The Eighth Congress of Romanian Mathematicians

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List of talks

I. Algebra and Number Theory

Special session: Local rings and homological algebra. Special session dedicated to Prof. Nicolae Radu

II. Algebraic, Complex and Differential Geometry and Topology

Special session: Geometry and Topology of Differentiable Manifolds and Algebraic Varieties

III. Real and Complex Analysis, Potential Theory

IV. Ordinary and Partial Differential Equations, Variational Methods, Optimal Control

Special session: Optimization and Games Theory

V. Functional Analysis, Operator Theory and Operator Algebras, Mathematical Physics

Special session: Spectral Theory and Applications in Mathematical Physics

Special session: Dynamical Systems and Ergodic Theory

VI. Probability, Stochastic Analysis, and Mathematical Statistics

VII. Mechanics, Numerical Analysis, Mathematical Models in Sciences

Special session: Mathematical Modeling of Some Medical and Biological Processes

Special session: Mathematical Models in Astronomy

VIII. Theoretical Computer Science, Operations Research and Mathematical Programming

Special session: Logic in Computer Science

IX. History and Philosophy of Mathematics

History and Philosophy of Mathematics 

(this list is in updating process)

RIT (Rochester Institute of Technology), USA
Title: Mathematics: Current State and Future Direction (details)
The current state of mathematics will be a chapter of tomorrow’s History of Mathematics. There are various opinions and conversations on the future of the discipline and many questions on what and who determines the mathematics of tomorrow. It is also important to understand how and why the economic environment, academic leaders, faculty and researchers influence the direction of this field. This presentation will address these questions and will entertain a debate on this topic. Moreover, case studies including experiences from Romanian universities will also be presented and discussed.
Ecole Polytechnique, France
Title: The 1874 controversy between Camille Jordan and Leopold Kronecker (details)
During the whole of 1874, Camille Jordan and Leopold Kronecker quarrelled vigorously over the organisation of the theory of bilinear forms. That theory promised a general and homogeneous treatment of numerous questions arising in various 19th-century theoretical contexts, and it hinged on two theorems, stated independently by Jordan and Weierstrass, that would today be considered equivalent. It was, however, the perceived difference between those two theorems that sparked the 1874 controversy. Focusing on this quarrel allows us to explore the algebraic identity of the polynomial practices of the manipulations of forms in use before the advent of structural approaches to linear algebra. The latter approaches identified these practices with methods for the classification of similar matrices. We show that the practices - Jordan's canonical reduction and Kronecker's invariant computation - reflect identities inseparable from the social context of the time. Moreover, these practices reveal not only tacit knowledge, local ways of thinking, but also - in light of a long history tracing back to the work of Lagrange, Laplace, Cauchy, and Hermite - two internal philosophies regarding the significance of generality which are inseparable from two disciplinary ideals opposing algebra and arithmetic. By interrogating the cultural identities of such practices, this study aims at a deeper understanding of the history of linear algebra without focusing on issues related to the origins of theories or structures.
Romanina Academy, Iasi branch, Romania
Title: Axiom of Choice in Finitely Supported Mathematics (details)
Finitely Supported Mathematics (FSM) is consistent with the axioms of the Fraenkel-Mostowski (FM) set theory representing an `axiomatization' of the FM permutation model of the Zermelo-Fraenkel set theory with atoms. The axioms of the FM set theory are those of Zermelo-Fraenkel with atoms (over an infinite set of atoms), together with the special property of finite support which claims that for each element x in an arbitrary set we can find a finite set supporting x. The finite support axiom is motivated by the fact that usually a syntax can only involve finitely many names/variables. Therefore, in the FM set theory only finitely supported objects are allowed. The FM set theory was constructed initially in 1930s, in order to prove independence of the axiom of choice and other axioms in the classical ZF set theory. In 2000s, the FM set theory found some applications in computer science. However, the consistence of the various weaker forms of choice (which were proved to be independent from the axioms of ZF) remained an open problem. We present our results regarding the consistency of various choice principles in FSM. It is known (from 1930s) that the full axiom of choice is inconsistent in ZF and FSM. We prove that the choice principles denoted generally by AC, DC, CC, PCC, AC(fin), Fin, PIT, UFT, OP, KW, RKW, OEP rephrased in terms of finitely supported objects are all inconsistent in FSM. Moreover, if the set of atoms is countable, the choice principle generally denoted by CC(fin) is also inconsistent in FSM. It is worth noting that such results are not easy to prove in FSM, even if various related results regarding these choice principles hold in the ZF framework. This is because nobody guarantees that ZF results remain valid in FSM. Therefore, all the possible relationship results between various choice principles in FSM have to be independently proved in terms of finitely supported object.
University of Kuweit, Kuweit
Title: Mathematical archaeology: Art Nouveau (details)
