EVENT SCHEDULE
9:15 AM - 9:30 AM
Welcome
9:30 AM - 10:30 AM
From existential definability and differential equations to transseries
Lou van den Dries (University of Illinois at Urbana-Champaign)
Definability, in particular existential definability, is a central concept in Julia Robinson’s work. I will discuss its dual role in relation to (algebraic) differential equations. This leads naturally to the idea of transseries. My work on this is joint with Matthias Aschenbrenner and Joris van der Hoeven, and has resulted in a complete theory of solving differential equations with initial conditions in the differential field of transseries. I will also briefly talk about the connection to Hardy fields that we have established recently.
10:30 AM - 11:00 AM
Break
11:00 AM - 12:00 PM
Decidability and definability in number fields and connections to Julia Robinson's work
Kirsten Eisentraeger (Pennsylvania State University)
In 1970, Yuri Matiyasevich, building on work by Martin Davis, Hilary Putnam and Julia Robinson, proved that Hilbert's Tenth Problem over the integers is undecidable. The analogue of Hilbert’s Tenth Problem over the rationals, and over number fields in general, remains open. A diophantine definition of the integers over the rationals, together with a standard reduction argument, would show that Hilbert’s Tenth Problem over the rationals is undecidable. A diophantine definition of the integers over the rationals seems out of reach right now, but one can also consider the problem of defining the integers inside the rationals or number fields with a first-order formula. Julia Robinson showed that this is possible, and we will discuss her proof as well as recent improvements and extensions of her result.
We will also discuss examples of sets that can be shown to be diophantine in number fields and in global function fields, such as the set of quadratic non-norms and the set of non-squares.
12:00 PM - 1:30 PM
Lunch Break (Food available for purchase onsite)
1:30 PM - 2:30 PM
Julia Robinson: Colleague and Friend
Martin Davis (Courant Institute of Mathematical Sciences, New York University)
Over a 25 year period, Julia and I worked on the same problem, mostly in parallel, but sometimes as collaborators. I will talk about this, including samples from our correspondence.
2:30 PM - 3:00 PM
Tea Break
3:00 PM - 4:00 PM
Around Hilbert's eighth and tenth problems
Yuri Matiyasevich (St. Petersburg Department of V.A. Steklov Mathematical Institute and Euler International Mathematical Institute of the Russian Academy of Sciences)
The name of Julia Robinson is inseparable from Hilbert's tenth problem. This was one of the 23 mathematical problems selected by David Hilbert in 1900, problems which the pending XX century was to inherit from the passing XIX century. The eighth had, at first sight, little in common with the tenth problem. However, when the latter was solved (in the negative) it turned out that the two problems are connected, in a way which was not anticipated by Hilbert.
In my talk I'll briefly outline the history of proving the undecidability of the tenth problem, the role played by Julia Robinson, my collaboration with her and Martin Davis, and a relationship between Hilbert’s tenth and eighth problems.
See below for public lecture details following the symposium. The UC Berkeley Hill Shuttle will depart MSRI at 4:25 PM for those wishing to attend this event.
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