Institute for Theoretical Studies

Report 2019/2020

ETH-ITS

Table of Contents

Foreword 5 The ETH Institute for Theoretical Studies 6 History and aims 6 Fellows at the ITS 6 Activities 7 Meetings, talks, mini-courses 7 The ITS Science Colloquium 8 Programme 2019–2020 9 Fellows’ seminar 10 Programme 2019–2020 10 Awards 11 Fellows’ report 12 Senior Fellows 12 Junior Fellows 17 Outlook 28 People at the ETH-ITS 29 Director 29 Administration 29 Board of Patrons 29 Advisory Committee 29 2013–2017 2017–2019 2019–2020 Fellows 2014–2020 30 Senior Fellows Junior Fellows (with current affiliation of former Junior Fellows)

Contact 31 Photo: Stefan Glock

Virtual ITS Tea Time in June 2020

4 Foreword

The academic year 2019–2020 started with a full house at the ITS with four Senior Fellows arriving in the summer of 2019. Their research interests spanned number theory, representati- on theory, symplectic geometry, and mathematical physics (and an additional Senior Fellow in statistics arrived during the year). Together with the nine Junior Fellows, there was an active Fall at the Institute with seminars, related mini-courses, ITS colloquia, Thursday teas, and even a group hike to Höllgrotten Baar.

The year 2019–2020 was my first as ITS director. Giovanni Felder, who had been director of the ITS since its foundation, handed over a wonderfully running Institute when his term ended on 31 May 2019. Livia Kürsteiner, who had experience with the ITS as an assistant coordinator, became the new ITS coordinator (after the retirement of Christina Buchmann). More than half of the Advisory Board completed their terms in the summer of 2019, so the board also now has many new faces and new directions.

The Spring semester at the ITS, of course, was very much af- fected by the Covid-19 pandemic. In addition to the closure of the building and the cancelling of lectures, workshops, and all events, some Senior Fellows returned to their home institutions (ahead of the travel bans). We moved activities online: virtual teas every Thursday, virtual seminars, and even a virtual ITS colloquium on the analysis of the spread of Covid-19 via genetic methods – by Tanja Stadler, a new Advisory Board member. We very much hope that the upcoming year 2020–2021 will be smoother (and we are planning for the Senior Fellows whose stays were shortened to return in the Spring of 2021).

Rahul Pandharipande, Director

The ETH Institute for Theoretical Studies is supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Foundation.

5 The ETH Institute for Theoretical Studies

History and aims The ETH Institute for Theoretical Studies (ETH-ITS) is an interdisciplinary Institute dedicated to research in ma- thematics, theoretical computer science and theoretical natural sciences. It was founded on 1 June 2013 on the initiative of former ETH president Ralph Eichler, with a generous donation of Dr. Max Rössler and the Walter Haefner Foundation. The aim of the Institute is to enable top theoretical scientists to be active for an extended period of time at ETH, interact with local researchers, and establish lasting scientific collaborations in an interdisciplinary context.

Fellows at the ITS The Institute hosts up to six Senior Fellows and up to twelve Junior and Advanced Fellows. Junior Fellows are talented young independent postdocs spending up to three years at ETH Zurich to work on research subjects of their choice. They are supported by a mentor, who is an ETH professor. Advanced Fellows are young researchers with some experience after their PhD, who have established themselves as leaders in their disciplines. The Junior and Advan- ced Fellows are selected by the director, with the assistance of the scientific Advisory Committee, by a nomination procedure: candidates are selected from a group of young researchers that are nominated by faculty members and senior researchers of universities and research institutions and are invited to apply.

Schedule for the selection of Junior and Advanced Fellows

Mid-September Target date for nominations, eligible candidates are invited to apply

Mid-October Deadline for application of nominated candidates

November Interviews with ETH members of the Advisory Committee

December Offers are made

Senior Fellows are leading international researchers in mathematics, theoretical computer science and theoretical natural sciences, spending up to a year at the Institute on a sabbatical leave from their home institutions. They dedicate their time to research and participate in the activities of the Institute and of ETH Zurich, for example by giving a course on research topics. They are invited by the Vice-President for Research and Corporate Relations of ETH Zurich on the recommendation of the Advisory Committee. Candidates are often suggested by members of the Advisory Committee or ETH faculty, but they can also apply directly.

www.ethz.ch/eth-its/fellows.html

6 Photo: Livia Kürsteiner

Rahul Pandharipande (right) introduces the ITS Colloquium speaker Claude Nicollier (left)

Activities

Meetings, talks, mini-courses In addition to our regular Fellows seminars, where the new Fellows introduce their research to the Institute, there were many other activities related to the ITS:

• Ivan Cherednik (Senior Fellow) gave a series of lectures in the ETH Math/Physics seminar on Knots and double affine Hecke algebras.

• Nicolai Reshetikhin (Senior Fellow) gave a lecture course at the FIM at ETH titled “Limit shape phenomenon in integrable models in statistical mechanics”.

Unfortunately, a major workshop on “Refined theories and beyond” in March 2020 with 27 confirmed participants from around the world was cancelled due to Covid-19. A lecture course at the FIM by Kannan Soundararajan at ETH on L-functions and multiplicative number in the Spring of 2020 was cancelled.

Yakov Eliashberg (Senior Fellow) had planned a lecture course at the ITS in the Spring of 2020 which was also cancelled.

