Romanian Mathematicians

Let’s Learn Maths Together

Multilateral Comenius Project

2011 - 2013

This project has been funded with support from the European Commission.

This publication [communication] reflects the views only of the author, and the Commission cannot be held

responsible for any use which may be made of the information contained therein.

Famous Romanian

Mathematicians

By Filip Chirica 7G Grade, ICHB, Romania

Alexandru Dobromir 7G Grade, ICHB, Romania

Adam Popescu 7F Grade, ICHB, Romania

Stefania Spandonide 7F Grade, ICHB, Romania

Andrei Stefanescu 7F Grade, ICHB, Romania

Maria Yaelle Toader 7G Grade, ICHB, Romania

Ana-Maria Tudorache 7F Grade, ICHB, Romania

.

Motto:

"All what is correct thinking is either

mathematics or feasible to be

transposed in a mathematical model.“

Grigore Moisil

“Gazeta Matematica” The most prestigious Romanian math publications The world longest running mathematical magazine of its kind Published monthly since 1895 by Romanian Mathematical Society.

Romanian Academy Founded on 1/13 April 1866 The quality of an academician is synonymous with absolute intellectual preeminence in modern Romanian society. There are 181 acting members, elected for life.

Three Symbols of Romanian

Mathematic World

“Learning maths, you learn to think. ”

Grigore Moisil

And …. Olympiads

Romania was the International Mathematics Olympiad promoter •Romania was host for IMO five times with a continuous increased number of contestants

•Over time 50 Romanian team attended IMO and won:

68 gold medals / 113 silver medals / 90 bronze medals / two honorable mentions.

•Olympics with the most gold medals (3 gold medals each) are: Ciprian Manolescu, Mihai Manea, Stephen Lawrence Hornet, Teodor Banica and the person who participated in the Olympics often (4 times - 2004-2007 ) is Adrian-Ioan Zahariuc. Moreover, Romanian mathematicians, internationally recognized professors at prestigious American universities such as Dan Voiculescu, George Lusztig and Daniel Tataru obtained, as students, the International Olympic gold medal.

Romanians love solving problems

• Any area of life is:

– to "solve a problem“

– to find a way out of a difficulty

– to find a way to achieve a goal that is not

directly accessible.

• To find the solution of such problems is a specific

feature of intelligence, and intelligence of the

human species is a distinctive peak.

• Mathematics, of all sciences, is that man learns

best how to solve a problem. And Romanians love

this.

What is a mathematician?

• A researcher

• A person who create new mathematical facts (concepts, methods, theories)

• A teacher with high degrees

The most Romanian mathematicians are all of these. They are not included in the 100 most famous mathematicians of the world, but they are international renowned for their contributions. We are proud of them.

Seven generations of Romanian

Mathematicians

First half of XIXth century – Janos Bolyai, Transilvania

(then was part of Austro – Hungarian Empire)

Middle of XIXth century - Spiru Haret and David Emmanuel (Ph. at Sorbonne)

Second half of XIXth century – Gh. Titeica, D. Pompeiu (Ph. at Sorbonne),

Al. Myller, Vera Myller (Ph. at Gottingen)

First half of XXth century – V. Valcovici, Traian Lalescu, Simion Stoilow

Middle of XXth century - O. Onicescu, P. Sergescu, Dan Barbilian,

A. Froda, Gh. Vranceanu

Second half of XXth century – Grigore Moisil, G. Calugareanu, Dramba

Contemporary – C. Calude, I. Cuculescu, Titu Andreescu, L. Tataru

János Bolyai

(1802 – 1860)

• Romanian (ethnic Hungarian)

• Born and lived in Transilvania,

Romania

• One of the founders of non-

Euclidean geometry (in parallel

and independent of Lobacevski).

The discovery of a consistent alternative geometry that might correspond

to the structure of the universe helped mathematicians to study abstract

concepts that later were applied in physics (Einstein theory).

János Bolyai and the Euclid’s Postulate

• Non-Euclidean geometry— a geometry that differs from Euclidean geometry in

its definition of parallel lines.

• By the age of 13, he had mastered calculus and other forms of analytical

mechanics. He became so obsessed with Euclid's parallel postulate that his

father wrote to him: "For God's sake, I beseech you, give it up. "

• János, however, persisted in his quest and eventually came to the conclusion

that the postulate is independent of the other axioms of geometry and that

different consistent geometries can be constructed on its negation. He wrote to

his father: "Out of nothing I have created a strange new universe".

