21st May 2025: Arshay Sheth (University of Warwick)
Title: Euler products inside the critical strip
Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. Indeed, the behaviour of Euler products inside the critical strip is very closely related to several major problems in number theory including the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip. We will end by giving applications of these results to questions related to Chebyshev's bias.
14th May 2025: Matyáš Kafka (Univerzita Karlova)
Title: Legendre polynomials in irrationality proofs
I will talk about Legendre polynomials and how they can be used to prove that a real number is irrational, mainly focusing on the irrationality of π and e. The talk is based on the topic of my bachelor's thesis supervised by Martin Čech.
30th April 2025: Krystian Gajdzica (Uniwersytet Jagielloński)
Title: The log-behavior of the A-partition function
The A-partition function p_A(n) enumerates partitions of n with parts in a given multiset A of positive integers. The main aim of this talk is to discuss the log-behavior for a wide class of A-partition functions. More precisely, we investigate the Bessenrodt-Ono type inequality, the log-concavity, the r-log-concavity, the higher order Turán inequalities and the Laguerre inequalities.
Finally, we also introduce a few non-standard generalizations of the A-partition function and present some of their basic properties.
23rd April 2025: Magdaléna Tinková (ČVUT)
Title: Non-monogenic simplest cubic fields
The simplest cubic fields are often studied under the assumption of monogenity. But what about the other cases? In this talk, we will show when non-monogenity happens and what such fields look like. Moreover, we will also indicate some nice relations among parameters and results related to indecomposables. The talk is based on joint works with Daniel Gil Muñoz.
16th April 2025: Vítězslav Kala (Univerzita Karlova)
Title: Finitely generated semifields
I will talk about an elementary problem concerning generating all elements of Z^n, and its background and motivation coming from classifications of certain semifields. The talk is based on joint works with Miroslav Korbelář and Lucien Šíma.
9th April 2025: Jérémy Dousselin (Université de Lorraine)
Title: A study of Hooley's class numbers
The study the class numbers h(d) has a rich history that goes back to the work of Gauss. He already understood that the class numbers h(d) behave erratically on average, and it was only about two centuries later that Hooley formulated the first conjecture for the average of class numbers over positive discriminants. To achieve this, Hooley studied a family of positive discriminants associated with a "small" fundamental unit. Our main result in this context is an asymptotic formula for complex moments of Hooley's class numbers, from which we can derive several results on the distribution of Hooley's discriminants.
19th March 2025: Mikuláš Zindulka (Univerzita Karlova)
Title: Negative bias in moments of the Legendre family of elliptic curves
Modular forms can be successfully applied to a variety of problems in number theory. Given an arithmetic function f, one may hope that the values f(n) are coefficients of a modular form. For example, the generating function for the Hurwitz class numbers H(n) turns out to be a mock modular form. In other words, it can be completed, by adding a suitable non-holomorphic piece, to a weight 3/2 harmonic Maass form. This led to the proof of some remarkable identities for these numbers.In this talk, I will show how to apply these techniques to the distribution of traces of Frobenius in the Legendre family of elliptic curves. The influential Negative Bias Conjecture states that the second moment has negative bias for every family with a non-constant j-invariant. I will give an expression for the higher moments of the Legendre family and prove that each lower order term is negative on average. The talk is based on a joint work with Ben Kane.
12th March 2025: Robin Visser (Univerzita Karlova)
Title: Sums of two units in number fields
Let K be a number field with ring of integers O_K. Let N_K be the set of positive integers n such that there exist units ε and δ in O_K satisfying ε + δ = n. In this talk, we show that N_K is a finite set if K does not contain any real quadratic subfield. In the case where K is a cubic field, we also explicitly classify all solutions to the unit equation ε + δ = n when K is either cyclic or has negative discriminant. This talk is based on joint work with Magdaléna Tinková and Pavlo Yatsyna.
5th March 2025: Siu Hang Man (Univerzita Karlova)
Title: Universal forms and Northcott property in algebraic fields of infinite degree
A totally positive definite, integral quadratic form is called universal if it represents all totally positive integers. Most existing literature concerns the case where the underlying field is a number field. In this talk, we shall discuss the notion of universal forms in an algebraic field K which has infinite degree over Q. We show that if K satisfies certain finiteness condition known as the Northcott property, then there are no universal forms over K. This is joint work with Nicolas Daans and Víťa Kala.
26th February 2025: Jongheun Yoon (Univerzita Karlova)
Title: Primitively 2-universal integral quadratic forms of rank six
A positive definite integral quadratic form is called primitively n-universal if it primitively represents all quadratic forms of rank n. In this talk, we prove that the minimal rank of a primitively 2-universal quadratic form is six, and we prove that there are exactly 201 primitively 2-universal quadratic forms of rank six up to isometry.
11th December 2024: Natália Bátorová (Univerzita Karlova)
Title: Arithmetic-geometric mean sequences and elliptic curves over finite fields
Starting with a pair of positive real numbers, one can define a sequence of ordered pairs, where the first coordinate represents the arithmetic mean and the second coordinate represents the geometric mean of the previous pair. In the talk, we extend this definition to finite fields using directed graphs. We explore the properties of these graphs and focus on AGM sequences over finite fields of order q = 3 mod 4 and q = 5 mod 8. Additionally, we show a relation between them and elliptic curves.
