Johannes Kepler was a mathematician, astronomer, astrologer and theologian. During his stay in Linz in 1614, Kepler worked on determining the volume of wine barrels. He struggled with his volume calculations in his Linz residence (Hofgasse 7 - discovered by the speaker in 2018), where he also discovered his third planetary law in 1618. Kepler's ingenious but today difficult to understand solution method - he used, among other things, an infinitesimal approach - is traced in the overview. In 1616, Kepler used decimal numbers for the first time in his German-language book on the calculation of wine barrels, inspired by the Swiss Jost Bürgi. Kepler's mathematically proven finding is remarkable: small dimensional changes in the Austrian barrel have no significant effect on the capacity. Changes in value near the maximum disappear - he laid the foundation for the rule of maxima and minima, which Fermat came up with 20 years later. The fatal effect of Gaussian error propagation in one of Kepler's important calculation steps is also explained. This effect was falsely blamed on Kepler in 1960 in 'Kepler's Collected Works, Volume IX - Mathematical Writings' as a serious arithmetical error. This false assertion persisted for more than 60 years! This year, the speaker was able to clear up this error and document it publicly.

We present two new methods for the recovery of $d$-variate exponential sums of order $M \in \mathbb{N}$ [1] defined by $$ f(\boldsymbol{t})=\sum\limits_{j=1}^{M} \gamma_j \mathrm{e}^{\langle \boldsymbol{\lambda}_j, \boldsymbol{t} \rangle}, \ \ \boldsymbol{t}=(t_1,\ldots,t_d) \in \mathbb{R}^{d}, $$ with complex coefficients $\gamma_j \neq 0$, distinct frequency vectors $\boldsymbol{\lambda}_j=(\lambda_{j1},\ldots, \lambda_{j d}) \in \mathbb{C}^{d}$ and $\langle \boldsymbol{\lambda}_j , \boldsymbol{t} \rangle=\sum_{\ell=1}^{d} \lambda_{j\ell} t_\ell$. In [1], we extend ideas from [2] to the multivariate setting. We show that the multivariate exponential recovery problem can be reformulated as a multivariate rational interpolation problem in the frequency domain. We present two approaches to solve this special multivariate rational interpolation problem by reducing it to several univariate ones which are then solved via the univariate AAA (adaptive Antoulas–Anderson) method for rational approximation [3]. Our first approach is based on using indices of the Fourier coefficients chosen from some sparse grid, which ensures efficient reconstruction using a respectively small amount of input data. The second approach is based on using the full grid of indices of the Fourier coefficients and relies on the idea of recursive dimension reduction. We demonstrate performance of our methods with several numerical examples. This is a joint work with Lennart Hübner (KU Leuven). \ [1] N. Derevianko, L. Hübner: Parameter estimation for multivariate exponential sums via iterative rational approximation. arXiv:2504.19157 [2] N. Derevianko, G. Plonka: Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients. Anal. Appl. 20(3), 543--577, 2021. [3] Y. Nakatsukasa, O. Sete, L.N. Trefethen: The AAA algorithm for rational approximation. SIAM J. Sci. Comput. 40(3), A1494–A1522, 2018.

I will present some geometric approaches to study the dimension and structure of the morphism and extension spaces over preprojective algebras of type A, and how these geometric interpretations lead to a classification of the weak exceptional sequences. This is joint work with Eric Hanson.

A (rational) Diophantine $m$-tuple is a set of $m$ positive integers (resp. rational numbers) such that the product of any two plus one is a perfect square. One of the generalizations, among many others, that has been considered is to look at $k$th powers instead of squares for a given $k\geq 3$. In the talk l will survey a bit the history of this direction of research on Diophantine $m$-tuples, in particular I will mention a beautiful result due to Y. Bugeaud and A. Dujella from 2003, and then present a new result for $k=3$ which uses the theory of elliptic curves and is recent joint work with D. Byeon.

