SwissMAP Perspectives Issue 4 | 2019

Special Edition: Equal Opportunities 04 20 30

Simplify your life by going large Universal Optimality of E8 and Awards and Grants Leech Lattices In this article, our member Susanne Reffert In this article, our member Maryna Viazovska We are pleased to announce the awards

explores how her professional path has lead discusses the E8 lattice and the Leech and grants received by our participants and her research to the subject of large-charge lattice who are among the most famous students. expansion and the exciting possibilities of mathematical objects. this relatively new direction.

Contents 08 22 31 Conversation with Prof. Anna Interview with Eva Miranda New Collaborators Beliakova Anna Beliakova is a full professor of In this short interview, Eva Miranda (UPC) We continue to expand and grow thanks to mathematics at the University of Zurich. tells us about the importance of organizing a new collaborators within SwissMAP. In this interview we discuss both the past, women workshop and what she hopes it will Welcome to Alessio Figalli and Corinna present and future of her scientific career. achieve. Ulcigrai! 12 26 32 Mentoring Program for women SwissMAP member stories: The Puzzle Corner PhD and Postdocs Olga Chekeres As part of the ongoing effort of SwissMAP SwissMAP-funded PhD student, Olga Test your math and logic skills with these to address the gender gap in our field, the Chekeres, completed her thesis in April 2019. puzzles, kindly put together by some of our Equal Opportunities Committee has recently Olga’s plans are to join the University of contributors. launched the SwissMAP Mentoring Program Connecticut, USA, for the next 3 years as an for Women PhD and Postdoc members. Assistant Research Professor. 14 28 Interview with Prof. Chenchang Upcoming Events Zhu Following our Women at the Intersection A brief overview of the events organised or of Mathematics and High Energy Physics co-organised by the NCCR SwissMAP. conference, we met with Chenchang Zhu to talk about Balancing Yin and Yang in the field of Mathematics.

2 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 3 it often involves physics I previously a handle on strongly knew only little about. This topic was coupled problems is no exception. It is centered on the important progress. study of special sectors of three- and four-dimensional conformal field the- The approach we follow ories (CFTs). The most surprising part is quite a time-honored was that the problem had nothing to one in theoretical phys- do with string theory or even super- ics, namely making use symmetry (even though it is possible of the symmetries of Simplify your life to consider superconformal field the problem and con- theories (SCFTs)). sidering special subsec- tors of the full theory by going Conformal field theories are, as the in which simplifications name suggests, invariant under con- occur. In our case, we formal transformations. This gives consider CFTs with a rise to special features. If we know global symmetry. Such the operator dimension and spin of a symmetry has by each local operator, conformal sym- Noether’s theorem an metry fixes the two-point functions associated conserved large up to normalization. If we further- charge, which can be more know the 3-point function used to slice the Hilbert coefficients, we can solve the theory space of states of the completely, in the sense that we can theory into sectors Susanne Reffert discussing large charge with Luis Alvarez-Gaume and Domenico Orlando at the Simons Center for Geometry and Physics. Photo by Jean-François Dars. write down all higher correlation labeled by their charge. functions. We concentrate on a suppressed by inverse powers of the al and the spacetime symmetries. subsector of fixed charge, where we charge. In the problems we have Part of this breaking is explicit and CFTs play an important role in take the charge to be very large. It studied so far, we were generally left due to fixing the charge, part of it is In theoretical physics, our best tool I never actively worked on this topic, theoretical physics, as they show turns out that it is possible to write a with only a very small number of con- spontaneous and due to the ground to calculate observable quantities is it lead me to become interested in up in a variety of contexts, such as low-energy effective theory in which tributions which are not suppressed. state itself. Spontaneous symme- perturbation theory. We know very the connections between integrable critical points characterizing second the inverse of the large charge acts as In other words, working at large try breaking gives rise to massless little about strongly coupled systems. systems and supersymmetric gauge order phase transitions, fixed points a controlling parameter, bringing us charge allows us to make the step Goldstone degrees of freedom. They Any new tool to access the strongly theories. Realizing such deformed in renormalization group flows, back to a perturbative regime. from an effective action with infinite- represent the quantum fluctuations coupled regime is therefore most gauge theories appearing in these and even quantum gravity via the Wilson’s notion of the effective ly many terms to an expansion which around the classical ground state and welcome. We found that working correspondence within string theo- AdS/CFT correspondence. CFTs are action in which any term compatible captures the low-energy physics in a encode the low-energy physics in the in sectors of large charge is way of ry came next. This topic, and brane scale-free, meaning they contain no with the symmetries of the problem handful of terms. effective action. If the global symme- compensating the effects of strong realizations of supersymmetric gauge characteristic length or energy scale. must appear is conceptually very try we started from was just a U(1), coupling. theories in general kept me busy for This means also that we have no di- then things are simple and we are left quite a while through my moves to mensionful small parameter in which with a single (relativistic) Goldstone I am a string theorist by training and Japan and then to CERN. to perform a perturbative expansion. boson in terms of which we write a in the course of my career I have I am a string theorist by training and in the course The dimensionless couplings in a CFT non-linear sigma model. The action gone through what resembles a To my own surprise, I ended up add- are generically of order one. While of my career I have gone through what resembles contains all the terms compatible random walk in the space of prob- ing a quite different line of research in two dimensions the special nature with conformal symmetry which are lems in formal theoretical physics. I to my collection. This happened in a random walk in the space of problems in formal of the conformal group allows us to not suppressed by the large charge. started out in string phenomenology spring 2015, just after having started use a host of analytic techniques, in theoretical physics. If we start from a larger, non-Abelian at a time when flux compactifications my new job at the University of Bern. higher-dimensional CFTs things are global symmetry group, we need to were a hot topic. I was most at- I was in Japan, visiting my long-time much more tricky. Of course, we have first determine the symmetry-break- tracted by their most formal aspect, collaborator Simeon Hellerman some methods at our disposal, such compelling. If we don’t have a way of We approach the problem semiclas- ing pattern. Since the ground state namely the algebraic geometry of together with my collaborator (and as large-N expansions, (small) epsilon truncating the effective theory, it is sically, solving the classical equations breaks Lorentz invariance, we are the compactification manifolds. husband) Domenico Orlando. For me, expansions and the conformal boot- however of very limited practical use. of motion at fixed charge and min- generally left with both relativistic Motivated primarily by my interest every new project tends to be a jour- strap. And it is possible to run Mon- Working at fixed charge allows us to imizing in order to find the low- and non-relativistic Goldstone bos- in the (much more formal) field of ney into the unknown, not just in the te-Carlo simulations on the lattice at do exactly that. In addition to using est-energy state at fixed charge. This ons, which are distinguished by their topological string theory, I went to obvious sense that we don’t know the strong coupling. But analytic results the constraints due to symmetry, we ground state has the special feature dispersion relations (linear versus Amsterdam to work with Robbert end result of a research project when are still few and far between and any also use the fixed-charge scaling to of being time-dependent. Working at quadratic in the momentum). Dijkgraaf for my first postdoc. While we start it, but also in the sense that new approach that can help us get discard those terms which are highly fixed charge breaks both the glob- Once we have written down the

