We’re delighted for Professor Maryam Mirzakhani, the first female recipient of the prestigious Fields Medal in mathematics. In this long overdue landmark, Professor Mirzakhani has been commended for her work in complex geometry.

The mathematics community is hopeful that this will encourage more girls and young women to pursue careers in the field.

The following articles by Maryam Mirzakhani, published in International Mathematics Research Notices, are free for a limited time:

Image: Maryam Mirzakhani by International Mathematical Union (IMU). Public domain via Wikimedia Commons.

Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies. […] Why is the difference between ensemble and time averages of such importance? Well, basically, because when you assume the processes to be ergodic, ensemble and time averages are identical. Let me give an example even simpler than the one Peters gives:

Assume we have a market with an asset priced at 100€. Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be 100€ – because we here envision two parallel universes (markets) where the asset price falls in one universe (market) with 50% to 50 €, and in another universe (market) it goes up with 50% to 150€, giving an average of 100€ ((150+50)/2). The time average for this asset would be 75€ – because we here envision one universe (market) where the asset price first rises by 50% to 150€, and then falls by 50% to 75€ (0.5*150).

From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen. Assuming ergodicity there would have been no difference at all.

Maryam Mirzakhani (Persian: مریم میرزاخانی ‎; born May 1977) is an Iranian mathematician, Professor of Mathematics (since September 1, 2008) at Stanford University. Her research interests include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. On August 12, 2014, Mirzakhani became the first woman as well as the first Iranian to have ever been awarded the Fields Medal.

Maryam Mirzakhani has made striking and highly original contributions to geometry and dynamical systems. Her work on Riemann surfaces and their moduli spaces bridges several mathematical disciplines|hyperbolic geometry, complex analysis, topology, and dynamics and influences them all in return. She gained widespread recognition for her early results in hyperbolic geometry, and her most recent work constitutes a major advance in dynamical systems.

4

CSA: The Ergodicity Exhibition

Developed from their Evolo skyscraper competition entries, Ergodicity, an exhibition hosted by Canterbury School of Architecture, presented thesis work from eleven Graduate Diploma students.

With over 70 percent of the worlds growing population soon to live within major cities, the exhibition reconsiders the effect of increasing densities. Projects developed their research and design to accommodate for a variety of topics affecting our urban areas today, including: population increase, the rising demand for resources, pollution, waste management, and the digital revolution.

The projects which were shown covered a wide range of locations and programmatic responses, but as a collective all questioned ‘what role can the Skyscraper play in improving our urban areas?’

Responses included approaches such as Tiny Tokyo by Carma Masson, a mixed-use community micro scraper based in the business district of central Tokyo. Tiny Tokyo re-evaluates the approach towards designing skyscrapers, using them as a tool for reviving local heritage and culture, whilst introducing relevance for the people they are designed for, rather than designing them as a corporate tool. 

The future of our history is a concept which has been explored within Luke Hill’s project titled Dis.Assemble. This project involves a complex network composed of 6 miles of disused rail systems buried deep beneath London’s streets which provides a subterranean industrial waste facility: its sole intention to ‘Dis.Assemble’ materials produced by the metropolis above.

Unused space has also been explored within Jake Mullery’s SYMCITY thesis, describing an architectural construct that occupies the ‘dead’ space between existing skyscrapers. 

A comedic thesis by Paul Sohi told the story of one man’s life growing and living in a world of 10 billion people, where 90% of society lives in urbanised cities. The comic explores what such a world would be like.

The launch night was attended by many and with special guest Peter Wynne Rees, chief planner for the City of London, the exhibition was an opportunity to showcase the work of students at the Canterbury School of Architecture ahead of the end of year summer show which starts on the 31st of May.