7 Photo: Barbara Frey

Olga Sorkine-Hornung during her talk at the ITS Science Colloquium in Fall 2019

The ITS Science Colloquium

The ITS Science Colloquium aims at exposing students and researchers in mathematics, theoretical computer science and theoretical natural sciences to new questions and research subjects of common interest to different disciplines.

There were three ITS colloquia in the Fall of 2019. Kannan Soundararajan (Senior Fellow) started the colloquium series with a lecture about the distributions of the last digits of primes and a gambler’s fallacy for consecutive primes with several new results and conjectures. The second colloquium, by Olga Sorkine-Hornung (ETHZ and ITS Advisory Board member), concerned the application of ideas in differential geometry to the problem of computer animation (with various examples of dancing monsters to illustrate her algorithms). Benny Sudakov gave the last colloquium of the Fall on questions related to the probabilistic method in combinatorics, new results in graph theory, and even applications to the lottery.

We had a full program of colloquia planned for the Spring of 2020 starting with an inspirational lecture in February by Claude Nicollier (EPFL) covering almost all aspects of his remarkable career in space (with special emphasis on his role in the successful repair of the Hubble telescope). There were more than 300 people in the audience, and a very lively question/answer period occurred afterwards (with many students asking about the selection process for the European Space Agency).

Unfortunately, all the planned colloquia after Nicollier’s were cancelled due to Covid-19. However, in May, Tanja Stadler (ETHZ and ITS Advisory Board member) gave the first ITS online colloquium via livestream from Basel on her work studying the spread of Covid-19 via a mathematical analysis of the genomes of virus strains. An article titled “Die Viren wurden von vielen Orten in die Schweiz importiert” reporting on Stadler’s lecture appeared in the Tages-Anzeiger on 12 May 2020. We expect to have several online colloquia in the Fall of 2020.

8 Photo: Livia Kürsteiner

Questions from the audience after Claude Nicollier’s talk

Programme 2019–2020

Kannan Soundararajan, ETH-ITS 10.10.2019 Primes, randomness, and the gambler’s fallacy and Stanford University

12.11.2019 Olga Sorkine-Hornung, ETH Zurich Interactive Modeling of 3D Shapes from Scans and 2D Sheets

12.12.2019 Benny Sudakov, ETH Zurich Probabilistic reasoning, packing, and decomposition problems

27.02.2020 Claude Nicollier, EPFL Where failure is not an option

Understanding the epidemic spread of Covid-19 through analysis 12.05.2020 Tanja Stadler, ETH Zurich of Sars-CoV-2 genomes

Videos of selected talks can be viewed on https://eth-its.ethz.ch/activities/videos.html

9 Photo: Livia Kürsteiner

Screenshot of the first virtual ITS Science Colloquium with Tanja Stadler

Fellows’ seminar

The aim of the Fellows’ seminar, organized by Junior Fellows Nina Holden and Pierrick Bousseau in Fall 2019, is to present the research of the Fellows of the ETH-ITS. It is open to all interested and the rule is that talks should be accessible to other Fellows, which are typically from a different field.

Programme 2019–2020

15.10.2019 Dominik Schröder Hermitian Cusp Universality and Non-Hermitian Edge Universality

29.10.2019 Yi-Jun Chang Expander Decomposition in Distributed Computing

Zeta-functions of plane curve singularities, a new vintage in 19.11.2019 Ivan Cherednik a very classical field

03.12.2019 Nicolai Reshetikhin Limit shape phenomenon in statistical mechanics

10.12.2019 Stefan Glock High-dimensional equipartitions

10 Photo: Yakov Eliashberg

Former Senior Fellow Yakov Eliashberg

Awards

Pierrick Bousseau will receive the SwissMAP innovator prize for 2020.

Yakov Eliashberg, Professor at Stanford University, has been jointly awarded the 2020 Wolf Prize in Mathematics for his contributions to differential geometry and topology. He was at the ETH-ITS for a first stay in 2019 and will hopefully be back as a Senior Fellow some time in the near future.

Nina Holden received the SwissMAP Innovator Prize 2019 and the Bernoulli Society New Researcher Award 2020.

11 Graphics: Ivan Cherednik Ivan Graphics:

Covid-19 development in the Netherlands from 13 March to 22 May 2020

Fellows’ report Senior Fellows

Ivan Cherednik is a professor of mathematics at Univer- Since the motivic zeta functions are defined in terms of sity of North Carolina at Chapel Hill, a specialist in several Hilbert schemes, the focus is on Hilbert schemes and areas: representation theory, algebra, number theory, compactified Jacobians for plane curve singularities and geometry, topology, related physics, harmonic analysis, surface singularities. From the topological perspective, combinatorics. His latest theory of invariants of iterated these invariants are directly related to the Khovanov- torus links via double affine Hecke algebras (DAHAs) and Rozansky polynomials of algebraic links. In physics, the zeta-functions of plane curve singularities has attracted corresponding zeta-functions are presumably connected a lot of attention. The Riemann hypothesis for these inva- with Landau-Ginzburg theory, LGSM, for local super- riants is presumably connected with the classical theory potentials, and with the theory of spin chains. The most of zeta-functions and L-functions, but the passage from technically developed by now is the approach based plane curve singularities to isolated surface singularities on double affine Hecke algebras. The DAHA invariants is needed here. This was the main theme of his research presumably coincide with the corresponding Khovanov- and lectures at ETH-ITS during the Senior Fellowship (July Rozansky polynomials, when they are well-defined, and 2019–June 2020). He was also a Simons Fellow during with motivic zeta functions. DAHAs also provide a link to the first half of this period. SCFT, super-conformal field theory. Its connection with LGSM is one of the key themes in string theory.

