Tuesdays 15:40 in K3. In SIS it's listed as NMAG470. The seminar is intended for students, faculty and occasional visitors to give accessible talks on various areas of number theory.
Depending on the participants, the talks will be in Czech or English. If you're interested in attending the seminar or giving a talk, please email me ( Schedule28. 11. 2017: Tomáš Vávra, Periodic representations in number systems with an algebraic base After giving an introduction to $(\beta,A)$-representations, we will show that if $\beta\in\mathbb C$, $|\beta|>1$ is an algebraic number, then there exists an (integer) alphabet $A$ such that each element of the field extension $\mathbb Q(\beta)$ admits an eventually periodic $(\beta,A)$-representation. We will also show how the question whether a pair $(\beta,A)$ has this property is linked to fractal geometry.21. 11. 2017: Pavlo Yatsyna (Royal Holloway, University of London) Universal quadratic forms and interlacing polynomials
It is a classical result in number theory that any natural number can be
represented as a sum of four squares. Over the ring of integers of
$\mathbb{Q}(\sqrt{5})$, every totally positive integer can be written as
a sum of three squares. For a general quadratic form, Blomer and Kala
recently showed that the number of variables required in a real
quadratic number field is unbounded. I will present a similar result,
based on studying interlacing polynomials, which allows the degree of a
field extension to be arbitrarily large.14. 11. 2017: Pavlo Yatsyna (Royal Holloway, University of London) Salem numbers of trace −2, and a conjecture of Estes and Guralnick Estes and Guralnick conjectured necessary and sufficient conditions for a polynomial to appear as the minimal polynomial of a symmetric matrix with rational integer coefficients. They confirmed their conjecture for polynomials of degree up to 4. In this talk, I will show that there are counterexamples to Estes—Guralnick's conjecture for all degrees strictly larger than 5. One of the ingredients in the proof is to show that there are Salem numbers of degree 2d and trace −2 for every d≥12.31. 10. 2017: Martin Čech, Different approaches toward the proof of the prime number theorem We will show how complex analysis is used in number theory and discuss different approaches to the proof of the prime number theorem. These approaches will include the classical one introduced by Riemann, Hadamard and de la Vallée Poussin at the end of 19th century, and a more modern "pretentious" approach based on Halász's theorem proved in 1970's and recently further developed by Granville, Soundararajan and others.24. 10. 2017: Martin Čech, What are arithmetic functions and how to estimate them? Many questions in number theory, such as what is the average number of divisors of a natural number, can be stated in terms of arithmetic functions. Giving precise answers to these questions is very hard, which is the reason why analytic number theory studies their estimates. In the lecture, we are going to study basic properties of arithmetic functions and elementary techniques of estimating their rate of growth and the errors in the estimates.17. 10. 2017: Kristýna Zemková, Composition of quadratic forms over number fields The correspondence
between ideals in a quadratic number field and quadratic forms with
integral coefficients dates back to Gauss and Dedekind. But what about
quadratic forms whose coefficients are algebraic integers? In the talk I
will present my recent result on generalization of this correspondence
to some number fields.10. 10. 2017: Víťa Kala, Arithmetics of number fields and universal quadratic forms This will be an introductory talk (accessible to students) to several exciting topics of current research. The arithmetics of number fields has long played a key role throughout number theory, for example in solving diophantine equations. I will discuss some recent results on the additive structure of rings of integers of real quadratic fields and their relation to the study of quadratic forms (joint work with Valentin Blomer and Tomas Hejda). |

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### Number Theory Seminar

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Number Theory Seminar 2015-16