University of Primorska Faculty of Mathematics, Natural Sciences and Information Technologies
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Mathematical Research Seminar - Archive

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Datum in ura / Date and time: 7.6.24
(11:30 - 12:30)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Besfort Shala (University of Bristol)
Naslov / Title: A tour through multiplicative and probabilistic number theory
Vsebina / Abstract:

I will give a general overview of recent developments in number theory, with many old and new results being proved through the lens of multiplicative functions. Many deep unsolved problems about primes such as the Riemann Hypothesis and the Twin Prime Conjecture are intimately connected with the Möbius function. Viewing the latter as simply one instance of a multiplicative function satisfying certain properties and developing a general theory of multiplicative functions has been very fruitful – most classical results in analytic number theory have been reproved in this framework, without appealing to the analytic continuation of the Riemann zeta function (or more general L-functions). In particular, one may adapt a probabilistic viewpoint and consider so called random multiplicative functions taking the values +1 and -1 on the primes randomly. This was initiated by Wintner, who proved the analogue of the Riemann Hypothesis in this simplified setting. Later this was further developed by Harper, who considered finer distributional questions. If time permits, I will present results on the Twin Prime analogue in this probabilistic setting, based on ongoing joint work with Jake Chinis.


Datum in ura / Date and time: 3.6.24
(15:00-16:00)
Predavalnica / Location: FAMNIT-MP1
Predavatelj / Lecturer: Petr Golovach (University of Bergen)
Naslov / Title: Detours in Directed Graphs
Vsebina / Abstract:
 
We study the ''above guarantee'' version of the classical Longest Path problem on undirected and directed graphs called  Longest Detour where the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k. Bezáková et al. in 2019 proved that on undirected graphs, the problem is fixed-parameter tractable (FPT). We establish a connection between Longest Detour and Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Furthermore, the new approach yields a significantly faster FPT algorithm on undirected graphs.
 
Joint work with Fedor V. Fomin, William Lochet, Danil Sagunov, Saket Saurabh, and  Kirill Simonov