4th December 2024: Christopher Frei (TU Graz)
Title: Orthogonality of restricted primes with nilsequences
The randomness of arithmetic functions with respect to linear correlations can be measured by Gowers uniformity norms. We show that the von Mangoldt function of primes restricted to a fixed Chebotarev class varies randomly around its average, up to structure arising from congruences to small moduli. By the inverse theory of Green-Tao-Ziegler, we can achieve this by studying correlations with nilsequences. Under GRH, we get analogous results for primes with a prescribed primitive root. This is joint work with Magdaléna Tinková.
27th November 2024: Igor Balla (Masarykova Univerzita)
Title: The state of the art on equiangular lines
In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in R^r with angle arccos(α) and gave a good partial answer in the regime r ≤ 1/α^2 − 2. At the other extreme where r is at least exponential in 1/α^2, recent breakthroughs have led to an almost complete solution. In this talk, we will describe our recent progress on this problem, partly in joint work with Matija Bucić. In particular, we obtain upper bounds that unify and significantly extend or improve essentially all previously known results, thereby bridging the gap between the aforementioned regimes and determining the answer up to a factor of 2. Roughly speaking, our approach relies on new lower bounds on the second eigenvalue of a graph as well as new upper bounds on its multiplicity. In particular, we obtain the first extension of the Alon–Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to families of r+1 choose 2 equiangular lines in R^r. We will also mention some results in the complex setting.
20th November 2024, 14:30: Žaneta Lipertová (Univerzita Karlova)
Title: Non-unique factorization in number fields
In this talk we will study irreducible factorization in rings of integers of an algebraic number fields. We will show the use of factorization of a principal ideal into prime ideals in searching for irreducible factorization of an element that generates such principal ideal. We will define the class number and learn what it means that the class number of a ring of integers is 2 or 3 and what does it says about its irreducible elements. At the end we will present the characterizing Carlitz theorem.
20th November 2024, 14:00: Emma Pěchoučková (Univerzita Karlova)
Title: Conway's topograph
John Conway in his study of binary quadratic forms introduced the notion of a topograph: a graph in which he did not care about the edges or the vertices but about the rest - the blank space in-between. In this talk, I will introduce the topograph, how it is constructed and how we can use continued fractions to describe paths in this graph. Colorful illustrations will be included.
6th November 2024: Magdaléna Tinková (ČVUT)
Title: Non-decomposable quadratic forms over totally real number fields
Non-decomposable quadratic forms with integer coefficients were studied, for example, by Mordell (1930, 1937) and Erdős and Ko (1938). However, we know much less about them if their coefficients belong to the ring of algebraic integers of a totally real number field. Some of our new results are general, but one part is restricted to the case of binary quadratic forms over real quadratic fields. For them, we provide some bounds on the number of such non-decomposable quadratic forms, show that their number is rather large for almost all quadratic fields, or give their whole structure for several examples of these fields. We also show a relation between them and the problem of n-universal quadratic forms. This is joint work with Pavlo Yatsyna.
30th October 2024: Pavlo Yatsyna (Univerzita Karlova)
Title: Interlacing solution to the lifting problem
Alien to most, yet in Prague, it is number theory that often comes to mind when discussing UFO-related questions. In keeping with this tradition, we will delve into the lifting problem and one of the earliest attempts to address it using interlacing polynomials.
23rd October 2024: Robin Visser (Univerzita Karlova)
Title: The Effective Shafarevich Conjecture
Let K be a number field, d a positive integer, and S a finite set of primes of K. One of the crowning achievements of 20th century arithmetic geometry was Faltings's proof that there are only finitely many isomorphism classes of dimension d abelian varieties A/K with good reduction away from S. Whilst several effective algorithms have been developed to explicitly classify elliptic curves with good reduction outside a finite set of primes, effectively solving this problem in higher dimensions remains a challenge. In this talk, I will give a brief survey on some known methods for classifying abelian varieties, and will present some work in progress on classifying abelian surfaces over Q with good reduction away from 2.
16th October 2024: Subham Roy (Univerzita Karlova)
Title: Mahler measure and how to compute them
The (logarithmic) Mahler measure of a non-zero rational function P in n variables is defined as the mean of log |P| restricted to the standard n-torus. The Mahler measure has been related to special values of L-functions, and this has been explained in terms of regulators.
In this talk, we will explore these relations, and provide some interesting examples. In the second part, we will consider a counterpart of the Mahler measure first explored by Pritsker (2008), which is obtained by replacing the normalized arc-length measure on the standard n-torus with the normalized area measure on the product of n open unit disks. We will also make connections with a seemingly unrelated branch of mathematics: The theory of random walks in probability theory.
9th October 2024: Martin Čech (Univerzita Karlova)
Title: Moments of real Dirichlet L-functions and multiple Dirichlet series
During this talk, we will introduce the problem of computing moments of the Riemann zeta function or more general families of L-functions. We will then focus on the family of real Dirichlet L-functions, and introduce the method of multiple Dirichlet series to compute these moments.