From EWM Newsletter n° 21 (2012/2)

Corinna Ulcigrai was born in Trieste, Italy, in 1980. She obtained her diploma in Mathematics from the Scuola Normale Superiore in Pisa (2002) and defended her PhD in mathematics at Princeton University (2007), under the supervision of Ya. G. Sinai. In the academic year 2007/08 she spent a semester at the MSRI Institute in Berkeley, California, and a semester at the Institute for Advanced Studies in Princeton. Since August 2007, she has been a Lecturer and a RCUK Fellow at the University of Bristol. She is married to a mathematician and recently gave birth to her first child.

EWM: Can you describe your research area and what in particular you find most interesting to study?

CU: My research focus on the study of chaotic properties of parabolic dynamical systems. Dynamical systems are models of the many chaotic phenomena in nature which, despite obeying known evolution equations, elude deterministic prediction. The main goal of the study of dynamical systems, and in particular of ergodic theory, is to describe and quantify the chaotic properties of the system, in order to predict its asymptotic average behavior.

One of the features of chaotic systems, which makes their evolution complicated, is that trajectories of nearby initial points diverge. According to the speed of this divergence, dynamical systems are often grouped in three main classes: the system is called hyperbolic if nearby orbits diverge exponentially, elliptic if there is no divergence (or perhaps it is slower than polynomial) and {parabolic if it displays an intermediate form of chaotic evolution, i.e. nearby orbits diverge polynomially in time. Most of my work so far concerns the ergodic theory of parabolic systems, such as area preserving flows on surfaces or polygonal billiards.

An aspect that I like of research in dynamical systems is that it often at the crossroad between different disciplines, such as analysis, geometry and number theory. One of my areas of expertise is Teichmueller dynamics, a field that has seen a surge of activity in the last ten years. It’s great to be part of the community working in this field, since it is currently extremely active and topical, with three Fields medalists working in the field (M.Kontsevich, C.McMullen and J-C.Yoccoz) and many international workshops. The beautiful philosophy that has guided the research in the field is that the erogodic properties of the billiard flow on a rational polygon or on a translation surface are reflected by the orbit of the translation surface seen as a point in an abstract space, the space of all translation surfaces, under special family of deformations that act as renormalization operators (given by the Teichmueller geodesic flow).

EWM: You were recently awarded a prize at the recent ECM in Krakow, can you tell us something about the work that won this prize? How do you think that this prize will influence your career?

CU: The prize was awarded for solving a long-standing question on chaotic properties of area-preserving flows.

Flows on surfaces have been one of the fundamental examples of dynamical systems since the work of Poincare’ at the dawn of the field. The work concerns area preserving flows known as locally Hamiltonian flows. These class of surface flows received a lot of interest by Novikov and his school in the 80s, for their occurrence in solid state physics and pseudo-periodic topology.

In a series of papers, starting from my PhD thesis and concluded by a publication in Annals of Mathematics, I solved a conjecture by V.I.Arnold and an open problem posed both by G.Forni and A.Katok on the ergodic theory of locally Hamiltonian flows. Arnold conjectured in the 90s the presence of mixing, a strong chaotic property, in minimal components of locally Hamiltonian flows. Mixing was proved in the case of the torus (genus one) by Khanin and Sinai, in ’92. In my PhD thesis (Princeton University, 2007), I proved Arnold’s conjecture for any genus greater than two: as long as the flow has saddle loops, its minimal components are typically mixing.

Whether there is mixing when the locally Hamiltonian flow is minimal (in this case there are only simple saddles) was unclear and was an open question asked both by Katok and Forni. I first showed that for any positive genus minimal locally Hamiltonian flows are weakly mixing. Eventually, in the paper which appeared in the Annals in 2012, I showed that they are typically not mixing, thus completing the description of the mixing properties in locally Hamiltonian flows in higher genus.

I feel very honored by the prize. A certain degree of lack of self esteem is a trait that is probably shared by many women in mathematics. Receiving such a recognition is a great confidence boost. So far, the short term effect has been an increased number of invitations to give talks as well as a number of interviews requests from the media (from newspapers, to the Italian radio, including this EWM newsletter!). I find this sudden sense of celebrity a little daunting….

EWM: You have studied and worked in Italy, USA and UK. What can you tell us about your living/working experience in these environments?

CU: Even though I had a very positive experience in the USA, both mathematically and in general, I must admit that I am happy to be back in Europe. I realized how strong is the European common identity, despite the apparent differences between European national cultures and lifestyles, only when I first went to the USA. It is also good to be closer to my family, in particular now that my parents became grandparents.

Italy is a great country where to live, but for Mathematics the situation is unfortunately not good at all, in particular in view of the constant funding cuts, but also because of the lack of meritocracy . It’s very sad that many of my mathematician friends from Scuola Normale who stayed in Italy struggled to find a position and quite a few left Mathematics.

The UK are one of the European countries who has invested more in research. In our Department in Bristol almost half of academics come from outside the UK (“from oversea”, I should say…) and the average age is below forty. Both things are completely unheard of in Italian Universities.

One negative aspect of funding for research in the UK, though, is that it comes into cycles, as the UK economy, with its “boom and bust” cycles…. this is detrimental for academic excellence: compared to the USA, where hiring is regular process, an entire strong generation of mathematicians may be lost only because they graduated in the wrong year…

EWM: Has becoming a new mother had more or less of an effect on your work than you anticipated?

CU: The answer is surely more. As any mother certainly knows, before having a child I could not anticipate neither the greatness of the joy that having a healthy, big happy boy would bring, not the amount of time and energy that caring for him would take. Victor is now 6 months old and I am still on maternity leave (one of the good things of Europe!). In the past few months I had to struggle very hard to get done just few small academic jobs during Victor’s naps (few and short unfortunately), from marking my dynamical systems students exams to giving a couple of talks, including the one in Krakow, or writing the paper for the proceedings of the 6ECM. I’ve also enjoyed discussing Maths with collaborators, but I found it hard to concentrate especially if hearing Victor crying in another room… Now he started going to nursery part time and I will be back full time with January 2013, so we will see how it will be…

Finding a good balance between Mathematics and motherhood is my present challenge. I also think that finding a satisfactory way of conciliating a family with a career in academia is one of the reasons why there are few women in Mathematics. I keep asking to all women mathematicians I meet how they did it…. I had many email exchanges with other mathematicians mothers in my generation and found it really precious to hear about their experiences. I really think it is important to have role-models and I find inspiring to know so many great women mathematicians who have children and are great mothers too.

EWM: What are your hobbies and interests, when time permits?

CU: I like cooking: when spending time in the kitchen trying and creating new recipes, contrary to Mathematics, one can see the results of the creative effort after a short and finite amount of time! I also benefit a lot from practising yoga, especially since I am so often nervous and stressed out and I finding such an effective practice to relax. I also enjoy reading novels and going to the theaters, both to see operas and plays. Of course most of these hobbies are on hold right now, since Victor is now taking most of what used to be my free time. Now playing with him has become my best way of relaxing!