Ivan Cherednik has a lot of experience in applied ma- thematics: mathematical biology, non-linear optics and more. During his stay at ETH-ITS, he adjusted his theory of momentum risk-taking to highly efficient forecasting

12 tools for the spread of Covid-19. The match is almost with of M. Gromov, A. Floer, H. Hofer C. Conley and E. Zehnder. the accuracy of physics laws. The corresponding “2-pha- One of his main line of research in this area continues to se solution” for the curves of total numbers of (detected) be an interaction between fexible and rigid phenomena infections is as follows. It is a proper Bessel function for in symplectic topology. Until a few years ago the most the first phase, mostly under “hard measures”, and some exciting development was on the rigidity side of the sub- modified power functions for the second phase, which ject, beginning with the development of Gromov-Witten is based upon the switch to “soft measures”. Both solve invariants, various flavors of Floer homology theory, the corresponding systems of differential equations. His and more recently Fukaya categorical methods related starting point was that similar modes of epidemic ma- to Kontsevich’s homological mirror symmetry conjecture. nagement result in similar curves of the total cases in However, several unexpected flexibility results were very different countries. The figure of two modes and the proven in the last 7–8 years. Highlights here include corresponding functions provided for the Netherlands is Murphy’s theory of loose Legendrian knots, Cieliebak- very typical for practically all countries that reach phase 2. Eliashberg’s theory of flexible Weinstein manifolds, and Borman-Eliashberg-Murphy’s classification of overtwis- Professor Yakov Eliashberg spent the first three months ted contact structures on manifolds of all dimension. of his Senior Fellow appointment at ITS from March – June 2019. He will return for the second half of his term During his stay at the ITS, Yakov Eliashberg interacted in 2020/2021 academic year. Yakov Eliashberg holds a with several senior and junior faculty members, post- permanent position of Herald L. and Caroline L. Ritch docs and students at the ETH, as well as a number of ITS Professor of Mathematics at Stanford University. His re- visitors, including Grigory Mikhalkin from the University search interests include symplectic and contact topology, of Geneva, Kai Cieliebak from the University of Augsburg, several complex variables, and low-dimensional topology. Soren Galatius from the University of Copenhagen, Méla- nie Bertelson from ULB, Brussels, and Gael Meigniez from Yakov Eliashberg’s work in 1980s was at the origin of a the University of Bretagne-Sud. Eliashberg also organized new subject of symplectic topology, along with the work several seminar talks at the ITS.

13 Photo: Livia Kürsteiner

Nicolai Reshetikhin in the ITS garden

While at the ITS, Yakov Eliashberg worked on sever- this project, which lies at the interface of representation al research projects. He completed a joint paper with theory and quantum integrable systems, is completed in K. Cieliebak devoted to new applications of symplectic “N-point spherical functions and asymptotic boundary topology in the theory of functions of several complex KZB equations” (with J. Stokman) arXiv:2002.02251. In variables, and a joint paper with two young Japanese the paper “Integrability of Limit Shapes of the Inhomo- mathematicians N. Ogawa and T. Yoshiyasu devoted to a geneous Six Vertex Model” (with D. Keating, A. Sridhar) problem in symplectic topology of Weinstein (or affine) arXiv:2004.08971, he demonstrated that limit shapes for symplectic manifolds. the inhomogeneous six-vertex model in statistical me- chanics on a cylinder have infinitely many conservation Several other research and exposition projects are laws. He also made substantial progress in the study of still ongoing. In particular, jointly with D. Nadler and the geometry and representation theory of spin Calogero- D. Alvarez-Gavela, Eliashberg is involved in a fundamental Moser type systems. The papers are in preparation and project of the so-called arborealization program, with the are expected to be completed in the fall 2020. goal of reformulating symplectic topology as differential topology of the so-called arboreal spaces generalizing Reshetikhin also gave a number of lectures and collo- the classical notion of a smooth manifold. Jointly with quia: a mini-course on Mathematical Physics seminar K. Cieliebak and N. Mishachev, Eliashberg continues his (organizer by G. Felder) “Integrable and superintegrable work on a graduate textbook devoted to applications of systems on moduli spaces of flat connections”, ETH-ITS the h-principle in symplectic topology. seminar “Limit shape phenomenon in statistical mecha- nics”, lecture series at Skolkovo (Moscow) “Limit shapes During his stay at ITS, Professor Nicolai Reshetikhin in statistical mechanics”; Zurich colloquium: “Limit shape completed an important part of a project on superinteg- phenomenon in statistical mechanics”. In the Spring 2020, rable systems on moduli spaces of flat connections over he taught a Nachdiploma course “Limit shape phenome- surfaces. The paper is posted on the ArXiv: “Superinte- non in integrable models in statistical mechanics”. grable Systems on Moduli Spaces of Flat Connections”, arXiv:1909.08682 (with S. Arthamonov). Another part of