4 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 5 effective action in the form of a large- lar that cases with a moduli space of vors, we could compare to a super- charge expansion, we can start cal- vacua behave very differently from symmetric localization calculation culating the conformal data, namely theories with one discrete vacuum I which has again shown an amazing 4 operator dimensions and three-point described above. agreement and even allowed us to coefficients, from which the gener- estimate the exponential corrections 2 al n-point functions can be deter- There is in general little known about to the large charge expansion (see mined. The energy of the ground the strongly-coupled models we are Figure 2). 0 state at fixed and large charge Q in studying at large charge, so we have 35 2 particular gives via the state-opera- few results to compare our predic- Working on CFTs at large charge - 20 tor correspondence of CFT directly tions to. But whenever there are has been an extremely interesting 4 10 the conformal dimension of the results to compare to, be it on the lat- and enriching experience for me, - lowest-lying state of charge Q. The tice or from supersymmetric localiza- both scientifically and personally. 6 5 biggest contribution to the ground- tion in the case of SCFTs, the confir- On the one hand, this is the closest - 2 8 state energy comes from the classical mation of our predictions has been to “real-world” physics I have ever 1 ground state, while the vacuum ener- strikingly strong. When comparing come, and I am really enjoying it. - Im 10 20 30 40 50 gy of the relativistic Goldstones gives with numerical results from lattice On the other hand, the problem has τ Figure 2 - Plot of the universal part of the 3-pt function coefficient of N=2SQCD with 4 flavors as a subleading contribution. calculations, we found that our for- appealed to quite a varied group function of the gauge coupling for different values of the R-charge. The dotted lines represent the mulae derived at large charge even of people, getting me into contact exact numerical results from the localization computation, the solid lines are our large-charge work down to very small values of the with subfields I had known relatively predictions, the numbers on the right side give the value of the fixed charge. charge, which is highly unexpected little about before. Working on large Source: unpublished, based on results from “Universal correlation functions in rank 1 SCFTs”, S.Hel- Working on CFTs at lerman, Sh.Maeda, D.Orlando, S. Reffert, M.Watanabe, arXiv:1804.01535 large charge has been (see Figure 1). The largest charge charge not only got me talking to used on the lattice was 12, which is new people but has even lead to very Since our original paper in 2015, When we first started working on an extremely interest- by no means a very large charge, but interesting new collaborations out- Simeon Hellerman and his (now large charge four years ago, at least the agreement remains excellent side my usual string theory commu- former) student Masataka Watanabe, I had no idea what I was getting into. ing and enriching ex- even down to charge one. This has nity, such as the one with two lattice Domenico and I have continued to It’s been quite a ride and I’ve learned perience for me, both taken us very much by surprise, as it theorists, who recently verified our push the large-charge expansion so much! The large-charge expan- is far from the regime in which our prediction for the O(4) vector model forward in varying configurations of sion has exceeded our boldest hopes scientifically and per- effective theory is valid. to high precision. new and old collaborators. But it’s when we started on the project. I sonally. In the case of N=2 SQCD with 4 fla- been especially great to see that the am excited both to be part of and to topic has attracted a number of inde- pendent groups around the globe. In One way of using the large-charge 12 late summer of this year, I am co-or- There is in general little known about the strongly- method is to apply it to known CFTs, ganizing (together with Domenico, such as for example the Wilson-Fisher Simeon and Luis Alvarez-Gaume) a coupled models we are studying at large charge, fixed point in the infrared of the O(N) 10 1-month workshop at the Simons so we have few results to compare our predictions vector model in three dimensions. Center for Geometry and Physics, Another stance one could take is to to. But whenever there are results to compare 8 where we are hoping to bring togeth- simply assume that a certain CFT er a varied set of people interested to, the confirmation of our predictions has been exists and apply the large-charge in systems at large quantum number method to it. We have worked with 6 and related topics in order to explore strikingly strong. models motivated from both con- D(j, j) new approaches, connections and densed matter physics and particle 4 applications. watch its development and hope that physics. The large-charge expan- it holds many more exciting surprises sion even works for non-relativistic I believe that we’ve only scratched for us. Who knows where it will lead systems with Schrödinger symmetry 2 the surface of the power of the me next on my path of theoretical (as opposed to conformal symmetry). large-charge expansion and that the physics research? An example is the unitary Fermi gas method can be widely applied and which can be experimentally realized 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 developed in several new directions. in the laboratory via cold atoms in a Since we are in the unique position of trap. Connecting back to more for- j having a good theoretical handle on mal theory topics, the large-charge Figure 1 - Plot of the conformal dimension D(j, j) as a function of the charge j for the lowest oper- strongly coupled systems, the large- expansion can be applied also to ator of given charge in the O(4) vector model in three dimensions. The squares represent the data charge approach might for example Author: Susanne Reffert superconformal field theories at large obtained using Monte-Carlo calculations on the lattice. The solid line is the large-charge prediction. Source: “Conformal dimensions in the large charge sectors at the O(4) Wilson-Fisher fixed point”, allow us to explicitly check conjec- Professor, UNIBE R-charge. Here, we found in particu- D.Banerjee, Sh.Chandrasekharan, D.Orlando, S.Reffert, arXiv:1902.09542 tured (strong/weak) dualities.

6 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 7 - Could you say some words about I think this depends a lot on the per- your career? What were the deci- A very important point, not son. There are people who can just A Conversation with sive points and/or difficult choices? only in my career but in my be sitting in a chair, thinking about whole life, was the fall of something and get a great idea. For The first very important point, per- the Berlin Wall. me that doesn’t work. I need to work haps not only in my career but in my hard in order to get ideas. And then Anna Beliakova whole life, was the fall of the Berlin job at UZH. selecting the right one from all these Wall. At that time, in 1989, I was visit- ideas is again hard work. Sometimes, ing a friend in eastern Germany, and - Could you tell us about your field? you write a paper without years of suddenly the wall that we all thought What are the exciting things hap- working just because you profit from would be forever, fell down. I went to pening there at the moment? the thoughts you had before. But I the other side and asked a professor would not say that ideas come for at the Free University Berlin for a PhD I am a quantum topologist. This is free. position. Actually, at that point my an interdisciplinary domain, where English was not good enough to fully we are using ideas from mathemat- As an anecdote, I heard about a understand his answer, but I under- ical physics to produce topological mathematician for whom the envi- stood that it was rather positive than invariants of braids, links or 3-dimen- ronment was very important. Once negative. That is how I obtained my sional manifolds. At the beginning he got a great idea standing on a PhD position in Berlin, and had this of the century these invariants were specific mountain in the Swiss Alps. great opportunity to continue my categorized. This means we could lift And apparently, you can meet him studies. So this was a major event polynomial invariants to chain com- there every day because he’s still that determined both my personal plexes. The most exciting feature is waiting for the next great idea that and professional lives. that the surfaces connecting knots in will change his life. So people work in 4D induce maps between these chain different ways. The second turning point happened complexes, so we gain insides into when I was around 30 years old, 4-dimensional phenomena through What I really appreciate a lot in with my 3rd postdoc and two small categorization. My dream is to cate- mathematics are collaborations, children. I was living in Basel and gorize 3-manifold invariants. where we exchange ideas and moti- my husband moved to the region of vate each other. I do much better in

It is our job to show how important it is to further develop mathematics in order to have a leading position in technology and to be competitive with other countries.

Anna Beliakova is a full professor of mathematics at the Universi- collaboration, speaking with people, ty of Zurich. She was born in Minsk (Belarus) and completed her Zurich because he found a job there. - What were your most exciting or doctorate in 1994 under Professor Robert Schrader at the Free During the weeks, I was alone with rewarding research experiences? than when sitting alone in my office. University in Berlin. two small kids, and with very un- And this is where I have a lot of fun in clear career chances. At that point For me the latest one is always the my job. For me personally, collabora- Her areas of expertise are topology, in particular knot theory, as I decided to reduce my position in most exciting. After I’ve understood tion is the best way to produce new well as quantum invariants and their categorization. Basel to 20% and join my husband in the problem, I am usually not as inter- mathematics. Zurich. We put our kids in childcare in ested as I was at the beginning. So for In 2011 Prof. Beliakova developed a unified theory of quantum the canton Schwyz, which was com- the moment I am really excited about - Many people in the general public invariants for the 3-dimensional homology spheres in collabora- pletely unusual in those times. I had my current project: find a description have little or no idea about what tion with Le and Bühler. Later, she introduced horizontal tracks in to take an additional teaching job at mathematicians do. What, in your bi-categories, which she used in 2016 for the construction of the of 3-manifold invariants that admits a view, can we do to sell our subject Khovanov-Homology in rings. the Technical School in Rapperswil to categorization. pay for the childcare. Looking back, better? In 2007, Anna Beliakova founded the Junior Euler Society at the this was the right decision. It allowed - What is, in your view, the right bal- University of Zurich, which promotes students’ interest in mathe- me to work on my habilitation from ance between creativity and hard This is clearly a challenge because the matics. Since 2018 she is co-chair of the SwissMAP Equal Opportu- home, and after a while I got the SNF work in mathematics? problems we are working on are usu- nities Committee. professorship and finally my current ally not simple to describe. But on the