In November 2023, Clarivate Plc announced that it had excluded the entire field of mathematics from the most recent edition of its influential list of authors of highly cited papers because of massive citation manipulation, which in return influences the so-called “Shanghai ranking” of top universities (or those claiming to be top). While most mathematicians would probably not care, the exclusion is in fact the tip of the iceberg of a parallel universe of predatory and mega-journals whose main purpose is to offer publishing opportunities for whoever is willing to pay the right price. I will explain how the system works, why we should care, and what measures we can all take against. In preparation, I invite you to think about the following questions: How often have you been contacted in the past months to attend a conference not in your field / submit a paper to or edit a special issue in a journal you don’t know / review an article within 10 days or so? What do you know about the following journals: “Mathematics”, “Axioms” (published by MDPI), “Chaos, solitons, fractals” (Elsevier), “Journal of Difference Equations” (Springer)? This talk is related to my work as Chair of the Committee on Publishing (COP) of the International Mathematical Union.

Discussion on the lecture of the same title

Noga Alon's Combinatorial Nullstellensatz excludes that a polynomial vanishing on a finite grid contains a monomial with certain properties. This theorem has been generalized in several directions: Uwe Schauz (2008) excludes more monomials, and recently, Bogdan Nica (2023) improved the result for grids with additional symmetries in their side edges. Simeon Ball and Oriol Serra (2009) generalized some of the results to \emph{punctured grids}, which are sets of the form $X \setminus Y$ with both $X,Y$ grids. We generalize some of these results; in particular, we provide a common generalization to the results of Schauz and Nica. To this end, we establish that during multivariate polynomial division, certain monomials are unaffected, and we find Gr\"obner bases for certain vanishing ideals. This allows us to generalize Pete L.\ Clark's proof of the nonzero counting theorem by Alon and F\"uredi to punctured grids. This is joint research with John R.\ Schmitt and Henry Zhan (Middlebury College, USA). A talk with similar content was given at AAA107 (Bern) and the Conference on Rings and Polynomials 2025 (Graz).

Gravitational data from satellites in Earth’s orbit can be used to reconstruct the secular and periodic movements of mass on the surface, these effects can be the tides caused by the moon, seasonal variations in rainfall (a wet season) or the melting of glacial ice on the poles. All of these mass transports of course cause variations in the gravitational signal, but they also have to be considered as loads on the planet’s surface, which will cause an elastic deformation of the Earth’s body on short timescales. This presentation deals with the problem of incorporating these deformation effects into the existing differential equations. In the process, the Earth’s elastic response to a point load – modeled by a Dirac distribution – is described using an expansion into spherical harmonics. The resulting coefficients arising from this expansion are called Love numbers and play an essential role in the modelling. Finally, a weak formulation of the problem is also derived for future numerical implementation.

Automatic sequences are fairly efficient in producing nontrivial sequences but exhibit some regularity, making their application for cryptography suboptimal. On the other hand, for, e.g., the Thue-Morse sequence, it is known that normality can be established for polynomial subsequences. — We exhibit a family realizing the other extreme, which includes several well-known sequences.

Let $\{Ft_m\}_m$ and $\{C_n\}_n$ be the factoriangular number sequence and the Catalan number sequence (see https://oeis.org/A101292 and https://oeis.org/A000108 in OEIS), respectively. In this talk, after presenting the general idea for explicit resolution of a Diophantine equation and some standard tools for certain class of Diophantine equations, I will present an unexpected tools based on prime number properties which lead us to finding the intersection of factoriangular numbers and Catalan numbers. This result is from a joint paper [1] with Professors Florian Luca and Alain Togb\'e during my visit to the School of Maths of Wits University in August 2022. Thanks to EMS (European Mathematical Society), which granted me a collaboration grant for that visit. [1] F. Luca, J. Odjoumani, and A. Togbe. Catalan numbers which are factoriangular numbers. Ann. Math. Inform., 60:93–97, 2024.