14 Photo: Kannan Soundararajan Kannan Photo:

Senior Fellow Kannan Soundararajan

Kannan Soundararajan spent the period from 1 Sep- Soundararajan gave an ITS Science Colloquium in October tember 2019 to 31 March 2020 as Senior Fellow at ITS, on Primes and the Gambler’s Fallacy. He also spoke at the on sabbatical leave from his professorship at Stanford Number Theory Days meeting at ETH, and gave lectures University. He had originally intended to stay until the at Bristol, Oberwolfach, and Gottingen. end of July 2020, but Covid-19 forced a change in plans. Among the many wonderful memories of his stay at ITS, During this stay, he interacted with several mathema- Soundararajan recalls most fondly the warmth and ef- ticians at ETH, most notably Emmanuel Kowalski, Paul ficiency of the ITS staff, the welcoming and stimulating Nelson, Ian Petrow, Bill Duke (visiting FIM from UCLA), atmosphere of the ETH mathematics department, hikes and Ozlem Imamoglou. He also participated in the num- and travels in Switzerland, and learning (rudimentary) ber theory seminar at ETH, and enjoyed discussions with German. graduate students at ETH. Ming Yuan is Professor of Statistics at Columbia Uni- The outstanding working conditions at ETH-ITS enabled versity. He has a broad interest in the theory, methods him to complete two papers that had been in progress, and applications of statistics and its interface with other and also led to a fruitful new collaboration with Kowalski. quantitative fields. In particular, his recent research has One paper from the collaboration with Kowalski – Equi- primarily centered around the development and analysis distribution from the Chinese Remainder Theorem – has of flexible methods and algorithms for analyzing complex already been completed, and another is in progress. and high dimensional data – a class of problems that arise Soundararajan also began a new project with Gun and naturally in a multitude of scientific and technological Kohnen on large fundamental Fourier coefficients of half- fields, and draw interests from researchers in areas integer weight modular forms. Most of the work on this including computational harmonic analysis, electrical project was carried out at ITS, and the paper was com- engineering, high dimensional probability, machine lear- pleted shortly after the stay at ITS. He also made some ning, mathematical programming, theoretical computer progress on a book project based on his CBMS lectures science, among others. Yuan spent the spring of 2020 as a from summer 2019, but this is still in its early stages. Senior Fellow at the ETH Institute for Theoretical Studies.

15 Photo: Ming Yuan

Senior Fellow Ming Yuan joined the ITS in January 2020

While some of his planned research activities have been for these applications. Despite the impressive progress disrupted by the Covid-19 pandemic, Yuan was able to made in recent years, the current approach towards this carry out several research projects during the stay. goal is limited by the discrete nature of the classical notion of computational complexity and not suitable for Specifically, he completed a paper that develops deter- statistical problems that are often continuous. Yuan has ministic perturbation bounds for singular values and been working on a more coherent framework to address vectors of orthogonally decomposable tensors, in a spirit this challenge. Part of the progress was reported in an similar to classical results for matrices such as those by online lecture given at ETH Institute for Foundation of Weyl, Davis and Kahan. These bounds exhibit intriguing Data Science. differences between matrices and higher-order tensors. These results can be readily applied and provide a unified Finally, through collaboration with Peter Buhlmann from treatment to numerous different problems in statistics, ETH Seminar for Statistics, Yuan has started developing machine learning and numerical analysis, involving a series of statistical tools for identifying and correcting higher-order orthogonally decomposable tensors. for hidden confounding in causal inferences.

Another project Yuan has been working on aims to deve- lop an information-based approach that better accounts for both statistical and computational efficiencies and allows for the genuine trade off between the two. As more and more data are acquired, we want to address more and more complicated problems and these functions in- evitably get more and more complex. How to quantify the complexity of such tasks and learn these functions from data in an efficient way, both statistically and computati- onally, are often times the most fundamental challenge

16 Photo: Ulrike Rieß

Pierrick Bousseau (2nd left) and Ulrike Rieß (2nd right) at dinner with other conference participants in Beijing

Junior Fellows In a paper written with Shuai Guo, Honglu Fan, and Long- ting Wu, Bousseau proved quasimodularity results and an During his second year as ITS Junior Fellow, Pierrick holomorphic anomaly equation in relative Gromov-Witten Bousseau has focused on various aspects of complex theory. As a corollary, this established quasimodularity and real enumerative geometry. An unifying theme of properties of generating series of Betti numbers of mo- his research is the enumeration of algebraic curves in duli spaces of one-dimensional coherent sheaves on the algebraic varieties provided by Gromov-Witten theory. projective plane, thus proving a mathematical version of a conjecture based on string theory formulated by Huang Bousseau wrote a paper about the topology of moduli and Klemm around 2010. spaces of coherent sheaves on the projective plane. The result is a new algorithm computing topological invari- Hülya Argüz and Pierrick Bousseau wrote a paper pro- ants of these very classical objects in complex algebraic ving a correspondence theorem between counts of real geometry. This algorithm is of tropical flavour and its log curves in toric varieties and tropical curves, and in proof relies on recent advances on spaces of Bridgeland particular revisiting Mikhalkin’s correspondence theorem stability conditions, and on the Kontsevich-Soibelman between Welschinger invariants of toric surfaces and tro- wall-crossing formula in Donaldson-Thomas theory. A pical curves. It is one of the first examples of application remarkable fact is that this algorithm was previously of logarithmic techniques in real enumerative geometry. known to compute completely different invariants in Gromov-Witten theory. In a subsequent paper, the resul- Pierrick Bousseau, Andrea Brini and Michel van Garrel ting sheaf/Gromov-Witten correspondence was used to wrote a paper proving the log-local principle for toric give a proof of the Takahashi’s conjecture, a question open varieties with their toric boundaries. The log-local prin- for roughly 15 years about the multicovering structure of ciple is a general conjecture in Gromov-Witten theory. In relative Gromov-Witten theory for curves in the projective a follow-up work in progress, Pierrick Bousseau, Andrea plane meeting a given cubic curve in a single point. Brini and Michel van Garrel study the log-local principle

