Facets of Algebraic Geometry
A Celebration of Combinatorial Techniques

Abstracts

Rohini Ramadas (Warwick)

Title: Complex dynamics: degenerations and irreducibility problems

Abstract: This talk is about an application to complex dynamics of combinatorial algebraic geometry.

Per_n is an affine algebraic curve, defined over Q, parametrizing (up to change of coordinates) degree-2 self-morphisms of P^1 with an n-periodic ramification point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose vanishing locus parametrizes (up to change of coordinates) degree-2 self-morphisms of C with an n-periodic ramification point. Two long-standing open questions in complex dynamics are: (1) Is Per_n connected? (2) Is G_n is irreducible over Q?

We show that if G_n is irreducible over Q, then Per_n is irreducible over C, and is therefore connected. In order to do this, we find a Q-rational smooth point of a projective completion of Per_n. This Q-rational smooth point represents a special degeneration of degree-2 morphisms, and as such admits an interpretation in terms of tropical geometry.

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Evgeny Shinder (Sheffield)

Title: Gromov's cancellation question in birational algebraic geometry

Abstract: I explain some cancellation and non-cancellation phenomena in algebraic geometry. I will relate them to the structure of the Grothendieck ring of varieties and to the groups of birational self-maps of algebraic varieties, in particular the Cremona groups.

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Navid Nabijou (QMUL)

Title: Universality for logarithmic and tropical maps 

Abstract: Logarithmic maps model curves in a variety with fixed tangency orders along a divisor. Their moduli spaces are intricate and lie at the heart of modern enumerative geometry. 

I will discuss work in which we probe the "virtual" singularities of these moduli spaces. Our main result shows that the class of all such singularities is precisely the class of toric singularities. This establishes universality (also known as Murphy’s law) for logarithmic maps.

The proof illustrates a general technique: the use of logarithmic structures to probe moduli spaces via tropical geometry. In this case, insights from logarithmic deformation theory describe the local structure of the moduli space in terms of the combinatorics of tropical maps. The tropical arguments required to complete the argument are constructive and extremely concrete, as I will explain.

This is joint work with Gabriel Corrigan and Dan Simms.

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Practical

Registration
Please register by 1st April 2024.

Funding
There is limited funding available to support travel (but not accommodation) of PhD students. Please indicate the approximate amount required on the registration form.

Location
The event will take place in the School of Mathematical Sciences at QMUL (map). Stepney Green Tube is a 5-minute walk, Mile End Tube is a 10-minute walk. 

Talks (including Explain It To Me) will take place in the main lecture theatre (MB-MLT), on the ground floor, straight ahead as you enter the building. The Wine Reception will take place in the fifth floor common room (MB-504).

Organiser: Navid Nabijou (QMUL).

This event is generously supported by an LMS Celebrating New Appointments grant, and by the School of Mathematical Sciences at QMUL.