8 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 9 other hand, it is impressive how little This is not just a SwissMAP problem, program. This definitively deserves the public knows about mathematics. but one we see everywhere in sci- some recognition. If I tell someone I’m a mathematician, ence. For example, people in algebra the first reaction is usually, “Oh, you and geometry would usually not - As co-chair of the Equal Opportuni- should be very good in multiplying collaborate. The way mathematics ties Committee in SwissMAP, what big numbers!” If I say that I am doing was developed in the past was by plans do you have to encourage research they ask, “Are there any dividing fields and isolating problems female researchers and young fami- open problems in mathematics?” to be able to solve them. We are used lies in science? So we definitely need to change this to dig deeper and deeper holes, to be- view on mathematics. come more and more specialized to We’ve already had a few very good produce new results. This is a success- things implemented, like the pro- Another typical question that even ful strategy. But even deeper results gram that encourages professors to students of mathematics ask is, why may come from a broader perspec- hire women PhD‘s and postdocs, by use something in dimension N when tive; therefore it is important for us paying 50% of their salary through our real world is three-dimensional? to make an effort and to speak with SwissMAP. We are reimbursing Here I usually explain that when you people from neighbouring fields. In childcare costs during conferences for wake up in the morning in a good or SwissMAP we have this possibility by SwissMAP participants, etc. a bad mood, it might depend on your organizing conferences that border last conversation with your relatives, on different domains and by meeting In 2019 we launched a new mentoring the well-being of your friend, what colleagues from physics. program. The idea is to fix the “leaky you ate yesterday, and so on and so pipeline”, i.e. to encourage talented on. You could easily come up with a I have learnt a lot since the begin- young people especially women to list of 10 parameters just to explain ning of SwissMAP. For example, on continue their research career. On the how you feel today. And of course, a conference organised by Grigory SwissMAP website, we collect short a mathematical model that could Mikhalkin in Tessin last summer, I CV’s of the mentors, showing each predict how you feel tomorrow will heard lectures on enumerative geom- other’s experiences and areas of ex- not only depend on just 3 parame- etry and tropical knots and then used pertise as well as explain where they ters. Cambridge analysts for example, their methods in my research. I also can help especially well. In this way represented a person as a point in a learned about theoretical physicists our young researchers may choose seven-dimensional space. And appar- view on the categorization of 3-man- mentors that best suit their needs ently, this is a quite good approxima- ifold invariants in one of the SwissM- and profit from their experience. I’m tion of how we function by elections. AP conferences. really looking forward to this pro- gram. What we should make clear to people I think just making people talk to is that a lot of things around us are try. All these things exist and work each other is already a lot. In order Corinna Ulcigrai is also organising a based on mathematical models. And because the mathematics behind - What did SwissMAP change in for this cooperation to go further, women lunch in Zurich, where female they work, because the mathematics them is completely understood. your research life? we probably need a few more years mathematicians can chat, exchange was well understood and even sim- where we continue to talk to each experiences and also help each other. plified down to clear and fast algo- This is probably our job to stress this SwissMAP is great. I was never sup- other and organise more of such in- rithms. We are profiting from this point and to show how important it ported as much as during the years terdisciplinary activities. It is probably Finally, we are currently implement- since SwissMAP was created. We too optimistic to expect an immedi- ing a new hiring policy in SwissMAP have unique possibilities to organise ate result here. aiming to reduce bias in the hiring conferences, invite colleagues, hire process. It is important for us to make an effort and to speak with postdocs and even organise some Speaking to the public is quite sim- people from neighbouring fields. In SwissMAP we have this special activities for female mathe- ilar as speaking to colleagues from possibility. maticians that help them to develop neighbouring fields. This effort should their careers. This is an amazing be valued and appreciated more. For opportunity, and it has enriched my example, I have just read the book research quite significantly. “Love & Math” by Edward Frenkel. I every day. Like Siri which is nothing is to further develop mathematics really see this book as a serious effort Conversation with Anna Beliakova more than an artificial intelligence in order to have a leading position - What in your opinion could pro- to popularise a certain mathematical December 2018, Zurich which is again a good algorithm that in technology and to be competitive mote or encourage cross-project program. The author reached a broad works. We see this everywhere be it mathematical audience and in so do- with other countries. It’s really a key collaborations within SwissMAP? Interviewed by Maria Kondratieva money transfer or the medical indus- point. ing, contributed to the success of this On behalf of the NCCR SwissMAP

10 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 11 Presenting the

Problematic supervision Too much to do and little Can’t decide between an NCCR SwissMAP impacting research? time to do it? academic path and the industry?

Striking a balance between one’s tees can seek advice from more reers. The program is certain to have Mentoring Program scientific career and personal life is a senior SwissMAP participants, who a positive impact on the profession- problem every young researcher fac- have volunteered as mentors and al and personal lives of female PhD es. This is especially true for young can thus benefit from their knowl- and Postdocs within SwissMAP. women, many of whom might edge, skills and expertise. for women PhD and decide against entering the field because of the difficult choices one needs to make in order to become a As our Postdoc member Ulrike Riess scientist. (ETH Zurich) explained “At that time our first child had just been born and Postdocs things were getting somewhat diffi- As part of the ongoing effort of cult…During the talk I had with Anna SwissMAP to address the gender Beliakova, I received some very good gap in our field, the Equal Oppor- advice on how to organize childcare.” tunities Committee has recently launched the SwissMAP Mentoring Program for Women PhD and Post- The idea behind the program is doc members. that all PhD and Postdoc SwissMAP women freely choose a mentor, who does not necessarily work in the “It can be difficult to talk about what is on your mind - but it is PhD students and Postdocs often same institution or field. Participants face daunting challenges. Research- have access to the list of mentors most definitely worth it. Talking to someone who has had similar ers are often reluctant to seek help, and their profiles, in which each one experiences can be very motivating and beneficial. I felt that after believing that they should be able to of the eight mentors (four women resolve their problems on their own. and four men) has briefly intro- the discussion with my mentor, I was able to think much more It is not an easy decision to speak to duced themselves, described their clearly.” someone, however, it is precisely in career path as well as their hobbies these situations, talking to the right and interests. Participants then person, who may have had similar contact the mentor of their choice Ulrike Riess, SwissMAP experiences, can make the differ- directly to organize a first meeting. ence in finding solutions. Effective After this first meeting, they decide mentorship can play a critical role in whether or not to continue with the a researcher’s future career trajec- program. tory.