• Between 1820 and 1823 he prepared a treatise on a complete system of non-

Euclidean geometry. Bolyai's work was published in 1832 as an appendix to a

mathematics textbook by his father.

• Gauss, on reading the Appendix, wrote to a friend saying "I regard this young

geometer Bolyai as a genius of the first order". In 1848 Bolyai discovered that

Lobachevsky had published a similar piece of work in 1829. Though

Lobachevsky published his work a few years earlier than Bolyai, it contained

only hyperbolic geometry. Bolyai and Lobachevsky did not know each other or

each other's works.

Others Works

• In addition to his work in geometry, Bolyai

developed a rigorous geometric concept of

complex numbers as ordered pairs of real

numbers.

• Although he never published more than the 24

pages of the Appendix, he left more than 20,000

pages of mathematical manuscripts when he

died. These can now be found in the Bolyai-

Teleki library in Târgu Mureş, where Bolyai died.

Mathematical discoveries, like springtime violets in the woods, have their season, which no man can hasten or retard. J. Bolyai

Beginning of Maths in Romania

Only since 2 centuries in the world of maths.

Begun in 1830, mathematical education was regarded as a model of thinking, an innovative idea for that time. Gheorghe Lazar and George Asachi have implemented this by introducing the first math books in Romanian.

Their work was continued in the late nineteenth century by Spiru Haret, math teacher, considered the father of modern Romanian education reform and David Emmanuel, both with Ph. degree at Sorbonne.

Spiru Haret

DIMITRE POMPEIU

(4 October 1873 – 8 October 1954)

From a little Moldavian town to Sorbonne and

back to Bucharest

Dimitrie Pompeiu

University of Iaşi

University of Bucharest

He grew up and followed primary school in in a little Moldavian town, Dorohoi and then the secondary school in Bucharest

1893 – 1898 – primary school teacher

1898 - University of Paris (the Sorbonne) –graduated in Mathematics

1905 - Ph.D. degree in Mathematics with thesis “On the continuity of complex variable functions”, written under the direction of Henri Poincaré.

1934 – is elected member of Romanian Academy

- Chair at Universities from Iasi and Bucharest

Pompeiu’s Contributions

• His contributions were mainly in the field of mathematical analysis, complex functions theory, and rational mechanics.

• In an article published in 1929, he posed a challenging conjecture in integral geometry, now widely known as the Pompeiu problem.

• Among his contributions to real analysis there is the construction, dated 1906, of non-constant, everywhere differentiable functions, with derivative vanishing on a dense set. Such derivatives are now called Pompeiu derivatives.

Hausdorff - Pompeiu Distance

• In mathematics, the Hausdorff distance, also called Pompeiu–Hausdorff distance or metric was introduced in 1914.

• Hausdorff – Pompeiu distance measures how far two subsets of a metric space are from each other. It turns the set of non-empty compact subsets of a metric space into a metric space in its own right.

• Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance you can be forced to travel by an adversary who chooses a point in one of the two sets, from where you then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set.

Pompeiu’s theorem

Given an equilateral triangle ABC in the plane, and a point P in the plane

of the triangle ABC, the lengths PA, PB, and PC form the sides of a

(maybe, degenerate) triangle.

Proof 1. (Simple, but not classical)

Consider a rotation of 60° about the point C. Assume A maps to B, and B maps

to B '. Then we have PA=P’B and PC=P’C . Hence triangle PCP ' is

equilateral and PP’=PC . It is obvious that . Thus, triangle PBP ' has sides

equal to PA, PB, and PC and the proof by construction is complete.

Further investigations reveal that if P is not in the interior of the triangle,

but rather on the circumcircle, then PA, PB, PC form a degenerate triangle,

with the largest being equal to the sum of the others.

Pompeiu’s Theorem

Proof 2. (with complex numbers)

Let P(z), A(a), B(b) ,C(c) the four points with their afixes. Whereby ABC is equilateral => |a-c|=|b-c|=|a-b| By calculating we have (z-b)(a-c)=(z-a)(b-c)+(z-c)(a-b) Passing to absolute values and using 𝑥 + 𝑦 ≤ 𝑥 + 𝑦 we have:

|(z-b)(a-c)|=|(z-a)(b-c)+(z-c)(a-b)| <=> |z-b|=|(z-a)+(z-c)|<=|z-a|+|z-c) Now we obtain 𝑃𝐴 + 𝑃𝐶 ≥ 𝑃𝐵 , 𝑃𝐴 + 𝑃𝐵 ≥ 𝑃𝐶, 𝑎𝑛𝑑 𝑃𝐵 + 𝑃𝐶 ≥ 𝑃𝐴

OBS. We have equality in these relations then P is on the circumcircle. Then we have the Ptolemeu relation. This the explanation for the identity written at the beginning of the proof.

