MATH NEWS

Dennis DeTurck

Herman Gluck

Daniel Pomerleano

David Shea Vela-Vick

PHILADELPHIA — A team of mathematicians from the University of Pennsylvania has been named winners of the 2012 Chauvenet Prize, given by the Mathematical Association of America to the author or authors of an outstanding expository article on a mathematical topic. The prize was awarded at the Joint Mathematics Meeting.

Dennis DeTurck, Herman Gluck, Daniel Pomerleano and David Shea Vela-Vick authored the paper, "The Four Vertex Theorem and Its Converse," which was published in Notices of the American Mathematical Society in 2007.

DeTurck is dean of the College of Arts and Sciences, the Robert A. Fox Leadership Professor in the School of Arts and Sciences and a professor of mathematics. Gluck is also a professor of mathematics at Penn. Pomerleano received his bachelor's degree from Penn in 2007 and is completing his graduate studies at the University of California, Berkeley and Vela-Vick received his Ph.D. from Penn in 2009 and is a National Science Foundation postdoctoral fellow at Columbia University.

The award announcement praised the paper as a "carefully crafted survey [with] enough mathematical details to give the reader a sense of the proofs, but not so many to obscure the big picture."

"As far as we know, this is the first time anyone has won the Chauvenet for work they've done while they were students," DeTurck said; Pomerleano contributed to the paper as an undergraduate and Vela-Vick as a graduate student.

The team wrote the paper with a broad audience in mind, aiming to appeal to mathematicians with varied backgrounds. The subject was the four vertex theorem, which deals with the curvature of curves in the plane. It states that a simple closed curve in a plane, other than a circle, must have at least four "vertices," or points where the curvature has a local maximum or local minimum.

Maximum curvature, DeTurck said, can be imagined as the point on a road at which a driver has to turn the steering wheel most sharply to stay on track. Minimum curvature is where the smallest turn of the wheel is needed.

In its original form, the concept was proved by Indian mathematician Syamadas Mukhopadhyaya more than 100 years ago. Others with an interest in geometry have since built upon the theorem, and in 1998, when prominent Swedish mathematician Bjorn Dahlberg died, he left on his desk a manuscript proving that the converse of the theorem was also true.

To compose the paper, DeTurck, Gluck, Pomerleano and Vela-Vick worked to distill the proofs down to their essence.

"The goal was we wanted to say this in as clear a way as we possibly could," DeTurck said. "It's just a matter of presenting it to each other over and over again until you realize the crux of the idea."

All four authors have previously been honored for their work in mathematics.

DeTurck has won various awards including the SAS Ira Abrams Award, the Lindback Award and the Mathematical Association of America's Haimo Award for Distinguished Teaching. Gluck has been honored with a National Academy of Sciences-National Research Council Fellowship at Berkeley and the Institute for Advanced Study, an Alfred P. Sloan Research Fellowship at Harvard University, a NATO Senior Fellowship in Science at Zurich and Amsterdam, a Guggenheim Fellowship at Penn and the University of Bonn and with the Lindback Award and Dean's Award for mentoring undergraduates at Penn.

Pomerleano has won the Waldemar J. Trijitzinsky Memorial Award of the American Mathematical Society. Vela-Vick has received the Dean's Award for Distinguished Teaching by a Graduate Student at Penn and an NSF Postdoctoral Fellowship at Columbia.

Penn Mathematicians Win 2012 Chauvenet Prize

Michael Aschbacher

PASADENA - Michael Aschbacher, the Shaler Arthur Hanisch Professor of Mathematics at Caltech, will share the 2012 Wolf Prize in mathematics, Caltech officials announced Wednesday.

The $100,000 award recognizes his role in classifying types of mathematical objects called finite simple groups. "His impact on the theory of finite groups is extraordinary in its breadth, depth and beauty," the award citation said.

"The classification of finite simple groups is one of the crowning achievements of modern mathematics," Hirosi Ooguri, the Fred Kavli Professor of Theoretical Physics and Mathematics at Caltech, said in a statement. "It's wonderful that Michael is recognized as the principal architect of this work."