17 for toric surfaces with non-toric divisors and explore a Yi-Jun Chang arrived at the ITS as a Junior Fellow in July web of unexpected connections between log Gromov- 2019. In his first year, he continued working on various Witten theory of surfaces, open Gromov-Witten theory topics in distributed graph algorithms. of toric Calabi-Yau 3-folds, Gromov-Witten theory of toric Calabi-Yau 4-folds, and Donaldson-Thomas invariants of There is a recent line of work in distributed graph al- some quivers. gorithms in the CONGEST model that use expanders. All these distributed algorithms so far are based on the fol- More recently, Bousseau has been thinking about apply- lowing two tools: expander decomposition and expander ing techniques from enumerative geometry to positivity routing. Roughly speaking, an expander decomposition is questions in low-dimensional topology. a partition of a graph into well-connected clusters such that the number of inter-cluster edges is small, and ex- Bousseau spoke about his research during lecture series pander routing allows each vertex v in a well-connected given in Beijing for the Beijing-Zurich moduli workshop, cluster C to very quickly exchange deg(v) messages with and in Kyoto as part of the 5th KTGU workshop. He also any other vertices in C, not just its local neighbors. Chang presented his work at several international conferences: and his collaborator Thatchaphol Saranurak (Toyota in Cambridge for a workshop on mirror symmetry and Technological Institute at Chicago) gave the first efficient cluster algebras, in Nantes for a European conference deterministic distributed algorithms for both expander on tropical geometry, and in Bonn for a conference on decomposition and expander routing. As a consequence, perverse sheaves in enumerative geometry. In addition, several existing randomized distributed algorithms that he spoke at regular research seminars at University of use expanders can be derandomized. For example, a mi- Angers, Sheffield University, Warwick University, Geneva nimum spanning tree on a well-connected graph can be University, Heidelberg University, and at online seminars constructed in sub-polynomial rounds deterministically, organized by King’s College London, SISSA in Trieste, the and the triangle detection problem can be solved in sub- University of Edinburgh, and University of California San linear rounds deterministically. This work was accepted Diego. to the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2020).

18 Photo: Yi-Jung Chan Photo: Yi-Jung

Junior Fellow Yi-Jun Chang joined the ITS in July 2019

Since 2016, Chang has been working on a line of research graph problems. In the trivial BFS algorithm, each vertex that aims to understand the distributed complexity of lo- v has to be awake for all time slots listening to potential cally checkable graph problems. His latest work in this messages until the wavefront of the BFS hits v. Chang research direction is a joint work (arxiv:2002.07659) with and his collaborators Varsha Dani and Thomas P. Hayes Jan Studený and Jukka Suomela (Aalto University). They (University of New Mexico) and his PhD advisor Seth Pet- considered the following algorithm synthesis question. Is tie (University of Michigan) showed that it is possible to it possible to design an efficient algorithm that, given the reduce the number of awake time slots per vertex to sub- description of a locally checkable graph problem, outputs polynomial, at the cost of increasing the round complexity an asymptotically optimal distributed algorithm in the by a sub-polynomial factor. This work was accepted to LOCAL model of distributed computing? It is well-known the 2020 ACM Symposium on Principles of Distributed that this problem is undecidable for general graphs. More Computing (PODC 2020). recently, it was shown that this problem is PSPACE-hard even for paths and cycles with input labels or unlabeled Chang participated in several workshops in 2019. In July, trees. Therefore, to aim for polynomial-time algorithms, he gave a talk on distributed coloring in the Workshop it is necessary to focus on unlabeled paths and cycles. on Local Algorithms (WOLA 2019). In November, he gave Chang and his collaborators showed that all questions a talk on expander decompositions in the Shonan Mee- about solvability and round complexity of locally checka- ting on Distributed Graph Algorithms. He was a keynote ble graph problems on unlabeled paths and cycles can speaker at the inaugural IGAFIT Workshop for Algorithms be decided in polynomial time, except for some specific Postdocs in Europe (AlgPiE by IGAFIT 2019). questions on paths that are NP-complete.

