Mathematical research seminar Archives - UP Famnit

Harmonic analysis and counting field extensions

nina.chiarelli — Thu, 22 Oct 2026 05:35:03 +0000

Datum / Date: 26.10.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Ratko Darda (University of Primorska) Naslov / Title: Harmonic analysis and counting field extensions Povzetek / Abstract: The Hermite–Minkowski theorem states that there are only finitely many finite extensions of Q, up to isomorphism, with bounded discriminant. We are interested in counting them. Focusing on Galois abelian extensions, we present an approach to this question using harmonic analysis. While the approach extends naturally to the context of finite group schemes and more general height functions, we restrict in this talk to finite abelian groups and the discriminant. The talk is based on a joint work with Takehiko Yasuda.

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Pathographs and some undecidability results

nina.chiarelli — Sat, 17 Oct 2026 07:11:21 +0000

Datum / Date: 12.10.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Nicolas Trotignon (CNRS, LIP, ENS de Lyon) Naslov / Title: Pathographs and some undecidability results Povzetek / Abstract: We define pathographs, that is a generalization of graphs. Pathographs have vertices and edges, as graphs, but they also have «ur-path», that are special edges (that we will later subdivide). There are also spokes, that are adjacencies between vertices and ur-paths, and rungs, that are adjacencies between ur-paths. A realization of a pathograph G is any graph G’ obtained from G be replacing each ur-path by a path of length at least 2. Furthermore, for each spoke v-p there must be at least one edge between v and the path P arising from the ur-path p, and for each rung pq, there must be at least one edge between the paths P, Q arising from p and q respectively. There are no other vertices or edges. We consider the problem whose instance is a pathgraph G and a finite set of graphs F. The question is whether there exists a realization of G that contains no graph from F as an induced subgraph. We prove that this problem is undecidable (by reducing the Wang tiles problem to it). This result has several motivations. Pathographs are flexible enough to encode several graph containment problems. Some of them seems easy though it not known whether an algorithm exists for them, and our result show that this is not trivial. Also, many proofs in structural graph theory relies on a fact that some pathograph cannot be realized without introducing a forbidden induced subgraphs, so our result show that automatic provers for such lemmas do not exists in general. This is a joint work with Daniel Carter.

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Modular forms on higher-rank groups and their associated L-functions

nina.chiarelli — Sat, 17 Oct 2026 06:58:45 +0000

Datum / Date: 19.10.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Rafail Psyroukis (University of Primorska) Naslov / Title: Modular forms on higher-rank groups and their associated L-functions Povzetek / Abstract: In this talk, I will introduce higher-rank modular forms and discuss some of my work in the Hermitian and orthogonal cases. In particular, I will explain connections between Fourier–Jacobi forms and L-functions. Time permitting, I will also report on ongoing projects on algebraic modular forms and Hermitian theta series.

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Automorphisms of Cubic Surfaces and the Clebsch Map

nina.chiarelli — Mon, 05 Oct 2026 15:07:54 +0000

Datum / Date: 5.10.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: George Savvoudis (University of Primorska) Naslov / Title: Automorphisms of Cubic Surfaces and the Clebsch Map Povzetek / Abstract: It is a classical result that a non-singular cubic surface with 27 lines can be constructed from a non-conical 6-arc in the plane via a Clebsch Map. We investigate the action of automorphisms of the cubic surface on the points in the preimage of the Clebsch map, with the aim of better understanding the relationship between cubic surfaces and the non-conical 6-arcs.

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Metric and ultrametric inequalities for directed graphs