The SwissMAP mentoring program The mentoring program offers all is an important step towards cre- SwissMAP women PhDs and Post- ating a more encouraging environ- Author: Sophie Schmidhuber docs impartial, confidential support ment for women researchers and On behalf of the NCCR SwissMAP at all stages of their careers. Men- providing with support in their ca-

12 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 13 inspired me during this conference. Balancing Yin and Yang in She really awakened this thing inside As a young woman professor, I have had to work extra to of me. Before that, I was just enjoying life and I didn’t think too much about earn respect and trust in my ability, my proofs, and my the field of Mathematics: my future, then Karen really made me insight. think seriously about what I wanted to do in the future. I could see Karen, advisor and secondly, because I was I’m still continuing on this path. this amazing woman professor who not yet fully established. Now I real- Interview with Prof. was there, who organized panel dis- ised that my advisor Alan Weinstein, I believe that the greatest pressure cussions and other very good women at the same time, being an outstand- and challenge I face is the balance professors, women postdocs, who ing mathematician, has also a very between career and the family. I also shared honestly their life, both pro- gentle and supportive character. He I feel this applies to most women Chenchang Zhu fessional and personal, with us. There is incredibly fair towards his students professors. were difficulties, but they were man- with regards to gender. So, I didn’t ageable and these women all seemed feel at all being different among his - Do women need to be much better to enjoy what they did! I thought to - Going back to your school days, degrees outside and I would still get students. Today, I realised that it is than men to get to the same place? myself, yes, it is possible to become a what or who drew you initially to up at 8:00 am and work until 10:00 not the case in general. I was really woman professor! In fact, I returned mathematics? pm. I actually managed to finish all lucky to have Alan as my advisor! When I entered the job market the Chenchang Zhu received her PhD to WAM again the following year and the high school math textbooks in pro-women movement, as well as the in mathematics from the Univer- I was really thankful to Karen. sity of California at Berkeley in I would say when I changed elemen- one summer break! - Do you think the criticism was presence of the equal opportunities tary schools. I was around 10 years accentuated by the fact you were a officers in committees, were already 2004 under prof. Alan Weinstein. - Can you speak about your experi- old, I remember I was not such a Although in these small classes there woman or, does everybody experi- very established. I actually I felt it was She was an assistant professor ence as a woman mathematician, popular kid, I ranked around the were mostly boys, I was not affected ence this on the road to professor- quite more balanced compared to at the Fourier Institute, Grenoble particularly at different stages of middle of the class. Then, there was by this. The boys were quite nice to ship? the previous generation of women as of 2006. She got tenured in at your life? mathematicians. I think it was much the Mathematics Institute of the a math’s test and somehow I got a the few girls in the class. Also, I feel perfect score, I don’t know how! A that compared to other peers who At that time I didn’t think it was tougher for them. On the other hand, Georg-August-Universität Göttin- I guess the period of graduate school woman teacher then came up to me say that when they were younger particularly because I was a woman. as I mentioned above, I do feel, as a gen since 2013. was a beautiful time for me: to learn and explained that she held a small they sometimes felt a gender bias However, now when I look back I woman, sometimes, I need to work to live in a new culture and explore Her research interests include: class specially for the kids who had coming from boys and even from the think it was half and half. I feel math’s extra to convince people to trust me Poisson geometry, Symplectic ge- very high scores and invited me to teachers, in my case it was different. ometry, higher structures in join the class. In those classes we It is probably because in China, at differential geometry, such as high- did some exercises together, I found least in my generation, the one child er stacks, gerbes, Lie algebroids, those small classes and the people policy meant that in my family I was I believe that the greatest pressure and challenge I face is the balance between career and higher Lie groupoids and simplicial very nice, they were very intelligent. like “the little princess”, and I never family. I also feel this applies to most women professors. manifolds. I was certainly very happy to attend felt second best. I would say that for this class, especially as it was a new that generation of girls in China, we Following our Women at the school for me. That was how this felt very much equal to boys. I did not Intersection of Mathematics and whole thing started and then I began care so much about this. life. There was a lot of going on for High Energy Physics conference to participate in the Mathematical society tends to have more “male cul- or take my words, even comparing to me, and I was at the moment when Olympiad in this small group. - Did you have any women role ture” compared to other areas, such male colleagues with the same age. that took place in April 2019, we I had to figure out what to do in life models who inspired your career? as art or biology. met with SwissMAP scholar Prof. for the future. Then, the period of Afterwards, it was kind of by chance. - Did you ever benefit from men- Chenchang Zhu, to talk with her being a postdoc, I spent 2 years at When I was in high school my favour- I went to the WAM (Women And - … and as a young woman profes- toring or are there any specific about her experiences as a wom- ETH Zurich and then another 2 years ite rock singer Wong Ka Kui died in Mathematics) in IAS Princeton. It is sor what challenges have you had moments when you believe it could an mathematician and about the as a young assistant professor maître Hong Kong, I think he died June 30th very similar to the recent SwisssM- to face? Do you feel more pressure have been helpful? impact of the conference. de conférences in Grenoble. That time 1993. I was really sad and at that time AP women’s conference, only it or expectations? for me was mostly a period when I you couldn’t go to Hong Kong from was more aimed at students. It was I benefited much from mentoring became independent from my PhD the mainland but I knew that the hosted by Karen Uhlenbeck, who by As a young woman professor, I have from Karen Uhlenbeck. During my advisor and I began to work with International Mathematical Olympiad the way, recently became the first had to defend hard make a sound, graduate school years it was unfor- many new collaborators. It was amaz- was going to take place in Hong Kong woman to receive the Abel Prize in to work extra to earn respect and gettable how Karen encouraged so ing how during that period there was in 1994 and I was determined to go Mathematics! trust in my ability, my proofs, and my many women mathematicians. She much criticism! I would say it was there. I’ve never worked so hard in insight. With time and patience, I’m served as a role model to us. We firstly, due to the fact that I was no my life. I remember it was around 40 I would say that it was Karen who happy that some are convinced, but could see there were women pro- longer under the protection of my

14 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 15 fessors and how they could also be was about candidate profile reading are present. The subject was there at probably advice for both men and speakers at ICM, and how they work in a committee. I remember some- a moment when the ratio of men and women. Secondly, to be persistent, their path over there concretely! one said “if a profile contains a lot women in education matched more to stick to it. I would say before you of activities, interactions and collab- evenly. That I find specific to mathe- become established you will meet a We also discussed about all the orations, if it’s a man, people tend matics, because of this somehow all lot of criticism and you need to be problems we could face and how we to think, he’s very active and if it’s a the definitions of smartness… and aware that you are likely to encoun- could go about them. How to manage women they tend to think she’s very also missing early women role mod- ter it. Women sometimes tend to family life and a career. I found WAM sociable.” This is so true! I was just els… that is why it is great that Karen leave if they think “I am not so good extremely successful and helpful. going through some profiles and I won the prize, I was really happy with this” and how do we judge if They gave advice and we all shared noticed that even I was unconsciously when I found out. we are good at this? Often through experiences. doing so. After noticing this, I went feedback from outside. This outside back and restarted, avoiding reading - … and specifically on the subject feedback plays a major role - that’s - You attended the recent SwissM- the applicant’s name and only read- of recruitment? unavoidable. AP Women at the Intersection ing the profiles. I had to re-educate of Mathematics and High Energy myself. As I’ve said, maths has a lot of Yang I would say to women mathemati- Physics conference. Could you tell energy in it. Firstly, we need more cians and young women especially, us about the benefits of attending? - Are there any challenges and pos- women, who will automatically bring that one needs to stick at it a bit sible solutions specifically related some other kind of culture. Second- more. It’s totally normal for some- In terms of content structure, math- to the field of mathematics with ly, we certainly need to have more one to receive criticism early in their ematically speaking, I benefited a regards to gender equality? of these women conferences, like career - especially when you are a lot. It was a rich mixture of a variety the SwissMAP conference or what woman. Try to digest this criticism of topics. I’ve never learnt so much. I think mathematics is a very clas- Karen had in the US, and also summer and don’t give up because of it. Be Each section was followed by a half sical subject created at a time of a schools for young women mathe- more sure about yourself and try to hour topic discussion lead by a top male dominated society. I think the maticians. I also think it is important find support within the community. figure in the area, where we were first women mathematicians known not to let mathematics change us, free to ask questions. to the general public date back to but precisely we need to stand up for around 100 years ago, whereas the ourselves and bring something new Somehow the energy amongst par- first man goes back to 2000 years to the mathematics culture, to have ticipants was different. For example, ago. So, because of this the challenge more of a balance. I think that it is when someone is too prominent or is that it has already a lot of Yang good to have a balance of both ener- aggressive, you don’t dare to ask energy inside. What is definition of gies. When you have energy balance questions, but if someone is more being smart and intelligent in mathe- you can progress more efficiently. low key and calm then your brain matics? What is the definition of be- is stimulated to ask questions. You Credit: © Noel Tovia Matoff ing valuable and deep in mathemat- - What advice would you give do not feel intimidated or, afraid to ics? What is definition of a talent and young women considering a career ask questions is inappropriate or not is something to support and grow ference and in terms of giving wom- a star in mathematics? Its all, already as researchers in mathematics? welcomed. In my opinion during this something on top of it and Yang is en visibility and recognition, I think 99% or 99.9% male based opinions. before this conference I did not know I think first one would need to make most of the participants and now I’ve If you take biology for example, a sure it’s something that you really met women from both mathematics more recent science, more women like. Follow your heart but that is I think it is important not to let mathematics change us. We and physics and we had deep discus- need to stand up for ourselves and bring something new to sions. The atmosphere invited you to the mathematics culture, to have more of a balance. exchange ideas.

