Titles and abstracts

Peter Beelen, Technical University of Denmark

Title: An overview on the theory of maximal curves

Abstract: This talk is about maximal curves, that is to say, curves defined over a finite field that attain the Hasse-Weil bound. Such curves are used for applications in coding theory, but are also of independent interest and have a rich structure. The aim of this talk is to give an overview of the history of maximal curves, point out several open problems and to talk about recent results for several specific families of maximal curves.

Sigrid Grepstad, Norwegian University of Science and Technology (NTNU)

Title: Connecting bounded remainder sets and cut-and-project sets

Abstract: In this talk we study the connection between bounded remainder sets in Rd and cut-and-project sets in R. I will try to illustrate how this connection can be exploited to provide a geometric and intuitive proof of why two Jordan measurable bounded remainder sets of the same measure are necessarily equidecomposable in a certain sense.

Trygve Johnsen, UiT The Arctic University of Norway

Title: Higher weight spectra of Reed-Muller codes RMq(2,2)

Abstract: We determine the higher weight spectra of q-ary linear Reed-Muller codes Cq=RMq(2,2) for all prime powers q. This is equivalent to finding the usual weight distributions of all extension codes of Cq over a study of all field extensions of Fq. To obtain our results we will utilize well-known connections between these weights and properties of the Stanley-Reisner rings of a series of matroids associated to each code Cq. This is joint work with R. Ludhani, S.R. Ghorpade, R. Pratihar.

Vitezslav Kala, Charles University

Title: Universal quadratic forms and Northcott property of infinite number fields

Abstract: Universal quadratic forms generalize the sum of four squares about which it is well known that it represents all positive rational integers. In the talk, I’ll start by discussing some results on universal quadratic forms over totally real number fields. Then I’ll move on to the – markedly different! – situation over infinite degree extensions K of Q. In particular, I’ll show that if K doesn’t have many small elements (i.e., “K has the Northcott property”), then it admits no universal form. The talk is based on a recent joint work with Nicolas Daans and Siu Hang Man.

Jürg Kramer, HU – Berlin

Title: Arithmetic intersections of line bundles with singular metrics

Abstract: In our talk, we will present an extension of arithmetic intersection theory of adelic divisors on quasi-projective varieties introduced by Yuan-Zhang to the case where these divisors are not necessarily arithmetically nef. The key tool to realize this extension is the concept of relative finite energy established by T. Darvas et al.. In particular, our theory will allow to compute heights on mixed Shimura varieties, e.g., the arithmetic self-intersection number of the line bundle of Siegel-Jacobi forms on the universal abelian variety. This is joint work with José Burgos Gil.

Adrien Morin, University of Copenhagen

Title: Weil-étale cohomology and the ETNC for constructible sheaves

Abstract: Let X be a variety over a finite field. Given an order A in a semisimple algebra A_Q over the rationals and a constructible étale sheaf F of A-modules over X, one can consider a natural equivariant L-function associated with F. We will formulate and prove a special value conjecture at s=0 for this L-function, expressed in terms of Weil-étale cohomology, provided that the latter is “well-behaved”; this is a geometric analogue of the equivariant Tamagawa conjecture for a Tate motive over a global function field, and generalizes the results of Lichtenbaum on special values at s=0 for zeta functions of smooth proper varieties, and the work of Burns-Kakde in the case of the equivariant L-functions coming from a finite G-cover of varieties.

Örs Rebák, UiT The Arctic University of Norway

Title: Special values of Ramanujan’s theta function φ(q)

Abstract: In his notebooks, Ramanujan determined some values for his theta function φ(q). In his lost notebook, Ramanujan provided an incomplete value for φ(exp(-7π√7)), which was recently completely evaluated. We present a sketch of the proof. In joint work with Berndt, we develop general cubic and quintic analogues of Ramanujan’s now completed septic formula. It turns out that some of the values are expressible in terms of trigonometric function values. As corollaries, we are able to determine several new values of φ(exp(-π√n)).

Damaris Schindler, Göttingen University

Title: Density of rational points near manifolds

Abstract: Given a bounded submanifold M in Rn, how many rational points with common bounded denominator are there in a small thickening of M? Under what conditions can we count them asymptotically as the size of the denominator goes to infinity? I will discuss some recent work in this direction and arithmetic applications such as Serre’s dimension growth conjecture as well as applications in Diophantine approximation. For this I’ll focus on joint work with Shuntaro Yamagishi, as well as joint work with Rajula Srivastava and Niclas Technau. 

Tuyen Trung Truong, University of Oslo

Title: Backtracking New Q-Newton’s method and connection to the Riemann hypothesis

Abstract: Backtracking New Q-Newton’s method is a relatively new Newton-type method for root finding and optimization. It has strong convergence guarantee for finding roots of meromorphic functions. In this talk, I will discuss about its connections to the Riemann hypothesis. In particular, a new equivalence to the Riemann hypothesis will be given. Observations from using this method to finding non-trivial roots of the Riemann zeta function are reported. This is in joint work with Thuan Quang Tran (Master’s student, UiO).  A paper is in preparation, and will be posted on arXiv. 

Pavlo Yatsyna, Charles University

Title: Even better sums of squares

Abstract: The celebrated result by Siegel tells us that in most totally real number fields, there are positive integers that cannot be expressed as a sum of squares. In this talk, we’ll explore this idea further, looking at the representation of integers in a totally real number field using a quadratic form with rational integer coefficients. Just like with sums of squares, we’ll see that in low-degree totally real number fields, this rarely occurs. These results appeared in the joint work with Vita Kala.