Chang also worked on energy-aware distributed compu- ting. Energy is a constrained resource for battery-powered wireless devices, and most of the energy is often spent on sending and receiving packets, not on computation. Breadth-first search (BFS) is one of the most fundamental

19 Photo: Stefan Glock

Stefan Glock working from home during the Covid-19 lockdown

Stefan Glock started as a Junior Fellow at ITS in Septem- average number of conflicts. One special case of their ber 2019. He works in combinatorics, a rapidly growing general result settled a problem that remained unsolved field of modern mathematics with close ties to theo- since 1994. retical computer science. At ETH, he established new research collaborations, in particular with the group of Eight papers which Glock has coauthored were accepted Professor Benny Sudakov. by journals in the last year, including one which will appear in the Memoirs of the American Mathematical In one recent paper, Glock and Sudakov discovered a Society. novel average degree condition for the existence of in- dependent transversals. Loosely speaking, this can be Glock presented his research at several occasions. For described as follows: suppose we are given n groups example, he was an invited sectional speaker at the of objects, and from each group we want to select one annual meeting of the German Mathematical Society in “representative”. However, there are certain restrictions Karlsruhe, and spoke at the Mathematics Colloquium of on which objects can be selected together, modeled by the University of Ulm. Stefan was also invited to talk at a graph whose vertices are all objects, and two objects a SIAM conference, a conference in Moscow (for which are connected by an edge if they have a conflict and he organized a minisymposium), and the European Con- therefore cannot both be representatives of their groups. gress of Mathematics in Slovenia. Unfortunately, due The challenge is now to find a set of representatives to the Covid-19 pandemic, these events were either that form an independent set in this graph. Previously, cancelled or postponed. sufficient conditions for the existence of such an “in- dependent transversal” were formulated in terms of In addition to his research, Glock supported the teaching the maximum number of conflicts one object can have. activities of the mathematics department of ETH, by However, this is quite wasteful, since there might just teaching an exercise class for the course “Graph The- be one object with many conflicts. The new insight in the ory”, supervising a semester project and supervising a work of Glock and Sudakov is that it suffices to bound the Master student.

20 Figures: Nicolas Curien Nicolas Figures:

The left figure shows a random planar map (RPM), which is the combinatorial object obtained by gluing together a collection of triangles uniformly at random. The right figure shows the RPM embedded (i.e., drawn) on the Euclidean sphereS ². It is possible to define embeddings which can be viewed as discrete approximations of conformal maps.

Nina Holden spent her second year at the ITS during the During the fall semester Holden arranged regular proba- 19/20 academic year. During this year she finalized a se- bility lunches at the ITS. The participants first eat pizza ries of papers with Xin Sun and other collaborators about and then one member of the probability group presents conformal embedding of random planar maps (RPM). A recent work or a topic of interest for the rest. In fall Nina RPM is a natural model for a discrete random surface also supervised a reading course for a master student studied in combinatorics, probability, theoretical physics, on Liouville quantum gravity and imaginary geometry. and geometry. In string theory and conformal field theory, RPM are viewed as discrete versions of the continuum Holden has presented her work at several occasions, random surfaces known as Liouville quantum gravity including seminars in Princeton, Lausanne, Münster, (LQG) surfaces. In recent years there has been a large and Bochum. She has also given talks at the SwissMAP interest among probabilists in establishing rigorous rela- General Meeting, a workshop on Mathematical Physics in tionships between these two natural models for random Les Diablerets, the One World Probability Seminar, and surfaces. Together with her collaborators, she proves that the Bernoulli-IMS One World Symposium. uniformly sampled RPM converge in the scaling limit to LQG under a particular discrete conformal embedding.

Holden has also worked on a project about critical LQG together with Juhan Aru, Ellen Powell, and Xin Sun. They prove that certain natural observables of a so-called criti- cal LQG surface decorated by a conformal loop ensemble (CLE) are given by independent Brownian motions. CLE is a random collection of loops which describes the sca- ling limit of interfaces in statistical physics models, and CLE-decorated LQG describes the scaling limit of such interfaces on a RPM.

21 Photo: Shoham Letzter

Shoham Letzter at a workshop in Banff, Canada

In her last few months as a Junior Fellow, Shoham Letz- non-adaptive algorithms – where all queries have to be ter continued her work with computer scientists Omri determined in advance. The group showed that for k at Ben-Eliezer from Tel Aviv University and Erik Waingarten least 4 this is no longer the case: while the complexity from Columbia, on a problem related to property tes- of the fastest adaptive algorithm does not depend on k ting. Property testing is a general name for a collection (it is, up to constant factors, of order log(n)), in the non- of problems where a large object is given – such as a adaptive case the complexity grows with k (in a previous sequence or a graph – and the aim is to determine, with work, joint with Clément Cannone, the group determined high probability, whether the object has a certain property the complexity of the latter problem). or is far from having the property. This can be motivated by real-life situations: given a very large object, it might Interestingly, the notion of property testing featured in be too time-consuming to read, let alone process, the Letzter’s recent work, together with Vytautas Gruslys, entirety of the object. Instead, it is often preferable to run where the two proved that given any complete graph on n fast randomised algorithms that are likely to analyse the vertices, whose edges are coloured red and blue, there is object correctly. a collection of approximately n^2/12 red or blue triang- les, no two of which have an edge in common. This result In this project, the object in question is a sequence of n confirms a conjecture of Erdös and is best possible, as numbers, and the aim is to determine if the sequence con- seen by a complete graph whose vertices are divided into tains many disjoint increasing subsequences of length k, two roughly equal parts, whose edges within the parts or if it has no such subsequence. A well-studied special are coloured red, and whose edges between the parts are case of this is when k equals 2, in which case this pro- coloured blue. A result about property testing, due to Alon, blem translates into distinguishing between decreasing Shapira and Sudakov, allowed the two to prove a ‘stability sequences and sequences that are far from decreasing. version’ of their result. Namely, they showed that a red- Interestingly, in this special case, it is well-known that blue complete graph either has significantly more than adaptive algorithms – which consist of a sequence of the above number of disjoint red or blue triangles, or it is queries that are allowed to depend on outcomes of pre- close to the aforementioned example. vious queries – do not outperform the more restrictive