- The panel discussions were very rich in content and it was great to conference people discussed with this constructive prominent energy see participants of different back- an opened mind. It was different to also in the direction of competitive- grounds interacting. Could you attending other types of high profile ness. Even the male participants in briefly tell us something about the conferences. the conference felt much more open gender equality issues which were Conversation with to ask questions - it changed the raised and were you surprised to Chenchang Zhu As I said in the panel discussion energy field. learn about any of them? April 2019, Geneva during the conference, you need to On the networking side of the con- I think one of the panel discussions Interviewed by Mayra Lirot have the balance of Yin and Yang. Yin NCCR SwissMAP Credit: Mayra Lirot

16 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 17 18 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 19 such as the Euclidean space, hyper- UNIVERSAL OPTIMALITY sphere, or hyperbolic space subject Following Cohn and Kumar, we call to the minimization of the energy of such a conguration universally op- pairwise point interactions. We can timal: LetC be a point configuration OF think of points in the conguration as in R^d with density rho, where rho>0. particles of some kind. We say C is universally optimal if Energy minimization generalizes WWit minimizes p-energy whenever E AND LEECH LATTICES the sphere packing problem in R^d, p: (0;infinity) -> R is a completely 8 in which we wish to maximize the monotonic function of squared dis- minimal distance between neighbor- tance. ing particles while fixing the particle Computations show that the univer- density. In the energy minimization sally optimal configurations are rare. 1. Introduction problem we fix the particle density A putative list of universally optimal delta>0 and assume that the ener- configurations on hyperspheres is The E lattice and the Leech lattice are Figure 2. Sphere packing in dimension 3 8 gy of interaction between a pair of given here [2]. Examples of univer- among the most famous math- particles depends only on distance sally optimal configurations are also ematical objects. These exceptional r between them. Suppose that this known in other metric spaces. References structures emerge in dierent areas of dependence is given by a function [1] H. Cohn, N. Elkies, New upper bounds mathematics and physics, here are , Annals of Math. 157 p(r). Then the p-energy of a congura- 4. Universal optimality in Euclidean on sphere packings I only a few to name: number theory, (2003) pp. 689-714. tion is the average interaction energy spaces classication of finite groups, theory [2] H. Cohn, A. Kumar, Universally optimal per particle. An energy minimization of Lie groups, automorphic forms, distribution of points on spheres, J. Amer. problem asks for a point conguration Currently, only three universally opti- coding theory, extremal graph theory, Math. Soc. 20 (1) (2007), pp. 99-148. of density delta with the smallest mal configurations at the Euclidean string theory and statistical physics. [3] H. Cohn, A. Kumar, S. D. Miller, D. Rad- possible value of p-energy. If exists, spaces are known: the lattice of inte- chenko, and M. S. Viazovska, What is the reason for their “celebri- The sphere such a configuration is also called a ger numbers in R1 [2], the E -lattice packing problem in dimension 24, Annals ty” status in nature? We cannot give 8 ground state. in R8, and the Leech lattice in R24 [4]. of Math. Volume 185 (2017), Issue 3, pp. a complete answer to this question. The hexagonal “honeycomb” lattice in 1017-1033 However, jointly with H. Cohn, A. We will concentrate on energy po- dimension 2 is also conjectured to be [4] H. Cohn, A. Kumar, S. D. Miller, D. Kumar, S. D. Miller, and D. Radchenko, tentials of a particular shape: we will universally optimal. A classical result Radchenko, and M. S. Viazovska, Uni- we have recently found that these lat- consider repelling interactions such is that the hexagonal lattice solves the versal optimality of the E8 and Leech tices have the following rare property lattices and interpolation formulas, arX- that the repulsion increases as parti- sphere packing problem [7], however – they are universally optimal [4]. iv:1902.05438 [math.MG]. cles get closer. Here is a more formal its optimality for potential energy [5] T. Hales, A proof of the Kepler conjec- mathematical description. Recall minimization still remains open. On , Annals of Math. 162 (3) (2005), pp. 2. E and Leech lattices ture 8 that a function g: (0;Infinity) -> R is the other hand, the computations 1065-1185. completely monotonic if it is infinitely suggest that there is no universally [6] F. H. Stillinger,Phase transitions in the Let us introduce the two lattices in differentiable and satisfies the ine- optimal configuration in the 3-di- Gaussian core system, J. Chem. Phys. 65 question a little closer. Both E and 8 qualities (–1)kg(k) >= 0 for all k >= 0. mensional Euclidean space and the (1976), no. 10, 3968-3974. Leech lattice belong to the “family” of 1. In other words, g is nonnegative, ground states depend on the energy [7] A. Thue, Über die dichteste Zusam- . We recall even unimodular lattices weakly decreasing, convex, and so profile [5], [6]. A similar situation has menstellung von kongruenten Kreisen in that a in the Euclidean space is Figure 1. Petrie projection of the shortest vectors of E -lattice into a 2-dimensional plane. , Norske Vid. Selsk. Skr. No.1 lattice 8 on. For example, inverse power laws been detected in other small dimen- einer Ebene a discrete full rank abelian subgroups, Source: Wikipedia (1910), pp. 1-9. are completely monotonic, as are sions. This leads to an assumption lattices are the lattices [8] L. Fejes Toth, Über die dichteste unimodular mension 24 and the only one among the Leech lattices are extremal for a decreasing exponential functions. By that the existence of a universally op- containing on average 1 point per unit Kugellagerung, Math. Z. 48 (1943), pp. them having no vectors of the small- much bigger family of optimization Bernstein’s theorem the complete- timal configuration in Euclidean space 676-684. [9] M. S. Viazovska, of volume, and the term even means The sphere est possible non-zero length \sqrt{2}. problems. ly monotonic functions of squared is an exceptional coincidence and , Annals that the length squared of each lattice packing problem in dimension 8 Both E and the Leech lattice enjoy distance are the cone spanned by the raises a question whether it occurs of Math. Volume 185 (2017), Issue 3, pp. vector in an even integer number. Lat- 8 the number of symmetries and ex- 3. Universal optimality Gaussians and the constant function only in dimensions 1, 2, 8 and 24. 991-1015 tices with these both properties can tremal properties. For example, the 2. It follows that if a periodic config- exist only in dimensions dividible by shortest vectors of each of these Universal optimality is a property of a uration is a ground state for every 8, they are rare in small dimensions lattices are the unique solutions to point conguration to be a simul- Gaussian, then the same is true for and come in huge numbers in dimen- the sphere kissing problem in dimen- taneous solution to a “universum” of every completely monotonic function sions bigger then 24. The E -lattice is 8 sions 8 and 24. Also these lattices optimization problems. This notion of squared distance (by monotone the unique even unimodular lattice in are solutions to the sphere packing was introduced by H. Cohn and A. convergence, because the potential is dimension 8, while the Leech lattice is Author: Maryna Viazovska problem in their respective dimen- Kumar [2] and applied to point config- an increasing limit of weighted sums one of 24 possible such lattices in di- Professor, EPFL sions. We have found that the E8 and urations in homogeneous spaces of finitely many Gaussians).

