Geometry and Mathematical Physics Seminar

The Geometry and Mathematical Physics Seminar usually takes place on Tuesday at 15:00 during term time.

Contact person: Cyril Closset

Spring 2022 seminars

Before Easter vacation (in March):

Title: Higgs Branches of 5d Superconformal Field Theories

  • Marieke Van Beest, University of Oxford
  • Tuesday 8 March 2022, 15:30
  • Watson-LT B (101)

Abstract: 5d superconformal field theories (SCFTs) are intrinsically strongly-coupled systems which can be accessed through geometric engineering in M-theory, or 5-brane-webs in type IIB string theory. In this talk, I will determine the Higgs branch of 5d SCFTs from their realization as a generalized toric polygon. This approach is motivated by a dual, tropical curve decomposition of the 5-brane-web. I will define a decomposition of the generalized toric polygon into a refined Minkowski sum, which, in the case of strictly toric polygons, reduces to the description of deformations of the Calabi-Yau singularities in terms of Minkowski sums. Moreover, successive decoupling of matter generates a tree of descendant 5d SCFTs. I will describe how the geometry of the Higgs branch is transformed under decoupling, and demonstrate a correspondence between Fayet-Iliopoulos deformations in 3d and 5d mass deformations.

Title: Derived projectivizations of complexes

  • Qingyuan Jiang, University of Edinburgh
  • Tuesday 15 March 2022, 15:00
  • Watson-LT B (101)

Abstract: In this talk, we will discuss the counterpart of Grothendieck's projectivization construction in the context of derived algebraic geometry. (1) We will first introduce the background, definitions, and study the fundamental properties. (2) Then, we will focus on complexes of perfect-amplitude contained in [0,1]. In this case, the derived projectivization enjoys special pleasant properties. For example, they satisfy the generalized Serre's theorem, a derived version of Beilinson's relations, and there are structural decompositions for their derived categories.(3) Finally, we will discuss some applications of this framework, including:     (3-i) applications to classical situations, such as derived categories of certain reducible schemes and irreducible singular schemes.     (3-ii) applications to Hecke correspondence moduli, focusing on the cases of surfaces.     (3-iii) applications to moduli of pairs and moduli of extensions, focusing on the cases of curves, surfaces, and threefolds. If time allows, we might also discuss the generalizations of these results to the cases of derived Grassmannians and some other types of derived Quot schemes.

 

After Easter vacation:

Title: Interpolations of monoidal categories by invariant theory

  • Ehud Meir, University of Aberdeen
  • Tuesday 3 May 2022, 15:00
  • Watson-LT B (101)

Abstract: In this talk I will consider algebraic structures such as lie, Hopf, and Frobenius algebras. I will show that under certain assumptions such structures can be reconstructed from their scalar invariants. I will then show how one can interpolate the category of representation of the automorphism groups of the structures by interpolation the invariants of the algebraic structures. In this way we recover the constructions of Deligne for categories such as Rep(S_t), Rep(O_t) and Rep(Sp_t), as well as some of the constructions done by Knop and by Khovanon-Ostrik-Kononov.

Title: On the Prym-Hitchin connection

  • Johan Martens, University of Edinburgh
  • Tuesday 10 May 2022, 15:00
  • Watson-LT B (101)

Abstract: I will discuss an algebraic approach to the Hitchin connection on bundles of non-abelian theta functions, which generalises to higher rank Prym varieties.  Combined with a Prym version of strange duality, this explains some exceptional behaviour of the monodromy of the standard Hitchin connection. This is joint work with Baier, Bolognesi and Pauly.

Title: Exploring the landscape of Fano varieties

  • Alexander Kasprzyk, University of Nottingham
  • Tuesday 17 May 2022, 15:00
  • Watson-LT B (101)

Abstract: Fano varieties are fundamental objects in algebraic geometry: they form the "atomic pieces" of geometry. Although we know that there are finitely many deformation families of Fano varieties in each dimension, despite over eighty years of work their classification eludes us. Recent advances, drawing on ideas from mirror symmetry, allow us to begin systematically mapping the landscape of Fano varieties. Mirror symmetry suggests that geometric objects come in "mirror pairs": in this setting, the mirror partner to a Fano variety is a Laurent polynomial. These mirrors satisfy many beautiful combinatorial properties, and are accessible to systematic classification. With the help of large-scale computation, we have started populating the classifications of Fano varieties in dimensions 3 and 4, finding hundreds of new Fano varieties in the process. Furthermore, techniques from data science and Machine Learning have exposed deep, previously unsuspected structure in this data. Although the mathematics required to explain these structures may take several decades to develop, this approach is allowing us to start exploring the fascinating landscape of Fano varieties today.

Title: Diagonal pencils and Hasse-Witt invariants

  • Ursula Whitcher, University of Michigan
  • Tuesday 24 May 2022, 15:00
  • Watson-LT B (101)

Abstract: Using a natural combinatorial generalization of the Fermat quartic and
the Batyrev mirror symmetry construction, we obtain a collection of K3
surface pencils of generic Picard rank 19 in Gorenstein Fano toric
varieties. We characterize point counts on these varieties over finite
fields using Picard-Fuchs equations and classical hypergeometric
functions. This talk describes joint work with Adriana Salerno.

Title: Wobbly bundles over curves

  • Christian Pauly, Université de Nice
  • Friday 10 June 2022, 14:00
  • Watson-LT A (G23)

Abstract: Very stable vector bundles over curves have been introduced by Drinfeld in the study of the Hitchin system on the moduli space of Higgs bundles. I will give some geometric properties of the locus of non-very stable or wobbly bundles. Recently they were described by S. Pal in terms of non-free rationalcurves on the moduli space of vector bundles.

 

Past (pre-pandemic) seminars 

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, the orbit space H/\Gamma has a canonical structure of Riemann surface with a hyperbolic metric, whereas if the action of \Gamma is not free, then H/\Gamma has a structure of ‘orbifold’. In the former case, there is a direct and very clear relation between \Gamma and the fundamental group\pi_1(H/\Gamma,x): a theorem of the theory of covering spaces states that they are isomorphic. When the action of \Gamma is not free, the relation between \Gamma and \pi_1(H/\Gamma,x) is subtler. A 1968 theorem of Armstrong states that there is a short exact sequence1->E->\Gamma->\pi_1(H/\Gamma,x)->1, where E is the subgroup of \Gamma generated by the elliptic elements. For \Gamma finitely generated, non-elementary and with at least one parabolic element, I will present full algebraic and combinatorial decompositions of \Gamma in terms of\pi_1(H/\Gamma,x) and a specific finitely generated subgroup of E, thus improving Armstrong’s theorem.

This talk is based on an ongoing joint project with Sibylle Schroll and Yadira Valdivieso-Díaz that aims at describing the bounded derived categories of skew-gentle algebras in terms of curves on surfaces with or bifold points of order 2.

Lagrangian tori and cluster charts

  • Marco Castronovo, Rutgers University
  • 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.