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Ann Varela: The Passing of a Mathematician

Sep 12, 2017 by

An Interview with Ann Varela: The Passing of a Mathematician

Michael F. Shaughnessy –

1) Ann, Maryam Mirzakhani recently died, sadly much too young.  First of all, what was your initial reaction?

It is truly a loss to us all when someone with the enthusiasm for life and learning that Mirzakhani possessed dies.  She personified what being a mathematician is all about; she attempted to explain problems that had not been solved before.  She wanted to comprehend something that had not been understood before.  Mirzakhani often fused unrelated techniques together to understand, visualize, and solve problems.  She incorporated methods from various branches of mathematics, such as algebraic geometry, topology, and probability theory.  She linked exceptional problem solving skills with the intuition and inquisitiveness of an established scientist.

2) Now, can you briefly tell us a bit about her life, and education and experiences?

Mirzakhani was born in Tehran, Iran in 1977.  She attended Tehran Farzanegan School, an all-girls school targeting gifted students and using a similar teaching style as college courses, that is, providing an in-depth study for each given topic.  At the age of 17, she became the first Iranian female to win a gold medal while competing in the International Mathematical Olympiad.  The following year, she broke another record by becoming the first Iranian student to attain a perfect score in the competition and to win two additional gold medals.

In 1999, Mirzakhani earned her Bachelor of Science, BSc, degree in mathematics from the Sharif University of Technology, located in Tehran, Iran.  Next, she came to the United States where she earned her Doctor of Philosophy, PhD, from Harvard University in 2004.  Mirzakhani’s thesis solved two problems that have been in question for many years, but it was how she connected those solutions in her thesis that was remarkable.  Mirzakhani devised a new proof for the formula that Edward Witten and Maxim Kontsevich discovered.  This proof deals with the number of times repetitive classes intersect on moduli space along with an asymptotic formula for the development of the number of simple closed geodesics on a compact hyperbolic surface.

In 2004, Mirzakhani’s degrees presented her with an opportunity to become a Research Fellow at the Clay Mathematics Institute in New Hampshire.  Mathematics is the main discipline of study at this private, non-profit foundation.  She was also an assistant professor at Princeton University.  Following her employment at Princeton, Mirzakhani became a full professor at Stanford University in 2008.

3) I understand that she won the FIELDS medal- what exactly is this award, and why is it exemplary?

The FIELDS medal is regarded as one of the highest privileges a mathematician can receive.  This prize is awarded only once every four years, and there are two to four recipients, under the age of 40.  Maryam Mirzakhani is the first female recipient of the FIELDS medal.  The FIELDS medal gives acknowledgement and appreciation to younger mathematical researchers who have made significant contributions to the field.

In addition to the medal itself, FIELDS medal recipients are awarded a cash prize of $15,000 Canadian dollars, which is, as of this writing, slightly less than $12,000 U.S. dollars.

4) Could you tell us a bit about her work?  Perhaps explain this idea of platonic solids.

Mirzakhani’s work was primarily focused on theoretical mathematics; more specifically, she was interested in studying the dynamics and geometry of complex surfaces.  It was, in fact, her work on the dynamics and geometry of Riemann surfaces and their moduli spaces that earned her the FIELDS medal.  The commemoration presented with her award stated that she made impressive advances in the theory of Riemann surfaces and their moduli spaces, and led the way to new limits in the area.

Mirzakhani’s thesis work focused on closed curves whose length cannot be shortened by distorting them.  Mirzakhani looked at what happens to the “prime number theorem for geodesics” when one considers only these types of closed geodesics on a hyperbolic surface that do not coincide.  Mirzakhani is shown drawing a geodesic form in Figure 1.

Figure 1

https://i2.wp.com/i.ytimg.com/vi/y03EPKn4z6I/hqdefault.jpg?ssl=1

https://i.ytimg.com/vi/y03EPKn4z6I/hqdefault.jpg

Platonic solids are three-dimensional shapes consisting of sides, which are regular polygons.  That is, the sides are of the same length and all angles are equal.  Initially, one may think that there are infinite possibilities for these specific geometric shapes; however, only five solids exist with these particular specifications.  The cube is a common platonic solid comprised of six squares.  See Figure 2.