-Text+photography by Taylor Grindley

The Encyclopedia of Mathematics (2002) defines ergodic theory as the “metric theory of dynamical systems. The branch of the theory of dynamical systems that studies systems with an invariant measure and related problems.” This modern definition implicitly identifies the birth of ergodic theory with proofs of the mean ergodic theorem by von Neumann (1932) and the pointwise ergodic theorem by Birkhoff (1931). These early proofs have had significant impact in a wide range of modern subjects. For example, the notions of invariant measure and metric transitivity used in the proofs are fundamental to the measure theoretic foundation of modern probability theory (Doob 1953; Mackey 1974). Building on a seminal contribution to probability theory (Kolmogorov 1933), in the years immediately following it was recognized that the ergodic theorems generalize the strong law of large numbers. Similarly, the equality of ensemble and time averages – the essence of the mean ergodic theorem – is necessary to the concept of a strictly stationary stochastic process. Ergodic theory is the basis for the modern study of random dynamical systems, e.g., Arnold (1988). In mathematics, ergodic theory connects measure theory with the theory of transformation groups. This connection is important in motivating the generalization of harmonic analysis from the real line to locally compact groups.
Is ergodicity at the root of all macroeconomic opinions?

Schools of macroeconomic thought differ widely in their policy preferences to achieve social optima. A broad chiasm exists between Keynesians and neoclassical economists with respect to monetary policy and fiscal policy preferences. While the following description is a summary, it will suffice to illustrate how different views on ergodicity explain the differences in these schools of thoughts.

Keynesians and allies believe that there are economic conjectures whereby monetary intervention can generate real growth (situations where the output gap is significant and inflation is below target for example). Neoclassical economists and their monetary allies believe that the gravity of market forces is so powerful that monetary surprises cannot yield real economic benefits.

On the monetary debate, neoclassical economists & monetarists believe that economies are ergodic as market forces ensure price adjustments that maintain the economy at potential at most times and thus any gains due to a monetary surprise today will be balanced by a price change that will annihilate those nominal gain. Keynesians and allies believe that a short-term gain will forever alter the development path of an economy, hence initial conditions matter. Depending on each perspective the economy either has a long run steady state or a path that can be altered at each short-term junction. While neoclassical economics believes in the ergodicity of economic systems, Keynesians and associates believe in path dependence.

With respect to the fiscal debate, neoclassical economists believe that changes in government expenditures cannot efficiently modulate economic activity and change potential output because agents’ behaviour is altered by the expectations of a balancing fiscal change in the future. Since the government must over time keep a reasonable balance, a tax cut that leads to a deficit heralds higher future taxes and leads agents to save the tax cut (Ricardian equivalence). Keynesians on the other hand feel that short-term stimuli may create a boost in the economy’s growth path whose value exceeds the amount of the stimulus. 

Who should we believe? Both schools of thought have a point. Unlike natural systems ergodicity does not apply always and everywhere with the same power. The challenge of wise economic management lies in the ability to recognize with a certain degree of certainty when a change in expected policy can yield positive results from those instances where a change in policy simply changes the timeframe of economic consequences.

The anti-black swan: oversignifying unlikely events and large deviations is as dangerous as undersignifying?

http://www.youtube.com/watch?v=f1vXAHGIpfc Time for a Change: Introducing irreversible time in economics Ole Peters

An exploration of the remarkable consequences of using Boltzmann’s 1870s probability theory and cutting-edge 20th Century mathematics in economic settings. An understanding of risk, market stability and economic inequality emerges.

The lecture presents two problems from economics: the leverage problem “by how much should an investment be leveraged”, and the St Petersburg paradox. Neither can be solved with the concepts of randomness prevalent in economics today. However, owing to 20th-century developments in mathematics these problems have complete formal solutions that agree with our intuition. The theme of risk will feature prominently, presented as a consequence of irreversible time.

