Geometry and Mathematical Physics Workshop 2019

We are pleased to announce that Loughborough University will host a Workshop in Geometry and Mathematical Physics in March 2019.
Loughborough University Institute of Advanced Studies

Start: 1pm on Wednesday 27th March 2019
Finish: 2pm on Saturday 30th March 2019








 

This meeting is being hosted by the Loughborough Centre for Geometry and Applications as part of the ongoing Semester in Geometry.
Financial support has been kindly provided by the Loughborough University Institute of Advanced Studies, London Mathematical Society scheme 1
London Mathematical Society
grant, and a grant from Foundation Compositio Mathematica.

Organisers:

  • Alan Thompson
  • Alexander Veselov

Schedule:

A campus map showing the locations of the buildings where the talks will take place may be found here:

Campus Map
https://compositio.nl/m/

Wednesday 27th March, all talks in Room U.0.06 (Brockington Extension).
 Time Speaker  Title 
 1:00pm   Martin Hallnäs Deformed Calogero-Moser-Sutherland operators: orthogonality of eigenfunctions and a physical interpretation
 2:00pm   Break  
 2:30pm Tyler Kelly Landau-Ginzburg Mirror Symmetry and open B-model invariants
 3:30pm Break 
 4:00pm Misha Feigin Supersymmetric V-systems

Thursday 28th March, all talks in Jennings Council Chamber (Hazelrigg Building).
 Time    Speaker Title 
 9:30am   Clélia Pech  Quantum cohomology and mirror symmetry for odd symplectic Grassmannians
 10:30am Coffee Break  
 11:00am  
 Michel van Garrel 2d scattering for local Calabi-Yau 4-folds
 12:00pm   Lunch 
 1:00pm Martina Balagovic  The affine VW supercategory
 2:00pm Break  
 2:30pm Hülya Argüz Mirrors to log Calabi-Yau's in dimension three
 3:30pm Tea Break  
 4:00pm Mark Gross  Open FRJW mirror symmetry
 7:00pm Conference Dinner 

Friday 29th March, all 
talks in Jennings Council Chamber (Hazelrigg Building).
 Time   Speaker   Title 
 9:30am Oleg Chalykh Diagonal quasi-invariants of Coxeter groups
 10:30am Coffee Break 
 11:00am Sibylle Schroll  Geometric models and derived invariants for gentle algebras
 12:00pm   Lunch  
 1:00pm Giovanni Felder Cyclotomic expansion and volume conjecture for knot invariants
 2:00pm Break 
 2:30pm Sara Filippini Orbital degeneracy loci and applications
 3:30pm Tea Break  
 4:00pm Michael Shapiro Cluster algebras with Grassmann variables (joint with V. Ovsienko)

Saturday 30th March, all talks in SCH.1.01 (Schofield Building)
 Time    SpeakerTitle 
 9:15am Andrea Appel  The Casimir connection of a symmetrisable Kac-Moody algebra
 10:15am Coffee Break  
 10:45am Iain Gordon Calogero-Moser cells and the Robinson-Schensted algorithm
 11:45am Break  
 12:00pm Anna Barbieri A Riemann-Hilbert problem for uncoupled BPS structures

Abstracts
Andrea Appel (Edinburgh) - The Casimir connection of a symmetrisable Kac-Moody algebra
The Casimir connection is a flat connection with logarithmic singularities over the root hyperplanes of a semisimple Lie algebra. It is equivariant with respect to the action of the Weyl group and produces monodromy representations of the corresponding Artin group (or generalised braid group). In the case of an arbitrary symmetrisable Kac-Moody algebra Felder et al. introduced a truncated and normally ordered Casimir connection, which maintains flatness but loses equivariance. In this talk, I will show how the latter can be restored, producing monodromy representations of Artin groups of Kac-Moody type. This is based on joint works with Valerio Toledano Laredo.

Hülya Argüz (Imperial College London) - Mirrors to log Calabi-Yau's in dimension three
An anti-canonical pair (Y,D) is a smooth projective variety Y of dimension n, together with a normal crossing divisor D in |-K_Y|. The complement Y\D is called a log Calabi-Yau variety. Mirrors to such varieties in dimension two were constructed by Gross-Hacking-Keel from a canonical scattering diagram. We outline steps to generalize this construction to higher dimensions. For this, we use an "asymptotic limit" of the toric scattering construction of Gross-Pandharipande-Siebert. In this talk I will explain the mirrors to some 3-dimensional Fano manifolds we found using these techniques, including mirrors to non-toric blow-ups of three-dimensional projective space. This is work in progress with Tom Coates, Mark Gross and Bernd Siebert.

