In the model theory of analytic functions, Existential Closedness problems aim to determine when systems of equations in several variables involving addition, multiplication, and some special transcendental functions (e.g. exponential or modular functions) have solutions in the complex numbers. This is a natural question at the boundary between complex geometry and model theory, and is an extension of the well-known problem of solvability of systems of polynomial equations in several variables, settled by Hilbert's Nullstellensatz. 

The first conjecture on Existential Closedness was proposed by Boris Zilber in the early 2000s in the exponential setting. It is known as the Exponential Closedness conjecture. In recent years, analogues of this conjecture for other functions, most notably the modular j-function, were formulated, and significant progress has been made towards them.

The name Existential Closedness comes from the model-theoretic property of structures, known as existential closedness, describing when quantifier-free formulas with solutions in a larger model already have solutions in the original model. In the setting of complex functions, the Existential Closedness conjecture does not imply that the complex field equipped with the given function is existentially closed in a first-order sense; instead, it is existentially closed in certain self-sufficient extensions.

The aim of this meeting is to bring together world experts on Existential Closedness and related questions to share ideas and updates and work together to make further progress on these problems.

Dates: 7--10 July 2026

Venue: Alan Turing Building, University of Manchester

Confirmed speakers

• Martin Bays (Oxford)

• Paola D'Aquino (Caserta)

• Sebastian Eterović (Vienna)

• Jonathan Kirby (East Anglia)

• Vincenzo Mantova (Leeds)

• Adele Padgett (Vienna)

• Giuseppina Terzo (Naples)

• Alex Wilkie (Oxford)

• Roy Zhao (Tsinghua)

• Boris Zilber (Oxford)

This workshop is supported by the EPSRC Open Fellowship EP/X009823/1.