Figure 2

https://i0.wp.com/www.sacredgeometryhealing.com/uploads/5/7/2/1/57219227/4006498_orig.jpg?resize=409%2C323

http://www.sacredgeometryhealing.com/uploads/5/7/2/1/57219227/4006498_orig.jpg

5)  What is a Koch snowflake and what is its relevance to mathematics?

Along with Dr. Marina Ratner, Mirzakhani was an expert when it came to studying rigidity.  They worked to describe shapes safeguarded by motions of space.  The Koch snowflake is an example of this mathematical model.  There is a repeating pattern of triangles along its edges.  The edge of this snowflake will look the same regardless of the scale at which it is viewed.  See Figures 3, 4, and 5.

Figure 3

http://slideplayer.com/slide/4774553/15/images/44/Koch+Curves+(2)+The+second-order+curve+K2+is+formed+by+building+a+bump+on+each+of+the+four+line+segments+of+K1..jpg

http://slideplayer.com/4774553/15/images/44/Koch+Curves+%282%29+The+second-order+curve+K2+is+formed+by+building+a+bump+on+each+of+the+four+line+segments+of+K1..jpg

 

Figure 4

https://i1.wp.com/www.artfulmaths.com/uploads/5/2/0/5/52054835/koch-snowflake-4.jpg?resize=413%2C413

http://www.artfulmaths.com/uploads/5/2/0/5/52054835/koch-snowflake-4.jpg

Figure 5

https://i0.wp.com/www.wikihow.com/images/0/0b/Draw-the-Koch-Snowflake-Step-7.jpg?resize=411%2C308

http://www.wikihow.com/images/0/0b/Draw-the-Koch-Snowflake-Step-7.jpg

6) The word “rigidity” seems to permeate her work. Why is this important?

Rigidity refers to reducing the apparently never-ending possibilities to a fixed number.  Something that is rigid cannot be distorted or twisted without destroying its fundamental nature.  Rigid objects are rare, and sometimes conjectural objects can be so rigid they do not actually exist.  The Koch snowflake is a rigid object because it retains its shape despite scaling.

Some mathematical objects remain unchanged under different types of movements.  For example, if one spins a spherical object, like a basketball, the shape is unchanged.  Mirzakhani’s work focused on shapes that are preserved under more complex types of motions, and in higher dimensional spaces.  Her work demonstrated that only the strictly regular and smooth surfaced shapes retain their shape in motion.  More specifically, her work focused on the effects of shearing, stretching, and compressing.

7) Apparently, there was an earlier mathematician whose work is relevant- Who was Theaetetus and how is this person relevant to Maryam’s work?

Theaetetus was a Greek mathematician from Athens who lived from 417-368 BC.  He studied irrational lengths.  For example, when considering a ratio (fraction), irrational numbers exist when the denominator divides the numerator of a fraction and the quotient yields a non-terminating decimal or a decimal with no existing repeated pattern.  Some common irrational numbers are the square root of 2 and pi.  Theaetetus’ study of irrational lengths is one section of Book X of Euclid’s Elements.

Theaetetus gave a mathematical description of all five solids and showed that there are specifically five regular convex polyhedra.  These are the platonic solids discussed earlier.  Theaetetus’ discovery is relevant to Mirzakhani’s work because both mathematicians studied rigidity.  Platonic objects and rigid objects are both uncommon.

8) What have I neglected to ask?

As a child, Mirzakhani had a passion for reading and making up stories.  She had all the makings of becoming a writer in the future.  However, Mirzakhani could not deny her desire to solve the puzzles that present themselves in the world of mathematics.  She likened solving mathematical problems to detective mysteries and several sources note that she would sometimes work on large sheets of paper and surround her proofs and problems with sketches and drawings that exhibited the development of her work.  Her daughter called these productions “paintings.”

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