Martina Balagovic (Newcastle) - The affine VW supercategory
We define the affine VW supercategory sW, which arises from studying the action of the periplectic Lie superalgebra p(n) on the tensor product of an arbitrary representation M with several copies of the vector representation V of p(n). It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group. The main obstacle was the lack of a quadratic Casimir element for p(n). When M is the trivial representation, the action factors through the action of the previously known Brauer supercategory sBr. Our main result is an explicit basis theorem for the space of morphisms in sW, and as a consequence we recover the basis theorem for sBr. The proof utilises the close connection with the representation theory of p(n). As an application we explicitly describe the centres of all endomorphism algebras, and show that they behave well under the passage to the associated graded and under deformation. This is joint work with Zajj Daugherty, Inna Entova-Aizenbud, Iva Halacheva, Johanna Hennig, Mee Seong Im, Gail Letzter, Emily Norton, Vera Serganova and Catharina Stroppel, arising from the WINART workshop.

Anna Barbieri (Sheffield) - A Riemann-Hilbert problem for uncoupled BPS structures
BPS structures locally describe the space of Bridgeland stability conditions of a CY3 category together with a generalized Donaldson-Thomas theory. On the other hand, Riemann-Hilbert problems are inverse problems in the theory of differential equations. After defining the notion of BPS structures I will introduce a Riemann-Hilbert problem attached to semi-classical and refined BPS structures in a simple case study. Part of this is joint work with T.Bridgeland and J.Stoppa.

Oleg Chalykh (Leeds) - Diagonal quasi-invariants of Coxeter groups
Let W be a Coxeter group in its natural reflection representation V. Quasi-invariant polynomials are a certain interpolation between C[V] and the invariants C[V]^W. They were first introduced in the context of the quantum Calogero-Moser system in our work with A. P. Veselov, with further interesting properties discovered in [Feigin—Veselov’02], [Etingof—Ginzburg’02], [Felder—Veselov’03], [Berest—Etingof—Ginzburg’03], [Berest—Chalykh’11]. I will discuss how one may define a “doubled” version of quasi-invariants. Computer experiments suggest some parallels with the theory of diagonal (co)invariants which attracted a lot of attention in connection with Macdonald’s positivity conjecture, geometry of Hilbert schemes, and Cherednik algebras.

Misha Feigin (Glasgow) - Supersymmetric V-systems
We construct N=4 supersymmetric quantum generalized Calogero-Moser systems associated with any root system and, more generally, any V-system. These models have superconformal symmetry algebra D(2,1; \alpha) for a suitable value of parameter \alpha which depends on coupling parameter(s). For type BC_N we also get N=4 supersymmetric trigonometric Calogero-Moser-Sutherland systems. The talk is based on joint work with Georgios Antoniou.

Giovanni Felder (ETH Zürich) - Cyclotomic expansion and volume conjecture for knot invariants
I will report on a circle of ideas I learned from Qingtao Chen relating generalizations of the cyclotomic expansion of quantum invariants of knots, due to Habiro, to the volume conjectures  for coloured Jones polynomials of Kashaev and Murakami-Murakami. An extension from the case of SU(2)  to SU(n), based on results of Q. Chen, K. Liu and S. Zhu and on work in progress with Q. Chen and H. Zhang, will be presented.

Sara Filippini (Imperial College London) - Orbital degeneracy loci and applications
In joint work with Vladimiro Benedetti, Laurent Manivel and Fabio Tanturri we consider a generalization of degeneracy loci of morphisms between vector bundles, called orbital degeneracy loci. These objects are modelled on orbit closures of algebraic groups in their linear representations. In a preferred class of examples we gain some control over their canonical sheaf. We show how these techniques can be applied to construct Calabi-Yau and Fano varieties of dimension three and four.