Aschbacher will share the prize with Luis Caffarelli at the University of Texas, Austin, who was recognized for work on partial differential equations.

They will receive the award from Israeli President Shimon Peres at a ceremony on May 13 at the Knesset in Jerusalem.

"Receiving an award such as the Wolf Prize is of course personally very satisfying," Aschbacher said in a statement. "The finite simple groups are the building blocks of finite group theory, playing a role somewhat analogous to that of prime numbers in arithmetic. As a result, the classification theorem is not only a beautiful and natural result, but it's also very useful."

Since 1978, The Wolf Prize has been awarded annually in the fields of agriculture, chemistry, mathematics, medicine, physics, and the arts. Among this year's winners is opera tenor Placido Domingo.

Past winners have included notable names such as Stephen Hawking in physics, violinist Isaac Stern and architect Frank Gehry in the arts. Previous winners from Caltech include Harry Gray, Ahmed Zewail, and Rudy Marcus in chemistry; Alexander Varshavsky, and the late Seymour Benzer, Edward Lewis, and Roger Sperry in medicine, Caltech officials said.

Aschbacher has recently garnered several awards for his work on finite simple groups. He was awarded the 2012 Leroy P. Steele Prize for Mathematical Exposition, and in 2011 he won the Rolf Schock Prize from the Royal Swedish Academy of Sciences. He also received the Cole Prize in Algebra and is a member of the National Academy of Sciences and the American Academy of Arts and Sciences.

From staff reports

Caltech professor wins Wolf Prize for mathematics

Professor Makinde

A Nigerian, Professor Oluwole Daniel Makinde,presently a Senior Professor of Applied Mathematics and Director of Post Graduate Studies at Cape Peninsula University (CPTU) in South Africa has emerged winner of the African Union Kwame Nkrumah 2011 Scientific Award for Basic Sciences, Technology and Innovation.

The award was presented on Sunday, January 29 at the African Union Commission Headquarters during the formal opening ceremony of the 18th AU Summit, Addis Ababa, Ethiopia. Professor Maryke Tine Labuschagne also bagged the Life and Earth Science Award.

Makinde, who obtained his B.Sc and M.Sc degrees from the University of Ife (now Obafemi Awolowo University), and a Doctorate from the University of Bristol has contributed immensely to the upliftment of previously disadvantaged groups in Africa especially in the area of mathematical science research and training.

The African Union launched the AU Scientific Award Programme on September 9, 2008. Renamed as the "African Union Kwame Nkrumah Scientific Awards" in July 2010, the award "is one of the holistic and deliberate measures taken by the Commission to maintain science and technology on top of Africa’s development, co-operation and political agenda. The objective of the programme is to give out scientific awards to top African scientists for their scientific achievements and valuable discoveries and findings."

President Goodluck Jonathan in a congratulatory letter to Makinde described his achievements as "a source of pride to all Nigerians, particularly worthy of emulation by the younger generation", adding that the winner is "a very gifted scholar, an achiever, and a man of impressive credentials."

Makinde a past winner of the Young African Mathematician award in 2003 was also appointed Secretary of the African Mathematics Union in 2009.

During his visit to the Nigeria in 2010, he was at the Covenant University.

Prior to joining CPUT, Prof Makinde headed the Applied Mathematics Department for more than ten years and became a Full Professor at the University of Limpopo in South Africa. Makinde in an interview published in CPUT Newsletter described mathematics as the "precursor of science and technology and the indispensable single element in modern societal development."

According to him, mathematics education is therefore indispensable in nation-building.

Before joining CPUT, Prof Makinde headed the Applied Mathematics Department for more than ten years and became a Full Professor at the University of Limpopo in South Africa.

His key focus area for research is in Computational and Mathematical Modelling of Engineering and Biological Systems.

Makinde: Nigerian Mathematician in Diaspora wins AU Science award

Mathematicians have launched an ambitious campaign to raise the profile of their discipline, by opening a museum of maths in the UK. Geoff Wain, who is leading the initiative, points out that every other subject has a variety of cultural and educational hubs for people to visit, so why not maths? “Where would you go to find out about mathematics?” he says. “There’s absolutely nowhere in this country, it’s very sad.”