20 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 21 The importance of organizing a women workshop: Interview with Eva Miranda

- Can you tell us about your impres- or in geometry and topology. sion of the 2017 Women in Geome- try and Topology in Zurich? People are sometimes a bit hesitant about attending a women’s confer- I remember that when I was first ence, as I was. I would like to change invited to be a speaker at the 2017 this. I want to bring women from Women in Geometry and Topology different countries together and give Workshop in Zürich, organised by them visibility and I also want men to SwissMAP, I must admit that I was ini- be present. I take this opportunity to tially slightly reluctant about the idea encourage people from all genders to of a workshop highlighting women come and to participate! (or any specific gender) in the title. is one of the strategic objectives as who work more in symplectic geom- in mathematics. The more we ad- I was, however, very impressed by - How did you get involved in the or- well as public engagement, and we etry, some in algebraic geometry and vance, the fewer there are. Although the event and its success. I was also ganization of this year’s workshop? decided to include this conference in also others who work in applications this has been a constant problem, particularly surprised by the number the list of activities addressing the to computer science. the situation is now desperate. In the of men who attended the talks and During the 2017 workshop in Zurich, gender gap. master class I am teaching this year actively participated by asking ques- SwissMAP member Anna Beliakova This year one of our public lecture on differential geometry, I have only tions. This was indeed unexpected for (UZH), suggested I organise the next - What are the similarities and new speakers will be Carme Torras (CSIC), one woman out of 11 students. I be- The Women in Geometry and me. workshop in Barcelona… Initially I features planned for this year’s who works in computer science. lieve we really have to do something Topology Workshop this year will was unsure. I remember then having program? We are very honoured as she is an about this and organising this work- be taking place in Barcelona, 25th - You say you were initially reluctant some very interesting discussions exceptional researcher and has been shop is a move in the right direction - 27th September and is organized to participate, are you convinced during the conference about how The plenary speakers from the last awarded many different prizes; she and I’m hopeful it will contribute to by GEOMVAP. Our Deputy Director now of the benefits of this type of to address the gender gap, how the workshop of Women in geometry and is a real role model! In fact, our goal making things change. and Co-Chair of our Equal Opportu- event? situation in Switzerland was different topology covered a range of different is to put forward different examples nities Committee Anton Alekseev from the situation in Spain or Italy - Who are you expecting will submit (UNIGE) is part of the Organizing I believe organising this type of and about how every country has applications for the contributed workshop is very important, not only a diverse scenario and social ingre- Committee. talks? for women in mathematics or, in this dients. Notwithstanding, there are Our goal is to put forward different examples of role mod- case geometry and topology, but in also many similarities. Being at the els on the table. We believe this is important as one of the In this short interview, Eva Miran- One of the new features for this other subjects too. As women, we are conference and seeing the benefits existing problems is the visibility and number of women in year’s workshop is that the call for da (UPC) also part of the organiza- used to being invited to conferences as well as the different conversations tion committee, tells us about the the mathematics. contributions is also open to all gen- and finding ourselves to be the only encouraged me to organise this year’s ders. The plenary speakers are female importance of organizing a women woman or one of the few women workshop. On top of that, in 2018 workshop and what she hopes it mathematicians but the call for speakers. I think it is very pleasant to we were awarded a special research topics. We are following the same of role models. We believe this that will achieve. contributions is open to all genders. bring a lot of women to the surface project focused on Geometry and guidelines this year: we have women is important, as one of the problems We want this because we believe that so that people stop thinking that Topology SGR932 from Generalitat who work more in topology, women that exists is the visibility of women things are going to change but for there are no women in mathematics (GEOMVAP), where gender balance

22 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 23 of view as this is a common effort. manipulation in assistive robotics: Re- We would like our panels to also be search challenges, ethics and fiction” composed of people who are not in She has a very interesting profile, a This type of conference firstly, provides great visibility for women researchers. Secondly, it mathematics. For example, we re- mathematician who has reinvented provides role models that can inspire younger generations. Finally, it sends out a message cently invited someone with a social herself. She is now working in robot- to the grant providers and to society. studies background to be part of the ics with robots to design clothes (ERC panel discussions as we want to have Advanced Grant Clothilde) and in this a wide perspective on the problems process of designing clothes there is a type of conference firstly provides place to visit and also that we have and solutions. One of our panel lot of mathematics. I don’t know how great visibility for women research- grants available to cover lodgings and discussions will be “From inequalities she finds the time, but she is also a ers. Secondly, it provides role models registration. Come and visit us! to equalities: how to break the glass science fiction writer. that can inspire younger generations. The link to the conference is ceiling in maths”. The moderator of Finally, it sends out a message to the www.crm.cat/2019/Women_GT the panel will be Marta Casanellas - What would you say are the long grant providers and to society. The (UPC). term effects of this type of event grant covering most of this event is and what in your opinion is the take from Generalitat and it was awarded - The public lectures are an impor- home message? to us as part of their strategic objec- tant part of the programme. Can tive on gender equality. We want to you tell us about the speakers? For the people who participate as tell the government to keep moving speakers, I think the visibility for in this direction and that what we are These public talks are open to non-ex- women is very important. I will give currently doing is not enough. perts and are not only for mathema- an example, it is something that can ticians. Both speakers are excellent happen too often when you are a I would like to add that one of the and have very original profiles. One member of a scientific committee and footprints that we are going to get of the talks will be given by Marta you’re asked to name and propose from this workshop is that we are Macho (UPV/EHU), recipient of the speakers of the field. In an instinctive going to offer the opportunity to par- Ekamunde Equality Prize and also way the first names that will cross ticipants who have a contributed talk chair of mathematics and gender at your mind are people whom you’ve to publish them in a Springer book of the collection Research Perspectives CRM Barcelona, which our research group is editing in 2019. So the con- ference will leave a trace behind not We want to have all genders involved because we believe that things are going to change only in our minds but also a printed but for this to happen we need the participation of everybody. trace in a book.

- Can you tell us about where the workshop will be taking place? this to happen we need everybody to the University of Bilbao. The title of actually seen giving talks. Perhaps participate. her talk will be “Sesgos de género en if you’ve never heard of a particular The three-day workshop will mostly la academia: cuando las matemáticas woman giving a talk you would not take place at CRM, which is the main Personally, I’d like to encourage all no funcionan” (Gender bias in aca- necessarily think of her. This type of research centre in mathematics in kinds of profiles for the contribut- demia: When mathematics doesn’t event is also a good way to promote Barcelona. We have decided that the ed talks. I don’t want an all-women work). Marta gives wonderful talks and give women visibility. People will public talks will be held in the cen- event. I want all genders to be and I am delighted to have her as a remember them and know which tre of Barcelona at the Institute of present to see how we can improve speaker. She has a very different type field they are working on. Catalan Studies because CRM is just the situation by talking to each other of approach when she gives talks, outside Barcelona. Our motivation to and cooperating to improve the I am sure that the participants will It is equally important for the next organise it in the centre of Barcelona situation. not be disappointed with her pres- generations. PhD students or Mas- was mainly to give more visibility to entation; she tackles the problems ter’s students who find themselves this part of the programme, which is - Have you also considered having from a different angle and she is very the only woman in their course will of general interest not only to math- male participants in the panel dis- provocative. be glad to participate in this women’s ematicians but also to the general cussions? workshop and be inspired by the role public. Conversation with Eva Miranda We also have a public lecture by models. June 2019, Geneva We plan to also have men because it Carme Torras, who I mentioned earli- I want to take this opportunity to ad- is very important to have all points Interviewed by Mayra Lirot er. Her lecture will be entitled “Cloth In summary, I would say that this vertise that Barcelona is a wonderful NCCR SwissMAP