mario.krajoski — Wed, 27 May 2026 22:00:00 +0000

Datum / Date: 1.6.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP6 Predavatelj / Lecturer: Vladimir Gurvich (RUTCOR, Rutgers The State University of New Jersey, USA) Naslov / Title: Metric and ultrametric inequalities for directed graphs Povzetek / Abstract: Consider an electrical network modeled by a digraph G each directed edge e of which is a semiconductor with a monomial conductance function ye* = fe (ye) = yer / μes if ye ≥0 and ye* = 0 if ye < 0. Here e is a directed edge, ye is the potential difference (voltage), ye* ≥0 is the current in e, and μe > 0 is the resistance of e; furthermore, r > 0 and s > 0 are two real parameters common for all edges. In particular, case r = s = 1 corresponds to the Ohm law, while r = ½, s =1 may be interpreted as the square law of resistance, which is typical of hydraulics and gas dynamics. We show that for every ordered pair of nodes a, b of the network, the effective resistance μa,b is well-defined. In other words, any two-pole network with poles a and b can be effectively replaced by two oppositely directed edges: from a to b of resistance μa,b and from b to a of resistance μb,a. Furthermore, for every three nodes a, b, c, the inequality μa,cs/r + μc,bs/r ≥ μa,bs/r holds, in which the equality is achieved if and only if every directed path from a to b in G contains c. In addition to case (i) s = r = 1, some limit values of parameters s and r are also of interest being related to classic triangle inequalities. Namely,(ii) the length/time of a shortest directed path,(iii) the inverse width of a bottleneck path,(iv) the inverse capacity (maximum flow per unit time)from a terminal a to a terminal b correspond to:(ii) r = s → ∞, (iii) r = 1, s → ∞, (iv) r → 0, s = 1$, respectively. These results generalize ones obtained in 1987 for monomial isotropic networks, which are modeled by undirected graphs. In this special case, effective resistances form a metric space when s ≥ r and an ultrametric one when s/r → ∞.Of course, for anisotropic networks, which are modeled by directed graphs, the symmetry, μa,b=μb,a may fail. For the linear isotropic networks triangle inequality is known since 1967 and it was extended to the linear anisotropic case in 2016.

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Self-reverse distance magic labelings

mario.krajoski — Wed, 20 May 2026 22:00:00 +0000

Datum / Date: 25.5.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Ksenija Rozman (University of Primorska and IMFM) Naslov / Title: Self-reverse distance magic labelings Povzetek / Abstract: According to a nonstandard definition introduced in 2021, a distance magic labelling ℓ of a regular graph of order n is a bijection from its vertex set to the set of integers of the arithmetic progression from 1-n to n-1 with common difference 2, such that the sum of the labels of the neighbors of each vertex is zero. Such a labeling is called self-reverse if, for any pair of vertices u and v, u is adjacent to v if and only if the vertices with labels -ℓ(u) and -ℓ(v) are adjacent. In this talk, we present the motivation for studying self-reverse distance magic labelings. We focus on self-reverse distance magic labelings in the case of tetravalent graphs. First, we examine some tetravalent distance magic graphs of small order and determine which of them admit a self-reverse distance magic labeling. We then provide several examples and a complete classification of all orders for which a tetravalent graph admitting such a labeling exists. The classification is obtained via a novel construction that produces a (tetravalent) distance magic graph from two given (tetravalent) distance magic graphs. We also discuss the existence of graphs admitting a self-reverse distance magic labeling among some well-known families of tetravalent graphs and conclude with possible directions for future research. This is joint work with Petr Kovář and Primož Šparl.

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New Directions in Manin’s Conjecture

mario.krajoski — Sat, 09 May 2026 22:00:00 +0000

Datum / Date: 8.5.26 Čas / Time: 12:30 – 13:30 Predavalnica / Location: ZOOM (link below) Predavatelj / Lecturer: Ratko Darda (Sabanci University, Turkey) Naslov / Title: New Directions in Manin’s Conjecture Povzetek / Abstract: One of the central questions in number theory and Diophantine geometry is to understand how solutions to polynomial equations are distributed among the rational numbers. The behavior is closely related with the geometry of the space – the variety – defined by the equations. Depending on certain properties of the variety, different mathematical theories are used to analyze it. The case where rational solutions are abundant is addressed by Manin’s conjecture, proposed by Yuri Manin and collaborators in the early 1990s. This conjecture provides a precise prediction for the number of solutions when counted according to their arithmetic size. Over the years, it has been widely studied and has become one of the central problems in the field.Recently, we have investigated a new environment for the conjecture – that of stacks – the geometric objects generalizing varieties. This perspective has led to unexpected applications beyond its original scope, offering insights into phenomena related to the occurrence of Galois extensions—a topic that, at first glance, seems unrelated to counting solutions of polynomial equations. ZOOM link: https://upr-si.zoom.us/j/62775855689?pwd=4xucK8QCXxPEQXPKFXk09kGkob59DZ.1

