2-Verma modules

We construct a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac--Moody algebras, and in particular of parabolic 2-Verma module.Comment: v2, substantial revision, introduction of a notion of dg-2-representation, 65p

Tensor product categorifications, Verma modules and the blob 2-category

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of $\mathfrak{sl}_2$. From the above we derive a categorification of the blob algebra.Comment: v2, 88 pages, reviewed versio

2-Verma modules and the Khovanov-Rozansky link homologies

We explain how Queffelec-Sartori's construction of the HOMFLY-PT link polynomial can be interpreted in terms of parabolic Verma modules for $\mathfrak{gl}_{2n}$. Lifting the construction to the world of categorification, we use parabolic 2-Verma modules to give a higher representation theory construction of Khovanov-Rozansky's triply graded link homology

Real Springer fibers and odd arc algebras

We give a topological description of the two-row Springer fiber over the real numbers. We show its cohomology ring coincides with the oddification of the cohomology ring of the complex Springer fiber introduced by Lauda-Russell. We also realize Ozsv\'ath-Rasmussen-Szab\'o odd TQFT from pullbacks and exceptional pushforwards along inclusion and projection maps between hypertori. Using these results, we construct the odd arc algebra as a convolution algebra over components of the real Springer fiber, giving an odd analogue of a construction of Stroppel-Webster

Biochar fracture resistance

Biochar is a brittle material that tends to break under mechanical stress. This could be an advantage if it is intended to obtain a char powder, but it is typically unwanted since it generates dust that induce biochar losses or even explosion risk. Mechanical stresses are typically observed inside pyrolysis reactors (Scala et al., 2006), during transport/storage and finally inside the soil (Spokas et al., 2014). There are few data in the literature regarding the mechanical strength of char (Capon et al., 1980). This study aims at assessing the impact of pyrolysis temperature and biomass species on fracture resistance. Please click on the file below for full content of the abstract