Schedule and abstracts

September 24, 2022.

(All times are Central European summer time.)

09:30 Welcome

10:00 - 10:20 Miklós Simonovits

Title: The algebraist Peter Vámos

Abstract: My personal and mathematical memories of my friend Péter Vámos.

10:30 - 10:50 Norman Fenton (online via Zoom)

Title: Smart data not big data: Improving critical decision-making with Bayesian networks

Abstract: Misunderstandings about risk, statistics and probability often lead to flawed decision-making in many critical areas such as medicine, finance, law, defence, and transport. The ‘big data’ revolution was intended to at least partly address these concerns by removing reliance on subjective judgments. However, even where (relevant) big data are available there are fundamental limitations to what can be achieved through pure machine learning techniques. This talk will explain how using causal probabilistic models of risk – based on a technique called Bayesian networks – provides powerful decision-support and accurate predictions by a ‘smart data’ approach. This combines minimal data with expert judgment.

11:00 - 11:30 Refreshment break

11:30 - 12:00 Pham Ngoc Ánh

Title: Peter Vámos and divisibility theory

Abstract: B. Bosbach initiated an abstract ideal theory of commutative arithmetical rings as Bezout monoids, a name coined later by Péter Vámos. Arithmetical rings had been introduced by László Fuchs in a seminal work published in the first volume of the Proceedings of the AMS. In this short talk we will discuss Vámos's contribution and influence in the realization problem of Bezout monoids, a natural generalization of valuation theory.

12:10 - 12:40 Vladimir Bavula

Title: Holonomic modules and 1-generation in the Jacobian Conjecture

Abstract: We show that the Jacobian Conjecture, the Conjecture of Dixmier and the Poisson Conjecture are questions about holonomic modules for the Weyl algebra A_n. Using this approach we show that the images of the Jacobian maps, endomorphisms of the Weyl algebra A_n and the Poisson endomorphisms are large in the sense that further strengthening of the results on largeness would be either to prove the conjectures or produce counter examples (the conjectures hold if and only if the images coincide with the algebras). A short direct algebraic (without reduction to prime characteristic) proof is given of equivalence of the Jacobian and the Poisson Conjectures (this gives a new short proof of equivalence of the Jacobian, Poisson and Dixmier Conjectures).

12:50 - 14:30 Lunch

14:30 - 15:00 Alberto Facchini

Title: Multiplicative lattices, skew braces

Abstract: Multiplicative lattices are complete lattices endowed with a further binary operation, not-necessarily associative. Left skew braces are algebraic structures defined five years ago in connection to set-theoretical solutions of the Yang-Baxter equation.

15:10 - 15:40 Endre Szemerédi

Title: On a Problem of Heilbronn

Abstract: We prove that in the unit square there are n points P₁, P₂, ..., P_n such that the minimum of the areas of the triangles P_iP_jP_k (taken over all selections of three out of n points) is greater than log n/n². Heilbronn conjectured that the minimum area is less than c/n², so we refuted this conjecture. This is a joint work with János Komlós and János Pintz.

15:50 - 16:20 Simone Virili

Title: Length functions on modules

Abstract: In this talk we will show that the bivariant length functions introduced by Hanfeng Li can be seen as classical length functions on the category of additive pre-sheaves [fp(R),Ab] of additive functors from finitely presented modules to Abelian groups. In particular, there is a bijection between Sylvester rank functions on fp(R) and (normalized) length functions on [fp(R),Ab]. As a consequence, when suitably extended to locally coherent Grothendieck categories, the theory of Sylvester rank functions is equivalent to that of length functions.

16:30 - 17:00 Refreshment break

17:10 - 17:40 Sylvia Wiegand

Title: Block-diagonalization over Pruefer domains

Abstract: It is a talk about joint work that Peter and I did about twenty years ago.

17:50 - 18:20 Roger Wiegand

Title: Monoids of modules

Abstract: I will discuss the monoid given by isomorphism classes of direct summands of direct sums of copies of some finite set of finitely generated modules.