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Towards the enumeration of EL-semihypergroups

mario.krajoski — Wed, 06 May 2026 22:00:00 +0000

Datum / Date: 13.5.26 Čas / Time: 14:00 – 15:00 Predavalnica / Location: FAMNIT-VP3 & ZOOM (link below) Predavatelj / Lecturer: Enrico Talotti (University of Nova Gorica, and University of Primorska) Naslov / Title: Towards the enumeration of EL-semihypergroups Povzetek / Abstract: We investigate finite semigroups equipped with a preorder relation that is compatible with the semigroup operation and we develop an enumeration methodology based on automorphism group actions. For a fixed finite semigroup S, the automorphism group Aut(S) acts naturally on the set Pre(S) of compatible preorders via pushforward; the orbits of this action correspond exactly to isomorphism classes of preordered semigroups. Using known classifications of semigroups of small order, together with exhaustive generation of compatible preorders, our method yields a complete enumeration of preordered semigroups up to order five. Finally, this framework opens a computational window toward the study of more complex algebraic systems, particularly EL-hyperstructures arising in hypercompositional algebra, where preordered semigroups serve as a fundamental tool for their construction. ZOOM link: https://upr-si.zoom.us/j/65127426860?pwd=kYjS6yFt6yeUcYAGGPcnJShbb5UI7r.1

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Quartic Symmetry, Hamiltonicity, and CFSG-Free Structure

mario.krajoski — Wed, 06 May 2026 22:00:00 +0000

Datum / Date: 11.5.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Dragan Marušič (University of Primorska and IMFM) Naslov / Title: Quartic Symmetry, Hamiltonicity, and CFSG-Free Structure Povzetek / Abstract: I will discuss a program bringing together quartic arc-transitive graphs, Hamiltonicity questions for cubic Cayley graphs, and CFSG-free methods in algebraic graph theory. The starting point is the use of auxiliary graphs, the so called hexagon graphs, associated with Cayley maps of $(2,s,3)$-groups. Large induced trees in these auxiliary graphs provide a route to Hamilton cycles/paths in the original Cayley graphs. I will then explain why quartic one-regular graphs are central in a possible generalization of this approach to $(2,s,4)$-groups. In the final part, I will discuss the Feng–Xu normality theorem for quartic Cayley graphs on regular $p$-groups and propose a CFSG-free proof in the special case where one generator has order $p$. This leads naturally to broader questions about direct proofs in algebraic combinatorics, including primitive groups of degree $2p$ and the Polycirculant conjecture.

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Curves on the torus with prescribed intersections

mario.krajoski — Wed, 06 May 2026 22:00:00 +0000

Datum / Date: 18.5.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Daniel Kral (Leipzig University and MPI MiS) Naslov / Title: Curves on the torus with prescribed intersections Povzetek / Abstract: Juvan, Malnič and Mohar [J. Combin. Theory Ser. B 68 (1996), 7-22] proved that for every surface S and every integer k, there exists N(S,k) such that any set of simple closed curves on the surface S, where any two are non-homotopic and intersect at most k times, has a maximum size at most N(S,k). In the case when S is the torus T^2, the problem has an interesting connection to number theory as the Riemann hypothesis yields that N(T^2,k)<=k+O(k^0.5*log k); Aougab and Gaster [Math. Proc. Cambridge Philos. Soc. 174 (2023), 569-584] have recently proven that this bound holds directly. We determine N(T^2,k) exactly for every k. In particular, we show that N(T^2,k)<=k+6 for all k and N(T^2,k)<=k+4 when k is sufficiently large. The talk is based on joint work with Igor Balla, Marek Filakovský, Bartłomiej Kielak and Niklas Schlomberg.

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Elementary proof of the prime number theorem

mario.krajoski — Wed, 29 Apr 2026 22:00:00 +0000

Datum / Date: 4.5.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Aleksander Simonič (University of Primorska) Naslov / Title: Elementary proof of the prime number theorem Povzetek / Abstract: In 1949, Selberg and Erdos independently published elementary proofs (no use of complex analysis and the Riemann zeta-function theory) of the prime number theorem. In this talk we will go through slightly simplified proof by Levinson, and see how we might present it to students who are studying Selected topics in number theory at FAMNIT.

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Prime Numbers: Emergence and Relevance of Bilinear Decompositions

mario.krajoski — Tue, 14 Apr 2026 22:00:00 +0000

Datum / Date: 20.4.26 Čas / Time: 15:00-16:00 Predavalnica / Location: FAMNIT-MP1 Predavatelj / Lecturer: Olivier Ramaré (CNRS and Institut de Mathématiques de Marseille) Naslov / Title: Prime Numbers: Emergence and Relevance of Bilinear Decompositions Povzetek / Abstract: This talk will retrace the main steps of the modern theory of prime numbers and in particular how the combinatorial sieve combined with the Dirichlet series theory to give birth to the modern representation of the primes via a linear combination of terms, some of which are "linear", while other ones are "bilinear". This will lead us to the recent developments of Green & Tao, Mauduit & Rivat, Tao, Helfgott, and Bourgain, Sarnak & Ziegler, and to some of mines.

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