Universiteit Leiden

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This Week’s Discoveries | 26 February 2019

Tuesday 26 February 2019
Niels Bohrweg 2
2333 CA Leiden
De Sitterzaal

First Lecture

Counting solutions to certain polynomial equations.

Erik Visse-Martindale (MI)

Last December, Erik obtained his PhD in number theory. He is currently teaching at the MI for three days per week and he spends the remainder of the week getting acquainted with the theory of machine learning, aiming to better understand the practice.

Consider the equation x^2 + y^2 = z^2. From a number theoretic viewpoint we are interested in its solutions in the integers, of which there are infinitely many. In contrast, the equation x^2 + y^2 = 3z^2 has only one. The subject of arithmetic geometry aims to explain the structure of solutions to such equations using geometric tools.
When confronted with infinitely many solutions, one may obtain more information by using a finer way of counting. We do this by requiring each of the variables to be bounded in absolute value by some B and then letting this B grow. For many classes of equations, even conjecturing the behaviour of this counting function would be considered bold, and proofs are certainly far out of reach.
In one chapter of my PhD thesis I focus on equations of the form ax^4+by^4+cz^4+dw^4=0. Under some geometric conditions, I provide theoretical evidence for behaviour that was experimentally observed by Ronald van Luijk a few years ago using computer simulations.

Second Lecture

Is it sexy to be smart? The evolution of cognitive abilities by sexual selection.

Carel ten Cate (IBL)
Carel is professor of Animal Behaviour at the Institute of Biology. His research interests concerns animal communication and cognition and involves comparative studies on the cognitive mechanisms involved in the learning and processing of vocal and visual signals in species ranging from birds and fish to humans. Many projects are at the interface of biology, cognitive science, psychology and linguistics.

Some animal species show behaviour that seems smart or intelligent. Whether it really is, is a prominent question addressed by researchers on animal cognition. Less attention has been given to the evolution of such behaviour. A so-far untested hypothesis is that cognitive abilities can evolve by sexual selection, i.e. by individuals preferring ‘smart’ partners. We tested this hypothesis and were able to show that potential partners become more attractive as mates when demonstrating ‘ smart’ behaviour. Being smart apparently can be sexy, at least for budgerigars…

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