Weronika Czerniawska, Tate's thesis and the Relative Riemann–Roch Theorem
Tate's PhD thesis developed Fourier Analysis on the so-called Ring of adeles and applied it to prove analytic continuation and functional equation of the Dedekind zeta-function. Using this theory, the Riemann–Roch theorem can be seen as an easy consequence of the Poisson summation formula. I will explain how to generalize this theory to obtain the Riemann–Roch theorem for morphisms of (both arithmetic and geometric) curves.

Slides



Paolo Dolce, Introduction to Diophantine approximation and a generalization of Roth’s theorem
Classically, Diophantine approximation deals with the problem of studying “good” approximations of a real number by rational numbers. I will explain the meaning of “good approximants” and the classical main results in this area of research. In particular, Klaus Roth was awarded with the Fields medal in 1955 for proving that the approximation exponent of a real algebraic number is 2. I will present a recent extension of Roth’s theorem in the framework of adelic curves. These mathematical objects, introduced by Chen and Moriwaki in 2020, stand as a generalization of global fields.

Slides



Łukasz Orski, Selberg Sieve and the Prime Number Theorem
In my talk I will present the elementary (i.e. not using complex analysis) proof of the Prime Number Theorem given by Atle Selberg and Paul Erdős in 1948. The main idea is the Selberg sieve which estimates the sum of weights of numbers in the considered set. It is important to find a proper weight i.e. the one which is easy to sum up with the sieve. The other difficulty is estimating prefix sums of sequences involving the Mobius function. Other formulas used in the proof are easy or well-known approximations and were discovered long before Selberg and Erdős.


Błażej Żmija, Ranks of universal quadratic forms over quadratic fields
Let \( K \) be totally real number field and \(\mathcal{O}_{K}\) be its ring of integers. We say that a quadratic form with the coefficients in \(\mathcal{O}_{K}\) is universal over \(K\) if it is totally positive definite and represents all the totally positive elements of \(\mathcal{O}_{K}\). For a squarefree positive integer \(D\) let \(R(D)\) denote the minimal rank of a universal quadratic form over \(\mathbb{Q}(\sqrt{D})\). The aim of the talk is to present results regarding the typical size of \(R(D)\). In particular, we will show that if \(\varepsilon >0\) is fixed, then for almost all the numbers \(D\) (in the sense of natural density) \(R(D)\) is greater than \(D^{\frac{1}{24} -\varepsilon}\). The talk is based on my joint work with V. Kala and P. Yatsyna.

Slides