Our conceptual understanding of randomness underwent a silent revolution in the late 19th century. Prior to this, formal treatments of randomness consisted of counting favourable instances in a suitable set of possibilities. But the development of statistical mechanics, beginning in the 1850s, forced a refinement of our concepts. Crucially, it was recognised that whether possibilities exist is often irrelevant — only what really materialises matters. This finds expression in a different role of time: different states of the universe can really be sampled over time, and not just as a set of hypothetical possibilities. We are then faced with the ergodicity problem: is an average taken over time in a single system identical to an average over a suitable set of hypothetical possibilities? For systems in equilibrium the answer is generally yes, for non-equilibrium systems no. Economic systems are usually not well described as equilibrium systems, and the novel techniques are appropriate. However, having used probabilistic descriptions since the 1650s economics retains its original concepts of randomness to the present day.

The solution of the leverage problem is well known to professional gamblers, under the name of the Kelly criterion, famously used by Ed Thorp to solve blackjack. The solution can be phrased in many different ways, in gambling typically in the language of information theory. Peters pointed out that this is an application of the ergodicity problem and has to do with our notion of time. This conceptual insight changes the appearance of Kelly’s work, Thorp’s work and that of many others. Their work - fiercely rejected by leading economists in the 1960s and 1970s - is not an oddity of a specific case of an unsolvable problem solved. Instead, it is a reflection of a deeply meaningful conceptual shift that allows the solution of a host of other problems.

The transcript and downloadable versions of the lecture are available from the Gresham College website:

Watch on currentsinbiology.tumblr.com

Biochemists uphold law of physics (EurkAlert)

Experiments by biochemists at the University of California, Davis show for the first time that a law of physics, the ergodic theorem, can be demonstrated by a collection of individual protein molecules — specifically, a protein that unwinds DNA. The work will be published online by the journal Nature on July 14.

The ergodic theorem, proposed by mathematician George Birkhoff in 1931, holds that if you follow an individual particle over an infinite amount of time, it will go through all the states that are seen in an infinite population at an instant in time. It’s a fundamental assumption in statistical mechanics — but difficult to prove in an experiment.

Using technology invented at UC Davis for watching single enzymes at work, Bian Liu, a graduate student in the Biophysics Graduate Group and professor Steve Kowalczykowski, Department of Microbiology and Molecular Genetics and UC Davis Cancer Center, found that when they paused and restarted a single molecule of the DNA-unwinding enzyme RecBCD, it could restart at any speed achieved by the whole population of enzymes.

Caption: RecBCD enzymes are unwinding DNA at different speeds. The bright ball at left is a bead, the bright strand is a stretch of DNA that shortens as it is unwound by the enzyme. The enzymes show ergodic behavior, supporting an important theory in statistical physics.

Credit: Bian Liu, UC Davis

Maryam Mirzakhani: ‘The More I Spent Time on Maths, The More Excited I Got:

Maryam Mirzakhani has become the first woman to win the Fields Medal, the most prestigious prize in mathematics. Mirzakhani, 37, is of Iranian descent and completed her PhD at Harvard in 2004. Her thesis showed how to compute the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Her research interests include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. She is currently professor of mathematics at Stanford University, and predominantly works on geometric structures on surfaces and their deformations.

What are some of your earliest memories of mathematics?

As a kid, I dreamt of becoming a writer. My most exciting pastime was reading novels; in fact, I would read anything I could find. I never thought I would pursue mathematics until my last year in high school. I grew up in a family with three siblings. My parents were always very supportive and encouraging. It was important for them that we have meaningful and satisfying professions, but they didn’t care as much about success and achievement.

In many ways, it was a great environment for me, though these were hard times during the Iran-Iraq war. My older brother was the person who got me interested in science in general. He used to tell me what he learned in school. My first memory of mathematics is probably the time that he told me about the problem of adding numbers from 1 to 100. I think he had read in a popular science journal how Gauss solved this problem. The solution was quite fascinating for me. That was the first time I enjoyed a beautiful solution, though I couldn’t find it myself.

What experiences and people were especially influential on your mathematical education?

I was very lucky in many ways. The war ended when I finished elementary school; I couldn’t have had the great opportunities that I had if I had been born 10 years earlier. I went to a great high school in Tehran – Farzanegan – and had very good teachers. I met my friend Roya Beheshti during the first week of middle school. It is invaluable to have a friend who shares your interests, and it helps you stay motivated.