24 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 25 supportive environment she found in some sense be treated as a gauge SwissMAP member stories: at Villa Battelle. theory. Research Area Gauge theories are geometric in character. From a physical point of Olga Chekeres Early on in her Bachelor’s Olga real- view they describe a particle subject ized that astrophysics was not ex- to an independent symmetry trans- actly what she was interested in and formation at each point of space- decided to orient herself towards time. Geometrically this means that theoretical physics. Her research for each point of space-time there Olga’s exceptional career trajectory spite of this promising and success- field is now mathematical physics is an “attached” space representing has been clearly led by remarkable ful start to her career, she ques- and she particularly enjoys working the internal structure of the particle. enthusiasm and determination. tioned her choice and what her real at the interface between mathemat- The natural mathematical descrip- From a very early age she was fas- interests were and began to explore ics and physics and believes in the tion for such a construction is the cinated with astronomy and astro- other possibilities. power of mathematics to explain theory of principal bundles, connec- physics and remembers that she and predict how the physical world tions and the curvatures of these was often reading books at the local Olga remembered the great interest functions. connections. library about stars and black holes she initially had in astrophysics. Re- and was very curious as to how it all viving this strong fascination on the More specifically, her research is A very important example of an worked. subject, she decided to prepare for focused on surface observables observable - that is to say a physical the entry exam for the Bachelor’s in in gauge theories. Gauge theories quantity that can be measured - is a However, once she finished school physics at the Moscow Institute of are at the very heart of our current Wilson loop. This is a gauge invariant the possibility of continuing her Physics and Technology (MIPT). The understanding of the fundamental quantity defined on a 1-dimensional Credit: Serguei Ganjour education in either physics or math- entry exam was a requirement for line. It was initially introduced in ematics did not really occur to her all prospective students. 1974 by Kenneth G. Wilson to solve Olga’s research plans are not re- SwissMAP member stories: Olga at the time. The reason was not lack the problem of confinement of stricted to the study of Wilson Chekeres (UNIGE, A. Alekseev of encouragement at home, nor ------quarks, which however still remains surfaces. She has several research Group) gender inequalities. Olga explained unsolved. In electromagnetism this projects related to Wilson surfaces, that in post-Soviet Russia equality But fate had other plans, she moved observable corresponds to a phase which she would like to implement SwissMAP-funded PhD student, between men and women prevailed. to Geneva with her husband and shift obtained by a charged particle in the near future. Her longer-term Olga Chekeres, completed her the- Furthermore, the education she daughter before she could sit for the transported in an electric field along research interests are to approach sis in April 2019, one year earlier received from her parents strongly MIPT entry exam. Upon arrival to a closed loop, measuring how much quantum gravity. Currently there ex- encouraged her to study and to Geneva, determined to continue her than planned. Supervised by An- the particle ‘fails’ to be the same ist several candidates for a unifying have a career. However, in the Rus- project, she was faced with a further ton Alekseev, she wrote two suc- on returning to the same point in theory that can incorporate quan- sian countryside where she grew up, challenge, that of learning French space. tum gravity, i.e. string theory, loop cessful publications: A. Alekseev, she had never been exposed to the to allow her to enter the University quantum gravity, causal dynamical O. Chekeres and P. Mnev “Wilson possibility of research as a career of Geneva. She pursued her stud- Olga’s work consists in constructing triangulations, causal sets, tensor surface observables from equiva- option. Though studying mathemat- ies and successfully completed her a Wilson surface observable, pass- models, etc. Olga believes that even- riant cohomology” and O. Chek- ics or physics was strongly recom- Bachelor’s, Master’s and PhD at the ing from a gauge invariant quantity tually in the correct theory space- eres “Quantum Wilson surfaces mended to her by her math teacher, University of Geneva in ten years. Credit: Alexandra Orlova; Taken at Pre-String- defined on a 1-dimensional line to time should be discrete, not contin- Math Summer School, July 2017 in Hamburg and topological interactions”. at the time she had the impression a gauge invariant quantity defined uous, in the same way that matter that the only possibilities this path Particularly admirable is also the laws of nature. The Standard Model on a 2-dimensional surface. In string consisting of particles is discrete. Olga’s plans are to join the Uni- would offer would be of working at way that Olga managed her per- of elementary particles is precisely theory such an observable would versity of Connecticut, USA, for the local school as a teacher. sonal life in an academic climate, a non-abelian gauge theory with correspond to a phase shift ob- We wish Olga all the best in her fu- the next 3 years as an Assistant juggling her research and being a gauge group G = U(1) X SU(2) X tained not by a point particle, but by ture endeavors! Research Professor. After finishing school, she decided mother of three children. Her sec- SU(3). It provides a unified model a charged 1-dimensional string trans- to study languages and economics, ond youngest, now two years old, for 3 out of 4 fundamental interac- ported in electric field along a loop. which were considered as fashion- was born in the middle of her PhD, tions - electromagnetic, weak and It can also describe surface defects able at the time, and completed a and her thesis defense took place strong - and is famous for its predic- within some gauge theory. Finally, Master’s degree on each of the sub- only two weeks before the birth tive power. The existence of its last the construction of a Wilson surface jects. Subsequently, not long after of her third child. A very important building block – the Higgs boson – observable allowed to formulate she started her professional career, aspect, without which her research was confirmed at CERN in 2012. Even Wilson surface theory as a separate Authors: Olga Chekeres, Mayra Lirot she became chief accountant at an could not have been accomplished, gravity in general relativity, which is 2-dimensional model, without any SwissMAP, UNIGE international company. However, in was the extremely encouraging and not part of the Standard Model, can gauge theory in the background.

26 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 27 2019-20 29 Jul - 07 Aug 2021 18 - 23 August ICMP 2021 11 - 22 March Geneva 18 - 20 February goMATH - Women in Mathe- New trends in geometry and mathematical physics We are happy to announce that Women at the Intersection of matics Ascona Geneva will be hosting the Mathematics and High Energy Zurich Anton Alekseev, Damien Calaque, next International Congress of Physics NCCR SwissMAP will be Alexander Veselov and Thomas Mathematical Physics in 2021. Geneva participating in the goMATH – Willwacher are organising the The event is expected to attract The primary goals of the work- Women in Mathematics event Events confernce on the occasion of the around 800 participants from shop are to promote the visibility organized by the ETH Zürich. 60th birthday of Giovanni Felder. around the world. of women in fields of research at 08 - 11 September the threshold of mathematics and 6th SwissMAP General Meeting theoretical physics. 22 - 26 July Young Topologists Meeting Villars-sur-Ollon Past Events: 2018 - 2019 2019 The program will include: the Between 2018 and 2019 SwissMAP Lausanne SwissMAP Innovator Prize held and participated in a number of The Young Topologists Meeting ceremony, colloquia covering successful events. is an annual event for graduate SwissMAP research directions, The joint activity of the ETH In- talks by junior participants etc. stitute for Theoretical Studies and students, recent PhDs and other SwissMAP resulted in the school junior researchers in topology Modular forms, periods and scattering (both pure and applied). amplitudes. The annual Workshop on Statistical Mechanics was held in Les Diablerets. The conference Inter- actions of low-dimensional topology and “higher” representation theory Feb - Apr May - Jul Aug - Sep Oct - Dec Jan - Feb brought together researchers and students interested in both topics. Along with the Univesity of Gene- va, SwissMAP organised the 2018 Mirimanoff Lectures. Our Club de Maths organised the UNIGE Olympi- 16 - 19 July 02 - 07 Feb 2020 ades mathématiques aimed at 11 to 15 Swiss Knots 2019 Winter School in Mathematical year old students. And in December, 01 Sep - 30 Jun Zurich Physics the Amsterdam-Brussels-Geneva-Paris Master Class in Mathematical Swiss Knots 2019 is part of a Les Diablerets Doctoral School was held on the topic Physics series of conferences on knot The annual Winter School in “Quantum Field Theory, Strings and 12 - 16 August Geneva theory and low-dimensional Representation Theory and Mathematical Physics will take Gravity”. SwissMAP is organizing a year- 18 - 21 March topology organized in Integrable Systems place at the hotel Les Sources in long master class at the University Deformation quantization, Bat- Switzerland. Zurich Les Diablerets in February 2-7, of Geneva in the academic alin-Vilkovisky formalism and Pavel Etingof, Giovanni Felder 2020. year 2019/2020 for master and index and Evgeny Mukhin are beginning PhD students. Geneva organising the confernce on the The aim of the workshop is to occasion of the 60th birthday compare different approaches of Vitaly Tarasov and the 70th to index coming from the birthday of Alexander Varchenko. deformation quantization, Batalin- Vilkovisky formalism and path For more detailed information integrals. please visit our Website: http://nccr-swissmap.ch/events

