Tropical Mathematics and its Applications

To register, please send an e-mail to b.smith9@lancaster.ac.uk or Rob.Silversmith@warwick.ac.uk.

If you are a UK-based PhD student requiring funding to support your travel, please let us know by 5th May.

Hannah Markwig: Tropical curve counting

In enumerative geometry, we fix geometric objects and conditions and count how many objects satisfy the conditions. For example, there are 2 plane conics passing through 4 points and tangent to a given line. Tropical geometry can be viewed as a degenerate version of algebraic geometry and has proved to be a successful tool for enumerative problems. We review tropical curve counting problems. In particular, we show how tropical methods can be applied for quadratically enriched counts, which can be viewed as generalizations that allow results over any ground field. 

Stefano Mereta: The space of valuated preorders and the congruence spectrum of S[x1,…,xn] as the spectrum of a ring 

In the late '60s, Hochster introduced the notion of spectral space and proved that every spectral space is the spectrum of a commutative ring. Unfortunately the proof of this fact is not constructive. More recently, Jun, Ray and Tolliver have proven constructively that every spectral space is the k-spectrum of an idempotent semiring, and that the congruence spectrum of an idempotent semiring is a spectral space.

In this talk we will introduce the notion of a valuated preorder on the monomials of K[x1,…,xn] for a valued field K and realise it, relying partially on work by Jóo and Mincheva, as the k-spectrum of a semiring, as the congruence spectrum of a polynomial semiring and, most importantly, as the spectrum of a commutative ring.

This commutative ring is constructed as the "unit ball" of respect a generalised Bézout valuation.

Alexander Esterov: Schön complete intersections

There is a number of "aesthetically similar" topics in combinatorial algebraic geometry, such as toric complete intersections, hyperplane arrangements, simplest singularity strata of general polynomial maps, some discriminant and incidence varieties in enumerative geometry and polynomial optimization, polynomial ODEs such as reaction networks, generalized Calabi--Yau complete intersections.

I will talk about a convenient umbrella generality for all of them, which still admits a version of the classical theory of Newton polytopes (but with so-called tropical complete intersections instead of polytopes).