Around 1880, R. Dedekind considered, for a group G and for elements x, y ∈ G, the unique element f of G satisfying the equality xy = yxf and thus invented what today is called the commutator of the ordered pair (x, y). Both Dedekind and G.F. Frobenius considered the set C of all of the commutators in a group G and asked themselves whether C is always a subgroup of G. Interestingly, finding examples of groups G such that C is not a subgroup of G is a nontrivial exercise, for the smallest order groups where this happens have order 96. Examples showing that C need not be a subgroup of G appeared soon after and people considered the subgroup G0 of G generated by the set C, calling G0 the derived subgroup (or, commutator subgroup) of G. Results ensuring that under suitable hypotheses G0 = C or G0 6= C started to appear and, for the last one hundred years or so, people tried to find necessary and sufficient conditions for the equality G0 = C to hold. My talk will tell the story of the long way towards the general solution of this problem for a finite group G. The solution was obtain jointly with G. L. Walls and is based on classic results and on ideas originating in the Art Nouveau period (circa 1890-1910): the CauchyFrobenius Lemma, Frobenius’ theory of complex characters, simple counting techniques and a measure on the set G defined by using commutators. These combine to give a formula for the number |H \ C| of those elements of a normal subgroup H of G that are not commutators in G. Many new and somewhat surprising results follow as consequences. For example, it is wellknown that every normal subgroup of G is the intersection of kernels of irreducible complex characters of G; but what becomes now transparent is that if the normal subgroup is the intersection of the kernels of more than half of the irreducible characters of G, then every element of that normal subgroup must be a commutator in G
ECKES Christophe
Archives Henri Poincaré Université de Lorraine, France
Title: The correspondence between Hermann Weyl and Erich Hecke (details)
The mathematician and mathematical physicist Hermann Weyl carries on a regular correspondence with the number theorist Erich Hecke until the death of the latter in 1947. Their letters, which are currently preserved at the Hermann Weyl Archives (ETH Zürich) and at the Erich Hecke Archives (University of Göttingen) have not been studied systematically until now. However, their correspondence appears to be a central element in order to describe their career and to get an overview of their scientific productions. Moreover, a careful study of these letters also reveals that Hecke and Weyl share the same conception of mathematical knowledge and that they belong to very close intellectual circles. In the first part of our talk, we will try to explain why this correspondence has been neglected by many historians of mathematics until now and we will underline its importance in order to avoid some biases in the description of Hecke’s and Weyl’s institutional trajectories. In the second part, we will describe anew Weyl’s exil after the nazis came to power. We must recall here that Weyl becomes permanent professor at the Institute for Advanced Study (IAS, Princeton) during the Winter semester 1933 and he remains there until his retirement in 1951. We will refer in particular to his correspondence with Hecke and Abraham Flexner (Founding director of the IAS). At that time, Hecke worries about the academic future of his colleague and friend Erwin Panofsky, who has just been dismissed from the university of Hamburg in April 1933. Panofsky will finally get a permanent professorship in art history at the IAS in October 1935. In the last part of our talk, we will describe Hecke’s situation in Hamburg from 1933 to 1945: his participation to the international congress of mathematicians in Oslo in 1936, his six month stay at the IAS in 1938 (he is invited by Weyl), his strong friendship with the danish mathematicians Harald Bohr and Jakob Nielsen, etc.
Brasov, Romania
Title: Aspects of parameter estimation (details)
Adaptive and optimization aspects of the density estimation inference for the probability distribution generating a set of observable values
IONITA Catalin
University "Politehnica" of Bucharest and CRIFST/DLMFS, Romania
Title: The Concept of a Real Definition and that of Real Numbers (details)
The problem we approach has three main features. One is pure mathematical, the other belongs to philosophy, and the later concerns history of mathematics. Despite the difficulties each of them posses, all three are emerging into a whole. By the real definition of something we mean the aggregate of all our modalities which are answering to ”how do we have to proceed to distinguish that something in things themselves”. In this setting, a real definition is unique. A real definition is part of any other definition of the same definiendum. It corresponds to what was called, long time ago, the real part of a definition (in contradistinction to the nominal part). A real definition does not say that other definitions would be wrong. On the contrary, the real definition has to be found, has to be formulated, and has to be proven that what was found is indeed the maximal real part of any other definition of the same definiendum. How do we have to prove that? It would mean to avoid the reduction of the definiendum to its pure nominal part. The problem is of an intrinsic philosophical interest only after the formal approach to a subject-matter is completely and exhaustively accomplished. That is the case of Euclidean geometry and of the system of real numbers. The unreasonable effectiveness and usefulness of contemporary mathematics cannot have another support than the real part they comprise from things themselves together with the real part of our actions upon things. Concerning the case of real numbers, we