28 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 29 AwardsAlessio Figalli GrantsMarcos Marino Corinna Fields Medal 2018 ERC Synergy Grant Ulcigrai Congratulations to Alessio Figalli, Professor at the ETH Zurich and new NCCR 2018 University of SwissMAP member for winning the Fields Medal at the ICM 2018 in Rio de Janeiro! Congratulations to our Zurich The Fields Medal is awarded every four years at the ICM to recognize outstanding member Marcos Mariño Corinna Ulcigrai mathematical achievement for existing work and for the promise of future achievement. (UNIGE) who received is an Italian math- Alessio Figalli is awarded for his contributions to the theory of optimal transport and an ERC Synergy Grant ematician working its applications in partial differential equations, metric geometry and probability. 2018 for the proposal on dynamical sys- “ReNewQuantum: tems. She obtained Recursive and exact new quantum theory”. The her Ph.D. in 2007 from Princeton Vincent Tassion grant is part of the EU’s research and innovation University with Yakov Sinai as her 2019 Rollo Davidson Prize program, Horizon 2020. thesis advisor. She was awarded Vincent Tassion received the 2019 Rollo Davidson Prize in recognition of his exten- the European Mathematical Soci- sive achievements in disordered systems and percolation. Vincent Tassion has been an ety Prize in 2012, and the White- assistant professor at the Department of Mathematics at ETH Zurich and a SwissMAP Benjamin Schlein head Prize in 2013. She worked participant since January 2017. His area of research covers phase transitions in statis- ERC Advanced Grant as a professor at the University tical physics models, such as percolation or the Ising model. In recent years, Vincent Congratulations to our of Bristol. She is a professor at Tassion has grappled with a number of unanswered questions in this area, including member Benjamin Schlein the University of Zurich. Her for models in three-dimensional space. (UNIZH) for receiving research interests include ergodic an ERC Advanced Grant theory, dynamical systems and Maryna Viazovska Alice Gasparini titled “Correlations in Teichmueller dynamics. Especial- 2019 Satter Prize CPEP Award - Second place Large Quantum Systems”. ly ergodic, mixing and spectral He aims at understanding

Congratulations to Prof. Mary- Congratulations to Alice Collaborators New properties of parabolic dynamical na Viazovska (EPFL) who re- Gasparini on winning second the macroscopic properties of quantum systems, as systems. cieved the 2019 Satter Prize for place of the CPEP Award for arising from a microscopic description based on the fundamental laws of nature. her groundbreaking work in dis- her work on General Relativity Alessio Figalli crete geometry and her spectac- and Cosmology Activities for ETH Zurich ular solution to the sphere-pack- high schools. Alessio Figalli is a ing problem in dimension eight. Ghazaleh Asghari professor at the ETH SwissMAP Equal Op- Zurich since 2016. Nina Holden and Lorenz Eberhardt portunities Excellence He earned a joint doc- 2019 SwissMAP Innovator Prize Scholarship torate in 2007 from the Nina Holden is awarded for her impressive work on probabilistic Ghazaleh is a Masters Scuola Normale Superiore topics related to mathematical physics, such as Liouville quantum student, she joined Prof. di Pisa and the Ecole Nor- gravity, random planar maps and their embeddings and scaling Cattaneo’s group at UZH male Supérieure de Lyon. In 2007 limits, random allocations and matchings, and trace reconstruc- where she will be until he was appointed Chargé de re- tion. Lorenz Eberhardt is awarded for his important insights into 2020. cherche at the CNRS, and in 2008 string theory on AdS3 and its dual 2d CFT. he went to the Ecole Polytech- nique as Professeur Hadamard. In Beatriz Navarro 2009 he moved to the University Yilin Wang andJunliang Shen SwissMAP Equal Oppor- of Texas at Austin as Associate 2018 SwissMAP Innovator Prize tunities Mobility Grant Professor. Then he became Full Yilin Wang is awarded for her remarkable and original work on We are glad to announce Professor in 2011, and R.L.Moore the Loewner energy of planar curves and loops. Junliang Shen is Beatriz Navarro, from Chair holder in 2013. His research awarded for his work on the moduli of sheaves: the construction Toronto University, as interests calculus of variations of virtual cobordism classes and the connection of the enumera- recipient of our Mobility and partial differential equations. tive geometry of elliptically fibered Calabi-Yau 3-folds to Jacobi Grant. Beatriz will be He recently won the Fields Medal forms. based at UNIGE. at the ICM 2018 in Rio de Janeiro.

30 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 31

PuzzleCorner 2. Rational Numbers All the rational numbers in [0, 1] with denominator up to 2019 in lowest terms have been written down in increasing order. The fraction preceding 736/737 has the form x/(x + 2) with x N. What is x? ∈ Test your math and logic skills with these puzzles, kindly put together by some of our contributors.

3. Count the zeros How many zeros are at the end of the number 100! ?

1. On the way home Every day, Zoe’s mother, Wilma, picks her up at the train station when she comes home from school, and then Wilma drives Zoe home. They always return home at 5:00 pm. One day, Zoe left school early and got to the train station an hour early. She then started walking home. Wilma left home at the usual time 4. Nonnegative Integers to pick Zoe up and they met along the route between the train station and their Is it possible to find two subsets A and B of the set of nonnegative integers house. Wilma picked Zoe up and then drove home, arriving at 4:48 pm. For how such that every nonnegative integer x can be written uniquely as a many minutes had Zoe been walking before Wilma picked her up? sum x = a + b with a A, b B? From Wearing Gauss’s Jersey, Dean Hathout, CRC Press, 2013. ∈ ∈

32 | © SwissMAP Perspectives | 2019 2019 | © SwissMAP Perspectives | 33 Answers 1. On the way home Zoe was walking for 54 minutes before she was picked up. 2. Rational Numbers Consider a grid of unit squares in the plane. If a/b < c/d are consecutive frac- tions in the sequence, the triangle with vertices (0, 0), (b, a), (d, c) contains no lattice points in its interior and no boundary lattice points besides its vertices.

Pick’s formula implies that its area is 1/2. On the other hand, it’s area is (bc–ad)/2 by the conventional formula. Therefore, bc–ad = 1. So, we obtain an equation for x; its solution is x = 1471. The sequence described in the problem is called the Farey sequence corre- sponding to N = 2019.

Editing & Design: 3. Count the zeros Amrin Design There are 24 zeros. Print & Distribution: Trajets Imprimerie & Copies Services

Contributors: Maria Kondratieva; Mayra Lirot; 4. Nonnegative Integers Susanne Reffert; Yes. Let A and B be the sets of all nonnegative integers whose binary expansion Maryna Viazovska; contains only 0’s at odd (respectively, even) positions, counted from the right. Sophie Schmidhuber; Olga Chekeres; Kaloyan Slavov; Shaula Fiorelli Vilmart; Le Club de Math;

Puzzle contributors: Original illustrations by Varia Timchenko and Paul Turner. All other design elements licensed under the designcuts.com extended license. No 1: Shaula Fiorelli Vilmart | Mathscope | http://mathscope.ch/ Copyright ©2019 Amrin Design, NCCR SwissMAP and all Contributors as identified. No 2, 4: Kaloyan Slavov | ETH Math Youth Academy | https://people.math.ethz.ch/~kslavov/ SwissMAP Perspectives (images excepted) is made available for non-commercial re-use with proper attribution under the terms of the Creative Commons Attribu- No 3: Le Club de Math | http://unige.ch/math/fr/clubmath/ tion-NonCommercial-ShareAlike 4.0 International License. To view a copy of this license, please visit http://creativecommons.org/licenses/by-nc-sa/4.0/

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