Gheorghe Titeica

1873-1939

• Founder of the Romanian

school of affine differential

geometry.

• He introduced a new class of

curves and surfaces that today

bears his name.

• Mate in Paris with H. Lebesgue

and Montel, he sustained his

master's degree thesis on the

framework of oblique curvature

with G. Darboux.

Gheorghe Titeica

Problem of the 5 lei coin (Titeica discovered it while pulling hung with a pencil outline of such coins during a meeting)

Statement Three congruent circles have a common

point H and twos longer intersect at points A, B, C.

Prove that the circle ABC is congruent with the

three circles and H is orthocenter of the triangle

ABC. (Long list, OBMJ, Izmir, Turkey, 2003)

The problem was the logo for

40th IMO, 1999, Romania

Titeica - Problem of the 5 lei coin

R- common length rays circles.

O1,O2 and O3 centers of the three circles.

Quadrilaterals O1AO2H, O2CO3H and O3BO1H are rhombus, with all sides congruent.

So O1B ∥O3H ∥ O2C and O1B ≡O3H ≡O2C. Then the quadrilateral O1BCO2 is parallelogram.

Therefore triangles ΔO1O2O3 ≡ Δ CBA ( S.S.S.). As O1H=O2H=O3H=O1A=R, observe that H circumcenter of triangle ΔO1O2O3. The circle (O1O2O3) has radius R.

Gheorghe Vranceanu

30 June 1900- 27 April 1979

• Famous Romania mathematician

• Best known for his work in differential geometry

• Romanian Academy full member from 1955 to 1979

• From 1964 to 1979 he was President of the Mathematics Section of the Romanian Academy.

• over 300 articles published in journals throughout the world

• “Gheorghe Vranceanu” school in Bacau, Romania.

Dan Barbilian- Ion Barbu 18 March 1895 –11 August 1961

• Distinguished Romanian mathematician ,

poet and professor at the University of Bucharest

• As a mathematician he is known for his publications in “Gazeta Matematica”

• As a poet, he is best known for his volume ”Second Game”

• The most important contributions - two papers that appeared

in Časopis Mathematiky a Fysiky.

• ”Barbilian” spaces in geometry were named after him.

• “Dan Barbilian-Ion Barbu mathematic and literature contest”

takes place yearly.

Romanian mathematician,

considered the father of

Romanian computer science.

His research was mainly in

the fields of mathematical

logic, (Łukasiewicz–Moisil

algebra), algebraic logic, and

differential equations.

Grigore Moisil

1906 - 1973

• Brilliant Math teacher, he is known not only for his

studies, but also for his jokes and for his great

sense of humor.

Professor Grigore Moisil, during a Math class, after

reading crap stuff written by a student, called the

student, kissed his forehead and said: "You can tell

everyone that Professor Grigore Moisil kissed your

ass, because that's not a head. "

Grigore Moisil - anecdotes

Grigore Moisil - anecdotes

During an interview, a reporter says:

- You know the truth hurts.

Moisil: I was never hurt by a math theorem

• An anecdote which is not understood by the listener, nor the narrator is

called psychological book.

• The marriage is the only escape for an unsuccessful man and a too

successful woman.

• - Hello, professor, do you believe in dreams?

- Of course! A while ago I dreamed that I became academician, I was in a

big conference room and I was the chair-man.

And when I woke up, I was an academician, I was in a big conference room

and I was the chair-man.

• During Moisil’s exams to become a professor, the commission evaluated

him. Professor Procopiu voted against Moisil because he considered Moisil

too young to get the “professor” title.

Moisil: It is a defect that I am correcting daily, said Moisil.

• A friend is telling to a math teacher:

- I am full of the math you are teaching; I am full up to my neck.

Moisil: But the math starts from the neck up!

.

• In 1960’s, Grigore Moisil participated at a National Meteorology Institute

meeting. Somebody stated:

- Based on the latest discoveries of Russian researchers, we are able to

forecast weather with a 40% probability..

