I will start with a motivation of what algebraic and model-theoretic properties an algebraically closed field of characteristic 1 is expected to have. Then I will explain how these properties forces one to follow the route of Hrushovski's construction leading to a a 'pseudo-analytic' structure which we identify as an algebraically closed field of characteristic 1 . Then I am able to formulate very precise axioms that such a field must satisfy. The main theorem then states that under the axioms the structure has the desired algebraic and analytic properties. The axioms have a form of statements about existence of solutions to systems of equations in terms of a 'multi-dimensional' valuation theory and the validity of these statements is an open problem to be discussed.

This is a joint work with Alex Cruz Morales.

# Past Special Seminar

1.Kremnitzer. I will explain an approach to constructing geometries relative to a symmetric monoidal

category. I will then introduce the category of normed sets as a possible analytic geometry over

the field with one element. I will show that the Fargues-Fontaine curve from p-adic Hodge theory and

the Connes-Bost system are naturally interpreted in this geometry. This is joint work with Federico Bambozzi and

Oren Ben-Bassat.

Part of the series 'What do historians of mathematics do?'

Ada Lovelace (1815-1852) is famous as "the first programmer" for her prescient writings about Charles Babbage's unbuilt mechanical computer, the Analytical Engine. Biographers have focused on her tragically short life and her supposed poetic approach – one even dismissed her mathematics as "hieroglyphics". This talk will focus on how she learned the mathematics she needed to write the paper – a correspondence course she took with Augustus De Morgan – which is available in the Bodleian Library. I'll also reflect more broadly on things I’ve learned as a newcomer to the history of mathematics.

In this lecture we describe the effective Chebotarev Theorem for global function fields and show how this can be used to describe the statistics of a polynomial map f in terms of its monodromy groups. With this tool in hand, we will provide a strategy to remove the remaining heuristic in the quasi-polynomial time algorithm for discrete

logarithm problems over finite fields of small characteristic.

Part of the series 'What do historians of mathematics do?'

"In this talk I present the history and proof of the four-colour theorem: Can every map be coloured with just four colours so that neighbouring countries are coloured differently? The proof took 124 years to find, and used 1200 hours of computer time. But what did it involve, and is it really a proof?"

Part of the series 'What do historians of mathematics do?'

I'm currently working on a biography of Charles Hutton (1737–1823): pit lad, FRS, and professor of Mathematics. No-one much has heard of him today, but to his contemporaries he was "one of the greatest mathematicians in Europe". I'll give an outline of his remarkable story and say something about why he's worth my time.

Part of the series 'What do historians of mathematics do?'

" Over the last four centuries a huge amount of mathematics has been published. Most of it has, however, had little or no influence. By way of contrast, some mathematics, although unpublished in its time, has had great influence. My hope is to illustrate this with discussion of manuscript sources and resources that have survived from Thomas Harriot (c.1560--1621), Isaac Newton (1642--1727) and Évariste Galois (1811--1832)."