Our school was close to a street full of bookstores in Tehran. I remember how walking along this crowded street, and going to the bookstores, was so exciting for us. We couldn’t skim through the books like people usually do here in a bookstore, so we would end up buying a lot of random books. Also, our school principal was a strong-willed woman who was willing to go a long way to provide us with the same opportunities as the boys’ school.

Later, I got involved in Math Olympiads that made me think about harder problems. As a teenager, I enjoyed the challenge. But most importantly, I met many inspiring mathematicians and friends at Sharif University. The more I spent time on mathematics, the more excited I became.

Could you comment on the differences between mathematical education in Iran and in the US?

It is hard for me to comment on this question since my experience here in the US is limited to a few universities, and I know very little about the high school education here. However, I should say that the education system in Iran is not the way people might imagine here. As a graduate student at Harvard, I had to explain quite a few times that I was allowed to attend a university as a woman in Iran. While it is true that boys and girls go to separate schools up to high school, this does not prevent them from participating say in the Olympiads or the summer camps.

But there are many differences: In Iran you choose your major before going to college, and there is a national entrance exam for universities. Also, at least in my class in college, we were more focused on problem-solving than on taking advanced courses.

What attracted you to the particular problems you have studied?

When I entered Harvard, my background was mostly combinatorics and algebra. I had always enjoyed complex analysis, but I didn’t know much about it. In retrospect, I see that I was completely clueless. I needed to learn many subjects which most undergraduate students from good universities here know.

I started attending the informal seminar organized by Curt McMullen. Well, most of the time I couldn’t understand a word of what the speaker was saying. But I could appreciate some of the comments by Curt. I was fascinated by how he could make things simple and elegant. So I started regularly asking him questions, and thinking about problems that came out of these illuminating discussions.

His encouragement was invaluable. Working with Curt had a great influence on me, though now I wish I had learned more from him. By the time I graduated I had a long list of raw ideas that I wanted to explore.

Can you describe your research in accessible terms? Does it have applications within other areas?

Most problems I work on are related to geometric structures on surfaces and their deformations. In particular, I am interested in understanding hyperbolic surfaces. Sometimes properties of a fixed hyperbolic surface can be better understood by studying the moduli space that parameterises all hyperbolic structures on a given topological surface.

These moduli spaces have rich geometries themselves, and arise in natural and important ways in differential, hyperbolic, and algebraic geometry. There are also connections with theoretical physics, topology, and combinatorics. I find it fascinating that you can look at the same problem from different perspectives and approach it using different methods.

What do you find most rewarding or productive?

Of course, the most rewarding part is the “Aha” moment, the excitement of discovery and enjoyment of understanding something new – the feeling of being on top of a hill and having a clear view. But most of the time, doing mathematics for me is like being on a long hike with no trail and no end in sight.

I find discussing mathematics with colleagues of different backgrounds one of the most productive ways of making progress.

What advice would you give those who would like to know more about mathematics – what it is, what its role in society has been, and so on?

This is a difficult question. I don’t think that everyone should become a mathematician, but I do believe that many students don’t give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.

• This interview is republished with the kind permission of the Clay Mathematics Institute.

 

 

 

Watch on taiganaut.tumblr.com

In the 78-year history of the Fields medal, winners of the prestigious mathematics award often compared to the Nobel Prize had one thing in common: all of them were men.

That changed with the August 12 announcement that the four 2014 Fields medalists include Maryam Mirzakhani, a professor of mathematics at Stanford University who is an Iranian woman. (She is also the first Iranian to receive the Fields.)

Mirzakhani’s research tackles big questions about different types of mathematical surfaces using disciplines with names like ergodic theory, Teichmüller theory, and hyperbolic geometry. It is completely alien to a lay reader. But that shouldn’t undercut the importance of Mirzakhani’s award.

Congratulations!