Last week, Wain and colleagues, as well as other interested fans of mathematics, gathered at King's College London to discuss their ideas for the museum, which is currently known as MathsWorldUK. So far, the plan is to have a number of zones covering topics such as numbers, shape and space, chance and infinity. The museum will also highlight the lives of historical mathematicians alongside those who use maths in their work today.

The museum will have a strong focus on interactivity, providing something for people of all ages to play and experiment with. “Mathematics as a theoretical thing with no concrete side to it is what can kill it off, I think,” says Wain. “Having things you can actually do is really important.” The museum gift shop could also sell shrunk-down versions of the exhibits, allowing people to take puzzles home with them.

Wain and colleagues are now looking to raise money to start the museum, with plans to approach a variety of companies and individuals. He says they are aiming high for funds of Ł50 million, with Ł10 million as the minimum needed to get the museum off the ground. Much of that money will go towards acquiring a building - so why not save on costs by integrating with the existing Science Museum in London?

“If you say maths is a part of science, the next thing is it never gets mentioned,” says Wain. He and others at the event last week also raised the point that maths is about more than just scientific number-crunching, as it also has cultural and entertainment value. “In a way it’s a game you play with logic, and amazingly it has these fantastic applications to almost every bit of the real world.”

Successful maths museums have already been established in other countries, such as the Mathematikum in Giessen, Germany, which first opened in 2002 and now attracts 150,000 visitors a year. This year will also see the opening of The Museum of Mathematics in New York, which began planning in 2008 and received funding from the likes of Google and some hedge funds.

Wain doesn’t know how long it might take to establish a similar museum in the UK, but he is enthusiastic about the demand from the public for more maths, having been involved in a mobile maths exhibit called the Pop Maths Roadshow during the 1990s. “It attracted a quarter of million people and enormous numbers of people wrote afterwards and asked 'when is it happening again?',” he says. Perhaps they will have an answer soon.

Mathematics, looking for a good home

George Szpiro (photo by Noga Szpiro)

Apologies to anyone who clicked on this story expecting to read about Christie Turlington or the latest Heidi Klum-Seal split news. But this is about Wall Street’s sexiest models – we’re talking about math.

The last few years have given us plenty of reasons to hate financial models. Models that promised to increase efficiency and manage risk became substitutes for common sense and justifications for greed. The real estate bubble was of course justified by them.

Yet people at hedge funds and trading firms, using models to mint money, remain passionate believers. Another supporter is George Szpiro, a mathematician turned writer who recently released a book called Pricing The Future, about the history of the Black-Scholes equation, the most famous model in finance and the one that launched this quantitative revolution (plus the Chicago Board Options Exchange). Szpiro, interviewed from his home in Jerusalem, explains why he still trusts models but why we should keep a close eye on the people who use them:

FORBES: How did you get interested in the Black-Scholes equation, and what is it meant to do?

SZPIRO: I was (and am) fascinated by the intellectual achievement. I liken it to Isaac Newton’s discovery of the Laws of Motion.

The Black Scholes-equation was meant to elicit the correct value of an option. Before Black, Scholes and Merton, values were based on gut feeling. Black Scholes ended the guessing game. Of course, in any model in physics and economics, there are assumptions. If the assumptions do not hold, the result is incorrect. For example, Newtonian physics ignores relativistic effects. Many other models assume that friction is zero, for example. So the model is only a starting point. Once one has the model, one can search further.

FORBES: Typically people who know the model think of Black, Scholes and Merton, the economists who wrote it. Who are some of the other people you discovered are important to its history?

SZPIRO: The first person to really use mathematical formulation to describe options was Louis Bachelier, who did his Ph.D at the Sorbonne in Paris, in 1900. His work was lost and only re-discovered by Paul Samuelson in the 1960s. Other people who are important to the story are Robert Brown (of Brownian motion fame), the 19th century French accountants Jules Regnault and Henri Levefre, and also Albert Einstein, the MIT-mathematician Norbert Wiener, the Russion probabilist Andrei Kolmogorov, the French mathematician turned soldier Wolfgang Döblin (who committed suicide rather than fall into the hands of the Nazis), the Japanese mathematician Kiyoshi Ito, and a host of physicists, chemists and mathematicians.