22 Photo: Barbara Frey

Rahul Pandharipande and Johannes Noller after the ITS Science Colloquium

Letzter assisted in the organisation of the 2019 Random Noller then turned his attention to the new fascinating Structures & Algorithms conference, which was held in window of gravitational wave science (https://arxiv.org/ ETH Zurich. She also participated in conferences in Banff, abs/1911.11671). They showed that the waves emitted af- Birmingham, and Oberworlfach, and presented her work ter a violent collision of black holes, which will be detecta- in the former two. ble with new generations of gravitational wave detectors, can carry imprints of new physics related to the observed Upon leaving ITS, in January 2020, Letzter joined the accelerated expansion of the Universe. This is an exciting University of Cambridge as a Royal Society University new way of linking tests from small, local objects like Research Fellow. Shortly afterwards, she accepted an black holes with the enormous length and distance scales offer to become a lecturer in University College London, probed by cosmology, hopefully leading to further insights where she is now based. as more data becomes available. In a similar vein, Noller derived current state-of-the-art observational constraints During his final year at the ITS, Johannes Noller worked on deviations from General Relativity on cosmological on several projects, both in continuation of his previous scales (https://arxiv.org/abs/2001.05469), by taking into work while at the ITS and with an eye on opening up new account several qualitatively new bounds from gravita- research avenues, also establishing new collaborative tional wave constraints for the first time. Together these links in the process. More specifically, with Pedro Fer- afore-mentioned papers form part of Noller’s main re- reira, Christopher Hill and Graham Ross, he established search programme, that aims to establish fully integrated the observational signatures of the so-called scale- observational and theoretical constraints on gravitational independent R^2 model of inflation (https://arxiv.org/ physics. Finally, also as part of this overall effort, with abs/1906.03415), a new candidate explanation for the Lavinia Heisenberg and Jann Zosso, Noller investigated formation of structure in the early Universe that acts as quantum corrections in Horndeski theories (https://arxiv. an initial seed for the formation of galaxies, stars and org/abs/2004.11655), one of the current most popular eventually planets as the Universe evolves. With Luca candidates for new physics linked to the accelerated Santoni, Enrico Trincherini and Leonardo Trombetta, expansion of the Universe, showing that a larger class

23 Photo: Michael Rieß

Ulrike Rieß working from home

of theories than previously expected is protected from (smooth) irreducible symplectic varieties. In a joint project catastrophically large quantum corrections. with Grégoire Menet (Grenoble), she described the struc- In the past year, Noller has also presented his work in ture of the Kähler cone for irreducible symplectic orbi- seminars and colloquia at the Stanford University, UC folds, thus generalizing the results previously only known Berkeley, the University of Portsmouth, the Albert Ein- for the smooth case. Furthermore, they also provided a full stein Institute in Potsdam and the University of Kent and at description of these wall divisors for an interesting class several conferences and workshops, including the “Gravity of irreducible symplectic orbifolds. and Cosmology” workshop in Warsaw, the COSMO 19 con- ference in Aachen, the “Lost in Gravity 2019” conference in The second main line of her research is the study of base Saint Flour and the GR 22 conference in Valencia, resulting loci of ample line bundles on irreducible symplectic vari- in several new and ongoing collaborations. Noller has re- eties. Building on works, where Rieß described base divi- cently moved to the University of Cambridge to take up sors for these line bundles and proved generic base point an Ernest Rutherford Fellowship there, but hopes to have freeness for certain polarized irreducible symplectic vari- frequent occasions to visit the ITS in the future. eties of dimension 4, she continues to investigate further aspects of these base loci. In particular, in a cooperation Junior Fellow Ulrike Rieß spent in 2019/2020 her third with Daniele Agostini (Berlin) she could extend the study of year at ITS. She is working in algebraic geometry with generic base point freeness to higher dimensions and they a focus on irreducible symplectic varieties. This class of could show base point freeness in many more examples of varieties has been of high interest ever since Beauville polarized irreducible symplectic varieties of dimension 4. described remarkable properties and gave two series of examples in 1983. Rieß supervised master’s student Mauro Varesco, who successfully completed his thesis in March 2020 and One principal line of work Rieß pursued, is an investigation subsequently started a PhD in Bonn. The results, which of irreducible symplectic orbifolds. This mildly singular he obtained under Rieß’ supervision are currently formed class of varieties is a generalization of the classical into his first publication.