shall prove that the real part of the axiomatic/categorical definition of real numbers (the later is standing, we sustain, as the best of the best of what can be said) consists of definitions 3,4,5,7 (Euclid, Book 5, ratios of quantities [magnitudes]) attributed to Eudoxus. We have to show two facts. (A) Those definitions are contained in any other definition of real numbers. (B) Those definitions comprise the maximal real part of any other definition of real numbers. The new point is that we have to refer not to geometry but to quantities, magnitudes and their relationships. The problem necessarily involves both history and philosophy of mathematics in their foundational aspects. The problem of the relationships between Eudoxus definition and that of Dedekind cuts was raised by Lipschitz (Dedekind-Lipschitz correspondence). Dedekind answer concerns the essence of continuity. Five years later, Cantor’s discussion on the reality of infinite sets and transfinite numbers foundationally completed the turning point of mathematics to its present state of the art. In that setting, the problem we have raised was somewhat obscured by the productiveness of the new points of view. The problem was inspired to us by Gregory Chaitin’s How real are the real numbers? (arXiv:math/0411418v3, 2004), while the historical viewpoint we follow was inspired by Solomon Marcus’ claim: a critical research passing over the merely notation of facts in what concerns history of mathematics (Solomon Marcus: Mathematics in Romania, Report to CRM-Piteti 2003, CUB PRESS 22, 2004).
MARCUS Solomon
Romanian Academy and Simion Stoilow Institute of Mathematics of the Romanian Academy, Romania
The impossibility theorem proved by Kenneth Arrow in 1951 refers to the aggregation of some individual preferences into a global one and shows that under reasonable requirements the existence of a dictator imposing his personal preference cannot be avoided. J. H. Conway’s cosmological theorem stated in 1986 refers to a specific discrete dynamical system as a sequence starting with a positive integer and so that to generate a term from the previous one have to read off the digits of the previous term, counting the number of digits in groups of the same digit: 1 is read off as “one 1” or 11; 11 is read off as “twice 1s” or 21; 21 is read off as “one 2, then one 1” or 1211. Roughly speaking, he Cosmological theorem asserts, among other things, that starting with any positive integer different from 22 the above discrete dynamical system will have no attractor and its asymptotic behavior will simulate the Mendeleev periodic table. The recent presence in Romania of Arrow and of Conway gave the opportunity of a debate of these results with huge scientific and philosophical impact, in conjunction with some results I have obtained, some of them in joint work with Gheorghe Paun.
NICULESCU Constantin
University of Craiova, Romania
Title: Tiberiu Popoviciu and his contribution to convex functions theory (details)
Fifty years ago Tiberiu Popoviciu published in Analele Univ. Iasi one of the most striking characterization of convex functions of one real variable. Our talk is aimed to discuss the importance of this result as well as the main role played by Popoviciu in promoting the study of convex functions in Romania.
University of Bucharest, Romania
Title: Petre Sergescu and the rebirthing of Bret's theorems (details)
\usepackage{amssymb,amsfonts,amsmath} \def\N{\mathbb N} \def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\R{\mathbb R} In 1815 Jean-Jacques Bret published a paper on the bounding of the positive roots of univariate polynomials over $\R,$. His results were less known than a bound obtained by J.-L. Lagrange and his proofs were not completely understood by his contemporaries. \smallskip The results of Bret were reconsidered by Petre Sergescu (1893--1954) in several papers on the upper bounds for positive roots. He explained Bret's proofs and obtained generalizations of his results. \smallskip We discuss the historical context of the first results on the upper bounds for positive roots and we present the contributions of Petre Sergescu on the reconsideration of the theorems of Bret.
VAIDA Dragos
Facultatea de Matematica si Informatica, Universitatea Bucuresti, Romania
The contributions of Dan Barbilian from various sources – mathematical articles (Euclid’s argument on the infinity of primes), courses (Theory of Galois) or conferences (Gauss and Hilbert) – provide a basis for the history and philosophy of mathematics in our country and a motivation for viewing mathematics as a part of culture. One presents an unknown essay of Barbilian on the experimental component in the mathematical investigation, a trend emphasized in the contemporary research. This essay indicates a mathematical realization of the Apollon-Dyonisos antinomy: concepts and ideas vs examples and concrete representations as an ur-phenomenon of the knowledge. The conventional image concerning Barbilian exclusively related to algebraic structures is thus corrected. An appendix on the symbolism of mirror suggests the existence of links between the mathematician (Dan Barbilian) and the poet (Ion Barbu). References to R. Guénon are relevant for the promoted theory of knowledge. The main case study of this paper deals with the Hilbert Theorem on the commutativity of the Archimedean po-division rings (D. Vaida: [2014] On partially ordered semiring-like systems A Tribute to Alexandru Mateescu. Gh. Paun, G. Rozenberg, A. Salomaa, eds., „Discrete Mathematics and Computer Science. In Memoriam Alexandru Mateescu (1952-2005)”, Editura Academiei, București).
Valahia University of Targoviste, Romania
Title: Some Aspects in the History of Mathematics in Romania (details)
We emphasize some facts about several Romanian mathematicians