FORBES: What was the greatest achievement of the equation?

SZPIRO: It made it possible to efficiently trade options, thus allowing people to buy and sell risk as if it were a commodity.

FORBES: How has Black-Scholes failed, or gone awry?

SZPIRO: I don’t think Black-Scholes failed as such. The equation is correct. The problem is that it has many parameters that need to be estimated. And when the estimates that are used are incorrect, the result is garbage (garbage in, garbage out). For example, one of the inputs is a stock’s volatility; it is very hard to estimate that. Also, the equation is based on the Gaussian “bell” curve. But stock prices do not behave strictly as the bell curve says they should. Catastrophic events (bankruptcies, tsunamis, bubbles) are more common than the Gaussian bell curve would have us believe.

FORBES: That sounds like a major failing to me. If you have a model that assumes data is correct, and it’s not, that’s a recipe for disaster. To me that’s an argument to rely less on math, more on common sense. Wouldn’t you agree?

SZPIRO: The model did not fail. It correctly describes the workings of the market. But you are right: if the data one uses is incorrect, the result is rubbish. It is a failing of the people who made inappropriate use of the model, who used incorrect data, not of the model itself. Regarding common sense: a resounding YES.

FORBES: Have the credit crisis, and risk models behaving badly, caused you to see Black-Scholes differently?

SZPIRO: No. The equation is totally correct. See above about garbage inputs. Also, many people use a calculator to compute the value of a stock option without knowing what they are doing. And then there is greed: brokers and banks are interested in their commissions, and do not always take the real risks into account.

One must not rely blindly on any model but make use of it wisely. Models should serve as a reference point. I also remind you of my previous answer concerning the inputs: garbage in, garbage out.

FORBES: What do you think of the fact a model meant to contain risk has created more of it? Was this all just a futile exercise?

SZPIRO: Again, the model as such has not increased risk. People who abuse the model created additional risk. This would be like faulting Isaac Newton’s Laws of Motion for car accidents. It is the drivers who are at fault, not Newton’s equations.

FORBES: We have rules of the road for drivers. Do we need similar rules for mathematical models?

SZPIRO: We need rules to steer market participants in the right direction and keep them on the right track. And villains must be kept off the streets.

FORBES: How?

SZPIRO: Strict enforcement of existing laws, creation of new laws if necessary, elimination of incentives that ignore long-term risks. Maybe institute a “malus” system to punish reckless managers, or at least keep their bonuses in escrow for a number of years to make sure that there were no hidden risks.

FORBES: What gives you this confidence in math?

SZPIRO: One must not use mathematical models blindly. One should make use of models to get an idea of the workings of the market and then use common sense when doing actual trading.

FORBES: Do you trade?

SZPIRO: No, I do not trade myself. I am interested in the history of ideas.

FORBES: Is Black-Scholes the sexiest model on Wall Street?

SZPIRO: As models go, Black-Scholes is very appealing. But I think the Bell Curve is also quite shapely… especially if it does not have fat tails.

This article originally said John Brown is he of Brownian motion fame. It has been corrected to say Robert Brown. Sorry about that, Robert.

Wall Street's Sexiest Model

The Rutgers Ist Annual Quant Summit at NASDAQ on Times Square last night to celebrate the role of quants in today’s financial markets showed that too much confidence in arithmetic models, even when invented by Nobel Prize mathematicians, are not foolproof solutions for avoiding risk.

After all the fascinating war stories were related by our fine panel, Bob Litterman, Leslie Rahl, Eli Ayache and Bruno Dupire , there was this almost embarassed admission that quant solutions to protect against risk and the dangers of volatility may not necessarily protect against the exaggerated level of confidence that existed in Wall Street in 2007, preior to the 2008 meltdown.

As Bob Litterman, former Goldman Sachs Risk Officer and chairman of the Kepos hedge fund put it, “We are not pricing risk appropriately– which led to the financial collapse. And if we don’t change our ways soon, it could lead to a much larger catastrophe in the future.”

