Logic Seminar - Anna Dmitrieva
| Dates: | 29 October 2025 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Anna Dmitrieva (University of Manchester)
Quasiminimality and generic functions
One of the well-known accomplishments of model theory is the study of the field of complex numbers, axiomatized by the theory of algebraically closed fields of characteristic zero. It possesses numerous nice properties, including strong minimality and uncountable categoricity. However, adding the exponential map to the structure makes it possible to define the ring of integers, preventing first-order axiomatization and many other properties. Nevertheless, there is still hope that the complex exponential field can be successfully studied using higher-order logic and, in particular, that it is quasiminimal, i.e. every definable subset is countable or co-countable.
In 2002 Zilber introduced the theory of a generic function, which is a first-order theory axiomatizing a function on an algebraically closed field of characteristic zero that satisfies versions of the Schanuel property and existential closedness. This theory possesses many nice properties, analogous to the ones we expect from the complex exponential field. As the main result, we prove that adding an entire generic function to the complex field gives a quasiminimal structure, and, moreover, this structure is unique up to an isomorphism.
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Frank Adams 1
Alan Turing Building
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