- Moisil: Then, why don’t you forecast it the other way around? That way

you would have a 60% probability.

Grigore Moisil - anecdotes

Alive Famous Romanian

Mathematicians

• Mathematicians with international reputations in the

scientific world work in Romania and many are currently

professors in the great universities of the world.

• In Romania, world-renowned mathematical schools were

created and are still active; some of their fields of study are

the theory of operators, complex analysis, the theory of

potential, the theory of manifolds, differential equations and

the theory of the optimal central, fluid mechanics and the

theory of elasticity, the theory of probabilities and

mathematical statistics.

Cristian S. Calude Born 21 April 1952-

Romania-New Zealand mathematician and

computer scientist

• Mathematical Student Prize, Faculty of Mathematics,

Bucharest University, Romania, 1975.

• Honorificum Membrum, Black Sea University, Bucharest, Romania, December 2002.

• Excellence in Research Award, University of Bucharest, Romania, 2007

• Dean's Award for Excellence in Teaching, Faculty of Science, University of Auckland, 2007.

• The “C. S. Calude Mathematics Regional Contest” takes place every year in Galati, Romania.

Titu Andreescu -

Coach for the Romanian IMO team in 1983

Although he coached Romanian IMO team and consultant of the Education Minister, in 1985 Titu Andreescu was not allowed to go at Helsinki, for IMO.

The reason?! His grandparents emigrated early last century, but his mother and grandmother returned to Romania after the World War I.

He is born in 1956

As a high school student, he excelled in

mathematics: in 1973, 1974 and 1975 he

won the Romanian national problem solving

contests organized by the journal Mathematical Review .

Titu Andreescu

In 1990, as the Eastern Bloc began to collapse, Andreescu emigrated with his mother to the United States.

National award of "Distinguished Professor”

Associate professor of mathematics at the University of Texas at Dallas.

Director of AMC (as appointed by the Mathematical Association of America), Director of MOP, Head Coach of the USA IMO Team and Chairman of the USAMO. He has also authored a large number of books on the topic of problem solving and Olympiad style mathematics.

Titu Andreescu

1993- 2006 Coach for the USA IMO team

Photo: US team, at IMO, 2002, in Scotland >>>>

1993- 2006 Coach for the USA IMO team

In 2006 he established a math camp for talented middle and high school students, AwesomeMath

Year-round program AMY and a mathematical circle, Metroplex.

In 2008 he founded Math Rocks! Program for kids with very high-level math skills.

Titu Andreescu Inequality

𝒙𝟏𝟐

𝒚𝟏 +

𝒙𝟐𝟐

𝒚𝟐 + … +

𝒙𝒏𝟐

𝒚𝒏≥( 𝒙𝟏+𝒙𝟐+ …+𝒙)

𝟐

𝒚𝟏+ 𝒚𝟐+ ...𝒚𝒏 𝒂𝒊 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓𝒔, 𝒙𝒊 > 𝟎, 𝒏 ≥ 𝟐

Proof:

• From Cauchy-Buniakovski - Schwartz inequality 𝑎𝑖

2 ∙ 𝑏𝑖2 ≥ ( 𝑎𝑖𝑏𝑖

𝑛𝑖=1 )2 ∙𝑛

𝑖=1𝑛𝑖=1

• By replacing 𝑎𝑖 =𝑥𝑖

𝑦𝑖 and 𝑏𝑖 = 𝑦𝑖 we obtain:

• 𝑥𝑖2

𝑦𝑖∙ 𝑦𝑖 ≥ ( 𝑥𝑖

𝑛𝑖=1 )2 ∙𝑛

𝑖=1𝑛𝑖=1

• And we divide by the second multiplier from left member.

“Learning maths, you learn to think. ” G. Moisil

Titu Andreescu

• After the success of USA team, U.S. journalists have called him "Bela Karoly of mathematics".

• Comparing with Romanian students, Titu Andreescu think American students "are not smarter, but more organized. Romanians are perhaps more creative, but Americans are more disciplined and work hard.

• "Know that American math Olympians are" well-rounded ", multilateral. Are good at sciences, sing - very well - an instrument, write well. Not nice, but well, "said U.S. professor.

Next Generation of Academicians? Romanians or abroad?

Romanian team at the International Mathematics Olympiad in Argentina, 2012