Zaha Hadid designs a Mathematics gallery at the Museum of Science in London

The new Mathematics gallery, scheduled to open in 2016,  will explore how mathematicians, their tools and ideas have helped to shape the modern world.

The gallery will be designed by Pritzker Prize recipient Zaha Hadid  who has imagined the space as wind tunnel for an aircraft, which will be supspended in the center of the hall, with the layout of the exhibits following the lines of airflow around it. Other mathematical concepts will be incorporated into the form of the display cases and other aspects of the gallery.

Design animation created by Zaha Hadid Architects of the designs for the new gallery.

The David and Claudia Harding Mathematics Gallery will open in 2016.

http://www.sciencemuseum.org.uk/

Kurt Gödel’s five citizenships: Austro-Hungarian, Czechoslovak, Austrian, German, American….

Kurt Gödel was born in Austria-Hungary in 1906. At age 12, when the Austro-Hungarian Empire was dissolved at the end of WW1, he automatically became a Czechoslovak citizen. He himself identified more with Austria and chose to become an Austrian citizen at 1923 . In 1938, Germany annexed Austria and Gödel became a German citizen. Afte the Anschluss in 1938,  Gödel automatically became a citizen of Nazi Germany. To avoid serving in the German army, Gödel and his wife left for the US in 1939. At the age of 42, in 1949 Gödel became an American citizen.

Prikry Forcing – Karel Prikry

Karel Prikry (1944- )
American. Mathematician (axiomatic set theory)

• 1944: Born in Chvalkovice, the Protectorate of Bohemia and Moravia which is now Czech Republic
• 1965: Awarded a Master’s degree from the Warsaw University
• 1968: Received his PhD at the University of California, Berkeley. Advisor: Jack Silver. Dissertation Changing measurable into accessible cardinals.
• 1980: Promoted to full professor at the University of Minnesota in Minneapolis.
• A method of forcing is named after him – Prikry Forcing.

For the interested reader, here’s the definition of Prikry Forcing:

For a set P of pairs (s,A) where s is a finite subset of a fixed measurable cardinal κ, and A is an element of a fixed normal measure D on κ. A condition (s,A) is stronger than (t, B) if t is an initial segment of s, A is contained in B, and s is contained in $latex t \cup B$. This forcing notion can be used to change to cofinality of κ while preserving all cardinals.

Geometry in Art – Tauba Auerbach’s large scale pop-up sculptures

American visual artist Tauba Auerbach (b. 1981) has created a large-scale pop-up book. Each page tranforms from a flattened surface to an elaborate 3D sculpture, six in total, based on geometric forms – the pyramid, sphere, ziggurat, octagonal bipyramid (gem), arc, and möbius-strip. She titled the book [2,3] which refers to the transformation from two to three dimensions.

Tauba Auerbach
[2,3]
New York, USA: Printed Matter, 2011
6 pp., 50.8 x 82.3 cm
Edition of 1000 signed and numbered copies

Łoś’ Theorem (Jerzy Łoś)

Behind the theorem: Jerzy Łoś (1920-1998)
Polish

QUICK BIO:

• 1920: Born  Lwów, in interwar Poland (now: Lviv, Ukraine).
• 1937: Entered  Jan Kazimierz University (now: Ivan Franko University) but his studies were interrupted by World War II in 1939.
• 1942-1943 Worked as a clerk at a sugar factory in Lublin [1].
•  1943-1944: Worked as a bookkeeper at Tarnagora [2].
• 1945: Resumed studies in 1945 at Marie Curie-Skłodowska University in Lublin, Poland.
• 1947: Awarded a Master’s degree in Philosophy.
• 1949: Received a PhD in sciences from the University of Wrocław under Jerzy Slupecki [3].
• 1955: Awarded a Doctor of Science in mathematics by  the Mathematical Institute of the Polish Academy of Science.
• 1964: Elected member of the Polish Academy of Scien

Throughout his career, Łoś research spanned over a broad range of research areas. His work has had significant impact in several fields and he published influential papers in philosophy, logic, algebra, probability theory, and mathematical foundations of economics.

Łoś was an innovator – his main strength was his ability to conceive fundamental new notions which would pave the way for entirely new directions in research. For example, Łoś was among the first logicians to study Model Theory for which he defined the notion of an ultraproduct of structures and formulated a fundamental theorem – Łoś’ Theorem. This eponymous theorem was first published in 1955. Since then, Łoś Theorem, also known as the Fundamental Theorem of Ultraproducts, has been considered to be the most powerful and useful tool in model theory [3]. The notion of ultraproducts, defined by Łoś, is also used in other branches of mathematics such as set theory and algebra.

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Let be a set and let be an ultrafilter on (i.e . We define the ultraproduct as where is a collection of structures indexed by .

Łoś’ Theorem:

 iff 

The proof is done by induction on the formula .

• The ultrapower is a special case of an ultraproduct in which all factors (i.e. ) are equal.

For more details about Łoś’ Theorem, go to: A_Lightening_Review_Los_Theorem_Hjorth   by Dr. Greg Hjorth (UCLA).

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References:

1. Stanislaw Balcerzyk. http://www.ipipan.waw.pl/GTME2004/Los1.pdf
2. idem
3. Leszek Pacholski. “In Memoriam: Jerzy Łoś, 1920-1998,” The Bulletin of Symbolic Logic. Vol. 6, No. 1 (Mar., 2000),  97-100.