Risk in Litterman’s mind, has many elements; it’s more than the amount of leverage used; It’s liquidity in the marketplace; its the quality of the counter-party on the other side of the trade, its the business involved. Just saying you are using “Value At Risk or VAR, an often used model, is not a sure protection against any of these risk factors going against you. That’s because the VAR dollar figure presented in the financial statements of Wall Street firms, does not present an adequate pricing of the dimensions of risk. No one can measure all of the many different dimensions of risk” and place an accurate projected figure on it.

Or as Elie Ayache, CEO of ITO 33 and author of “The Blank Swan, The End Of Probability” put it to the Rutgers MBA students; “There is no safe way to measure volatility.” Before the October 19, 1987 stock market meltdown of 23% on the Dow Industrials took , there had never been volatility of more than 40 with respect to stock prices. Most attempts to hedge risk took that VIX rate of 40 into consideration if they had time. But, as it turned out, the rate of volatility that day rose to 80– double the previous peak– throwing all strategies used for protection out the window. I personally knew the largest options trader on the Amex, who was wiped out by the volatility.

Leslie Rahl, managing partner of Capital Risk Advisers, who ran Citibank’s derivatives operation, underscored that mathematical models are only an approximate version of reality. Rahl reminded the audience of 250 Rutgers MBA students and faculty that before 2007 the worst decline in housing prices in the US had been 20%. So, whatever structured securities were used to hedge that anticipated decline of no more than 20%– could not be protected insurance if some securities were to decline 60% to 80% in value.

That bloodbath has resulted in new regulations on Wall Street, less borrowed money being used, and a substantial reduction in the use of structured finance notes that emply sophisticated derivative formulas.

Despite these very real problems, as the moderator of the panel, I spelled out how widespread quantitative methods were being used in corporate finance, in passive investing techniques like indexed ETFs, in essential risk control, and in the high frequency trading that has taken over 53% all stock executions, and now spreading to the buying and selling of bonds, commodities and currencies– as well as to vibrant geographic areas like Southeast Asia.

It was a 2 hour seminar for MBAs who are working towards a degree in Quantitative Finance. There were many impressive young Chinese who are ambitious to learn the ways of Wall Street, and seek their fortunes in the brave new world of quantitative finance– even with all its attendant dangers.

The Quants May Not Be Able To Prevent The Next Meltdown

To carry out the study, unemployment figures were provided by Spain's National Employment Institute.

Unemployment time series in Spain behave in a chaotic way according to a study at the University of Seville. Such chaos demonstrates the complex and unpredictable nature of the Spanish labour market in the long run. However, short term patterns can be predicted using complex mathematical models.

Olmedo explains that when a system is chaotic, its behaviour is "highly complex and unpredictable in the long run". This is the case because any small change is magnified by the system itself. She adds however that "in the short term, its behaviour can be predicted but non-linear models that capture the complexity of behaviour must be used for this."

To carry out the study, Spain's National Employment Institute (INEM) provided the country's unemployment figures over a 36-year period from 1965 to 2001. Through the use of two algorithms, the so-called 'maximum Lyapunov exponent' was calculated. This parameter measures the instability of a certain system. Positive results indicate instability and chaotic behaviour.

The results confirm the nonlinearity and chaoticity of the Spanish labour market. This, in turn, is the first step in characterizing unemployment time series and explaining their reality. Scientists are now working on the second phase of the study. This involves the development of short term predictions with the relevant mathematic models. The Sevillian researchers are currently working with artificial neural networks.

Chaotic models and the 'butterfly effect'

In economics, linear models have been traditionally used to characterise and predict unemployment time series. But, they tend to produce rather simple behavioural trends which have to be randomly disturbed to achieve more realistic results. For this reason the team opted for nonlinear models and concentrated mainly on chaotic models.

These mathematic models are capable of showing very different behaviours over time when dealing with infinitesimally small changes in initial conditions. An example would be the 'butterfly effect' which suggests that the flutter of one of these insects' wings could trigger a tsunami on the other side of the world.

Olmedo concludes that "the use of chaotic models allows us to obtain behavioural trends as complex as their own reality. However, we need to continue in our investigations to find better tools that help us in characterization and prediction."