Michel van Garrel (Warwick) - 2d scattering for local Calabi-Yau 4-folds
In this joint work with Pierrick Bousseau we consider Calabi-Yau 4-folds X given as the total space of certain rank 2 vector bundles over certain rational surfaces S. We show that the genus 0 local Gromov-Witten (GW) invariants of X equal certain log GW invariants of S, which in turn are calculated by an associated 2d scattering diagram. The log GW invariants admit a natural refinement, which translates to a conjectured refinement for the local invariants. 

Iain Gordon (Edinburgh) - Calogero-Moser cells and the Robinson-Schensted algorithm
This is joint work with Adrien Brochier and Noah White. I will explain a construction of Bonnafe-Rouquier about a partition of the elements of a finite Coxeter group which arises from the theory of rational Cherednik algebras. In the case of the symmetric group, this construction is related to the Galois theory of the (rational) Calogero-Moser phase space. Building on work of Aguirre-Felder-Veselov and of Halacheva-Kamnitzer-Rybnikov-Weekes, I will show that the construction yields a continuous version of the Robinson-Schensted algorithm and as a result confirms a conjecture of Bonnafe-Rouquier.

Mark Gross (Cambridge) - Open FRJW mirror symmetry
I will talk about joint work with Tyler Kelly and Ran Tessler. We propose a form of mirror symmetry for open FJRW theory associated to Fermat polynomials x_1^{r_1}+x_2^{r_2}. I will attempt to give some hint as to what the invariants we calculate are and what the shape of mirror symmetry is in this case.

Martin Hallnas (Gothenburg) - Deformed Calogero-Moser-Sutherland operators: orthogonality of eigenfunctions and a physical interpretation
Deformed (trigonometric A(n-1,m-1)) Calogero-Moser-Sutherland operators were introduced by Chalykh, Feigin and Veselov (1998) and Sergeev (2001), and, formally, they correspond to a system of n+m particles moving on a circle, reducing in the m=0 case to an ordinary Calogero-Moser-Sutherland system. However, this interpretation is problematic: m of the particles have negative mass and the known eigenfunctions, constructed in terms of super-Jack polynomials, are not square-integrable. I will present rather natural, albeit somewhat nonstandard, orthogonality relations for these eigenfunctions and sketch a physical interpretation in terms of second quantised (deformed) Calogero-Moser-Sutherland operators relevant for the fractional quantum Hall effect. The talk is based on joint work with Farrokh Atai and Edwin Langmann.

Tyler Kelly (Birmingham) - Landau-Ginzburg Mirror Symmetry and open B-model invariants
I will talk about what mirror symmetry looks like for Landau-Ginzburg models. Afterwards, I will explain what enumerative B-model invariants look like for Landau-Ginzburg models of the form (\C^n, \sum_i x_i^{r_i}) and what invariants will be mirror to the invariants that Mark Gross will give in the talk on Thursday. This is joint work with Mark Gross and Ran Tessler.

Clélia Pech (Kent) - Quantum cohomology and mirror symmetry for odd symplectic Grassmannians
Odd symplectic Grassmannians are two-orbit algebraic varieties which share a lot of common features with homogeneous varieties, and are one of the nicest examples of a wider family of objects called horospherical varieties. In this talk, I will explain some old and new results about them concerning their quantum cohomology, their derived category, and their mirrors. Part of this work is joint with R. Gonzales, N. Perrin and A. Samokhin.

Sibylle Schroll (Leicester) - Geometric models and derived invariants for gentle algebras
Gentle algebras are a class of tame algebras which naturally arise in various different contexts such as categorifications of cluster algebras, N=2 gauge theories and Fukaya categories of surfaces. In this talk we will describe a geometric model of the bounded derived category of a gentle algebra and we will show how this model relates to the Fukaya category of a surface with boundary and stops.  We will use the geometric model to give a complete derived invariant for gentle algebras.  (This is joint works with S. Opper - P.G. Plamondon and C. Amiot and P.-G. Plamondon). 

Michael Shapiro (Michigan State) - Cluster algebras with Grassmann variables (joint with V. Ovsienko)
We develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of “extended quivers” which are oriented hypergraphs. We describe mutations of such objects and define a corresponding commutative superalgebra. Our construction includes the notion of weighted quivers that has already appeared in different contexts. This project is a step towards understanding the notion of cluster superalgebra.

Photos