This talk asks which tropicalisations of subvarieties of the torus know the cohomology of the original variety? A motivating example are linear embeddings of complements of hyperplane arrangements. We prove that the tropicalisation knows the cohomology of the variety in a strong sense if and only if it satisfies local tropical Poincaré duality and the original variety is so-called “…

Let g be a finite-dimensional, simple Lie algebra over the field of complex numbers, and U be the quantum, untwisted affine algebra, associated to g. Via Lusztig's q-exponential formulae, it is well known that the affine braid group of g acts on any integrable representation of U. In particular, one obtains an action of the coroot lattice of g on such a representation.

In this talk, I will present an explicit formula for these lattice operators. Surprisingly, the answer is given as a normalized limit of a well-known series, which we call Chari-Pressley series, and hinges upon their rationality on integrable representations. Time permitting, I will discuss some applications towards the universal R-matrix of the quantum affine algebra, monodromy of trigonometric Casimir equations and geometry of Nakajima quiver varieties.

This formula and the rationality of CP series were obtained in my recent joint work with V. Toledano Laredo (arxiv:2501.02365).

A classical result due to Clebsch from the mid-nineteenth century confirms that every complex space sextic curve (given as an intersection of a quadric and a cubic surface in projective 3-space) has exactly 120 tritangent planes. In this talk we will show how to use combinatorial methods arising from tropical geometry to revisit this classical problem and perform the analogous…

Assigning characteristic classes to singularities is a powerful tool of enumerative geometry. In this talk, we survey various flavors of such classes, their computational techniques, and some applications. A challenge is that these classes are typically very difficult to compute explicitly. This motivates the search for structure theorems that constrain their possible values. One such structure theorem comes from…

We construct affine charts of a smooth projective toric variety which contain its non-
negative points, and which admit a closed embedding into the total coordinate space
of Cox’s quotient construction. We show that such positive charts arise from smooth
subcones of the nef cone. To each positive chart we associate an algebraic moment map,<…

Abstract: TBA