Axiom of Choice – A short story for the set theorist

In David Corbett’s short story Axiom of Choice, a mathematician explains to a detective how the bodies of his wife and his student came to be in his house. The professor explains how the student’s obsession with the Axiom of Choice and his perception of its philosophical implications regarding free will, brought on a depression which led to an affair with the professor’s wife.

The focal point is not on the murder itself, but on the professor’s character – somber and unemotional.

“Now, I can imagine what you’re thinking. Anyone who walks away from the scene you discovered in my house, then talks about his sex life, his lack of inhibition, his pleasures – not to mention set theory and Descartes, for God’s sake – you have to wonder: Is he demented?…I assure you, I am not deranged, nor do I lack a conscience.”

Get the book: http://www.amazon.com/dp/1453264329/ref=rdr_et_tmb

The short story is included in Corbett’s 2012 book Killing Yourself to Survive: Stories along with six other stories.

Andrzej Mostowski – Underground education

Andrzej Mostowski (1913-1975)
Polish
Mostowski Collapse

• 1913: Born in Lember, Austria-Hungary
• 1939: Received his PhD from the University of Warswaw. Directed by Kuratowski and Tarski.
• 1939-1944: Taught mathematics in the Underground University of Warsaw. Officially worked as an accountant.

Education in Polish was banned and punished with death during the German occupation of Poland. The Germans abolished all university education for non-Germans and closed all institutions of higher education.  Dspite the risks, teachers and professors a organized underground courses with small groups in private apartments. This clandestine education system spread rapidly and by 1944 there were 300 lecturers and around 3500 students in Warsaw alone. Andrzej Mostowski was one of the professors who risked his life teaching during the Nazi occupation. He was captured in 1944 by the Nazis and was to be deported to a concentration camp. With the help of two nurses, he escaped to a hospital.

“Many of Mostowski’s wartime results – on the hierarchy of projective sets, on arithmetically definable sets of natural numbers, and on consequences of the axiom of constructibility in descriptive set theory – were lost when his apartment was destroyed during the uprising. He had to choose whether to flee with a thick notebook containing those results or with bread. He chose bread.”

• 1944: Captured by the Nazi’s after the Warsaw Uprising, but escaped and was to be sent to a concentration camp, but escaped.

• post-war: Worked at the Warsaw University. Spent time at Princeton in 1948-1949 and made contact with Kurt Godel whom he had met before.
• 1953: Appointed head of the Department of Algebra at the University of Warsaw.
• 1958-1959: Unversity of California at Berkeley.
• 1975: Still an active researcher, Mostowski spent the summer at Princeton. Died unexpectedly when visiting Vancouver for a lecture.

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References

1. G N Moore, Biography in Dictionary of Scientific Biography, New York 1970-1990.

Tree space: phylogenetic trees

“Underground”

Visual artist Owen Schul and mathematician  Satyan L. Devados collaborated on a project with the goal to intertwine their disciplines in a direct manner. They decided to work with tree spaces, a mathematical concept related to the geometric relation between different branching configurations. The concept appear in in numerous areas of mathematics, including algebraic topology, enumerative combinatorics, geometric group theory, and biological statistics.

The result of the collaboration is a work titled Cartography of Tree Space, involving acrylic, watercolor, and graphite on 108cm x 108cm wood panels.  One of the pieces, titled “Underground”, is shown below

Sketch, computer illustration

In the end, the mathematician had more to say about the art, and the artist had more to inquire about the mathematics. Three visual pieces were produced, forming at triptych, with the following open mathematical questions:

1. What is the least number of associahedral duals needed to cover the tree space?

2. If we fix a caterpillar tree and relabel the leaves, we obtain the permutohedral dual polytopes. What happens when we fix another tree types and relabel its leaves?

3. The braid arrangement naturally appears as a scaffolding for tree space with rooted trees with five leaves. Does this naturally extend to higher dimensions?

4. Expanding on the notion of a map, what can be said about distances between two points in tree space under certain conditions (such as walks restricted to tree types or alternate notions of discrete metrics)?

“The particular object of our study is a configuration space of phylogenetic trees, originally made famous by the work of Billera, Holmes, and Vogtmann. Each point in our space corresponds to a specific geometric, rooted tree with five leaves, where the internal edges of the tree are specified to be nonnegative numbers. From a global perspective, this “tree space” is made of 105 triangles glued together along their edges, where three triangles glue along each edge. This results in 105 distinct edges, and 25 distinct corners.” – Prof. Devados

References

http://www.satelliteberlin.org/pages/treespace

The Incompleteness Theorem (Kurt Gödel)

Kurt Gödel (1906-1978)
Austrian/American

“The more I think about language, the more it amazes me that people ever understand each other at all.” – Kurt Gödel

• 1906: Born in Brünn, Austria-Hungary (now Brno, Czech Republic)

Ryoji Ikeda’s transfinite art

Japanese artist and electronic composer Ryoji Ikea explores the concept of infinity in a sonic landscape commissioned by the Park Avenue Armory in NYC. The large-scale installation features a 13.7m x 1.3 m projection screen which bisects the space of the exhibition hall with pulsating pattern to create an immersive audio/visual experience.

Here’s what the artist has to say:

“The purest beauty is the world of mathematics. its perfect assemblage of numbers, magnitudes and forms persist, independent of us. the aesthetic experience of the sublime in mathematics is awe-inspiring. it is similar to the experience we have when we confront the vast magnitude of the universe,  which always leaves us open-mouthed… this project explores the transfinite (the infinite that is quantitative and ordered) intersection that lies between such polarizations – the beautiful and the sublime;  music and mathematics; performance and installation; composer and visual artist: black and white; 0s and 1s’”

image © agent j loves agent a