjectural correspondence between Lagrangian tori in a real symplectic manifold and algebraic tori in a mirror variety. It is not clear what this mirror should be, but for coadjoint orbits work of Rietsch suggests a relation to Langlands duality. I will then explain how to partially verify this correspondence for Grassmannians. This point of view allows to answer purely dynamical questions about displaceability and abundance of Lagrangians.

Find out more

There is a complete list of the talks in the Geometry and Mathematical Physics seminar on talks@bham.