More information: Elena Olmedo. "Is there chaos in the Spanish labour market?". Chaos, Solitons & Fractals 44 (12): 1045-1053, December 2011.

Provided by FECYT - Spanish Foundation for Science and Technology

Mathematics confirm the chaos of the Spanish labor market

Dr. Mariel Vazquez, an associate professor
of mathematics at San Francisco State University

Title: Associate professor of mathematics, San Francisco State University

Education: Ph.D., Florida State University; B.Sc., National University of Mexico

Age: 40

Career Mentors: De Witt Sumners, Florida State University

“Math is so boring. Math is so hard.” Dr. Mariel Vazquez often hears these comments from elementary school students. It pains her because she always has loved mathematics.

Vazquez is an internationally known researcher in the emerging field of DNA topology. Vazquez, an associate professor of mathematics at San Francisco State University, studies how human DNA, the DNA of bacteria and the DNA of viruses become untangled. Her work could affect the design of antibiotics and anti-cancer drugs.

Earlier this year, Vazquez received a National Science Foundation Faculty Early Career Development, or CAREER, award for her research.

A native of Mexico and the daughter and granddaughter of engineers, Vazquez was drawn early on to math. “I loved math but didn’t think becoming a mathematician was a career option,” she says. In high school she became passionate about molecular biology and enjoyed working with DNA and learning about proteins and cells.

At the National University of Mexico, or UNAM, Vazquez pursued mathematics, earning a research fellowship to the Mathematics Institute. As she became more interested in pure mathematics, she started losing hope of finding a career connection to molecular biology. One day when she was a sophomore she saw a flier for a series of lectures about knot theory and the study of DNA. “A friend and I went. It was intimidating, and I felt overwhelmed by all that information,” she recalls. “I didn’t understand most of it, but that day I discovered what I wanted to do.”

She combined her interests of pure mathematics and molecular biology and focused on the study of the emerging field of DNA topology.

Vazquez worked with knot theorist Dr. Max Neumann on an undergraduate thesis titled “Applications of Knot Theory to the Study of DNA.” The thesis was based on the work of Dr. De Witt Sumners of Florida State University.

“While doing my thesis I went knocking on doors of mathematicians, trying to learn more,” she recalls. She met Sumners twice before graduating. His work on the applications of knot theory to DNA fascinated her and became her doctoral topic. Vazquez pursued her Ph.D. in mathematics at Florida State University, where Sumners was her adviser.

“She is creative, brilliant, tenacious, intuitive, artistic, personable and courageous,” notes Sumners. “When I saw [her work], I knew she was the real thing, and I was lucky to get her as a student. It is always great to have students that are smarter than you are.”

“The students she mentors always seem to pick up some of her own mathematical clarity and highly professional approach,” says Dr. Rainer Sachs, emeritus professor of math and physics at Berkeley.

Vazquez was an academic visitor in the biochemistry department at the University of Oxford in England in 2006 and 2007. She was a visiting scholar at the University of California, Berkeley in 2008. Vazquez was an academic visitor at the Cancer Research Center in Salamanca, Spain, and an academic visitor at the molecular biology department in Barcelona, Spain.

Vazquez, 40, began working on the National Science Foundation grant in 2009. Last April, after nine reviewers meticulously pored over the material, Vazquez learned that she would receive almost $600,000 for research. The selection committee included biology scholars, math scholars and interdisciplinary scientists.

Vazquez plans to travel to Oxford where she will meet with her biology collaborator and spend time in the lab refining the biological questions that can be answered with her methods. She will work with an international group of mathematicians, biologists and computer scientists toward a goal of understanding the mechanism of an enzyme essential for DNA replication in the bacterium Escherichia coli.

In addition to her work and family, Vazquez has enjoyed organizing a math circle for children between first and third grade as part of the San Francisco Math Circles. Through her CAREER award she also will collaborate with the California Academy of Sciences museum to present DNA topology research to children and to the public. She has created a curriculum that’s not stuffy, Vazquez says. She knew she was on the right track when she heard one child say, “This isn’t math. This is fun.”

Emerging Scholars: The People’s Mathematician — Mariel Vazquez