Maryam Mirzakhani has become the first woman to win the Fields Medal, the most prestigious prize in mathematics. Mirzakhani, 37, is of Iranian descent and completed her PhD at Harvard in 2004. Her thesis showed how to compute the Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces. Her research interests include Teichmüller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. She is currently professor of mathematics at Stanford University, and predominantly works on geometric structures on surfaces and their deformations.

Thesis Points? I dont even know anymore.

Ergodic Literature vs the non-linearity of memory? 

Ergodic literature refers to texts that require a reader to make a different or greater than normal effort. This is usually because they are non-linear in some way, which theoreticians relate to the possibilities of hypertext. An ergodic text re-interprets the idea of ‘plot’, plays with layout or typography, requires the reader to find a ‘key’ to unlock the meanings of the text or introduces an unreliable narrator or digression. Examples in print include Miguel de Cervantes's Don Quixote.Lawrence Sterne's Tristram Shandy and Julio Cortázar's Rayuela, or Hopscotch, and in electronic literature Michael Joyce's Afternoon: A Story. The term itself was coined in this specific sense in 1997 by Espen Aarseth in Cybertext: Perspectives on Ergodic Literature.

Hypertext and meaning

Aarseth appropriated the word ‘ergodic’ from physics. It combines the Greek words ‘ergon’ and ‘hodos’, which mean ‘work’ and ‘path’ respectively, and Aarseth writes that, “In ergodic literature, nontrivial effort is required to allow the reader to traverse the text”. This is why cybertext, or hypertext fiction, is seen as the epitome of ergodic literature. The reader’s expectations are foiled, and there are many possibilities to branch out and explore different manifestations of form,plotcharacter, and movement. Each reading yields a different story, and the reader is ‘productive’, rather than passive. Ergodic literature thus challenges the very notion of authorship or authority, and is sometimes seen as liberating or empowering. Michael Joyce’s Afternoon and Twelve Blue illustrate these principle through the use of hyperlinks and multiple possibilities, some hidden and radical, in plot and character.

Theoreticians

The term was first coined by Norbert Weiner in his 1948, Cybernetics; or, Control and Communication in the Animal and the Machine, and suggested byVannevar Bush's office of the future with its hyperlinked microfiche. Ergodic literature, while often modernist has been described in postmodern studies, for example, by Gilles Deleuze and Félix Guattari in their writings about ‘rhizomes’, Jacques Derrida and his decentered text, and Roland Barthes's 'writerly text'.Sherry Turkle and Jean Baudrillard write about the simulation of ‘reality’, or realities.

Memory:

Frenchman Marcel Proust. 

Whence could it have come to me, this all-powerful joy? I sensed that it was connected with the taste of the tea and the cake, but that it infinitely transcended those savours, could, no, indeed, be of the same nature. Whence did it come? What did it mean? How could I seize and apprehend it?

I drink a second mouthful, in which I find nothing more than in the first, then a third, which gives me rather less than the second. It is time to stop; the potion is losing it magic. It is plain that the truth I am seeking lies not in the cup but in myself. The drink has called it into being, but does not know it, and can only repeat indefinitely, with a progressive diminution of strength, the same message which I cannot interpret, though I hope at least to be able to call it forth again and to find it there presently, intact and at my disposal, for my final enlightenment.

Notes: This is what I’m interested in… but I have no idea how that relates to my project. 

4

In the midst of a lot of crap going on, take a moment to feel super proud of this woman: Maryam Mirzakhani.

At the relatively young age of 37 she has just become the first woman ever to win the Fields Medal (basically the Nobel Prize of mathematics). She has been preceded by 52 male winners.

Mirzakhani is Iranian by birth, and in high school she became the first Iranian student ever to achieve a perfect score in the International Mathematics Olympiad. She studied at Harvard and teaches at Stanford. Her work is primarily in geometric structures, including Teichmuller theory, hyperbolic geometry, ergodic theory, and symplectic geometry. x

Text
Photo
Quote
Link
Chat
Audio
Video