24 Photo: Dominik Schröder

Dominik Schröder giving a talk in Aarhus, Denmark

In June 2019 the “Japanese European Symposium on Dominik Schröder started as a Junior Fellow at ITS in Symplectic Varieties and Moduli Spaces”, which Rieß is September 2019 shortly after obtaining his PhD from IST regularly (co-) organizing, took place in Zürich. This annu- Austria. During the last year his research was focused al conference on topics related to irreducible symplectic on universal aspects of large random matrices, as well varieties, brings together young researchers in this area possible applications for the analysis of deep neural from all around the world (with a focus on Europe and Ja- networks. pan). This year’s edition of the conference was postponed to 2021, while there will be a reduced digital replacement. In two joint works with László Erdös and Giorgio Cipolloni from Vienna, Schröder obtained a central limit theorem After a time, where Rieß tried to significantly restrict travel (CLT) for the linear statistics of non-Hermitian random activities due to her small son, she restarted presenting matrices with independent entries. Since the 1980’s and her work in many different places this year. Trips for fundamental work of Ginibre and Girko the eigenvalues conferences and invited talks included Beijing, Lausanne, of such matrices have been known to be asymptotically Cambridge, Paris, and many more. After the beginning of uniformly distributed in a disk of the complex plane. In the the Corona pandemic, she started using a graphic tablet last two decades the deviations from this ‘circular law’ in during video calls to intensify long distance collaborations. the sense of a CLT have been studied in a series of works This will certainly still be very useful, even when restric- when either the test functions are analytic or the matrix tions of scientific travel activities are no longer needed. elements are Gaussian. Schröder and his collaborators finally solved the general case by analysing weakly cou- pled Brownian motions and managed to prove that the deviation from the circular converges to a variant of the Gaussian Free Field (GFF). The GFF has been subject of intensive research in the last decade and the emergence in the context of large random matrices is a further ma- nifestation of its universal significance.

25 Photo: Livia Kürsteiner

Albrecht Klemm (left) and Rahul Pandharipande at the annual committee meeting in 2019 at Villa Hatt

Besides several continuation projects on the purely ma- for practical applications. In an ongoing project Schröder thematical aspects of random matrices, Schröder has is working on the neural tangent hierarchy, a system of also started working on possible applications to machine differential equations describing the gradient descent dy- learning, and more specifically neural networks. Together namics of the NTK with the aim of explaining the observed with Vanessa Picollo, a MSc student at ETH, Schröder performance discrepancy. extended recent results on non-linear random matrices occurring in neural networks. In each layer of a neural Schröder presented his research on several occasions, network the outputs of the previous layer are multiplied including an Oberwolfach workshop, and research semi- by weights upon which a bias is added and a non linear nars organised by the universities of Geneva, Erlangen ‘activation function’ is applied. Initially both weights and and Melbourne. biases are chosen randomly and are then optimised in the training phase. Previous works by Pennington-Worah and Benigni-Péché had derived analytic expressions for the expected training error of random data in the case without bias, a restriction which could be removed in the MSc thesis supervised by Schröder.

Recently Schröder also began working on the neural tangent kernel (NTK) which recently has received much attention in the theoretical study of deep neural networks. In the so-called infinite-width limit deep neural networks with non-linear activation functions simplify to linear mo- dels with the NTK kernel. While being more susceptible to a mathematical analysis this limiting linear model ac- tually performs worse than its finite width counterpart

26 Photo: Barbara Frey

Members of the ETH-ITS enjoying lunch after the hike in September 2019

27 1 2

Outlook The ETH-ITS is welcoming three new Junior Fellows in September 2020.

1 Alexandru Gheorghiu received his PhD in Computer Science from the University of Edinburgh in September 2018, under the supervision of Elham Kashefi. His main research interests are in quantum computation and cryptography, with a focus on verification of quantum computations. He is also interested in computational com- plexity theory and foundations of quantum mechanics. In his dissertation, he investigated and designed robust protocols for the verification of quantum computations.

2 Christoph Kehle received his PhD in mathematics from the University of Cambridge in 2020 under the supervisi- on of Mihalis Dafermos. He is interested in analysis and partial differential equations, specifically in the context of general relativity. His thesis revealed a novel connection of Diophantine approximation arising from small divisors to general relativity, more precisely, the Strong Cosmic Censorship conjecture.

28 People at the ETH-ITS

Director Rahul Pandharipande

Administration Livia Kürsteiner Barbara Frey

Board of Patrons Martin Haefner, Walter Haefner Foundation Dr. Max Rössler Prof. Dr. Ralph Eichler

Advisory Committee

2013–2017 ETH members Gianni Blatter, Matthias Gaberdiel, Rahul Pandharipande, Tristan Rivière, Angelika Steger, Nicola Spalding* External members Noga Alon (Tel Aviv and Princeton), Luis Alvarez-Gaumé (CERN), Artur Ekert (Oxford), Shafira Goldwasser (MIT and Weizmann Institute), Gerhard Huisken (MFO), Elon Lindenstrauss (Hebrew University), Martin Hairer (Warwick) *replaced by Ueli Maurer in 2015

2017–2019 ETH members Alessio Figalli, Matthias Gaberdiel, Ueli Maurer, Rahul Pandharipande, Manfred Sigrist, Angelika Steger External members Noga Alon (Tel Aviv and Princeton), Robert Brandenberger (McGill), Artur Ekert (Oxford), Gerhard Huisken (MFO), Elon Lindenstrauss (Hebrew University), Martin Hairer (Imperial College), Claire Voisin (Collège de France)

2019–2020 ETH members Alessio Figalli, Renato Renner*, Manfred Sigrist, Olga Sorkine-Hornung*, David Steurer*, Tanja Stadler* External members Robert Brandenberger (McGill), Ben Green (Oxford University)*, Albrecht Klemm (University of Bonn)*, Nathan Linial (Hebrew University of Jerusalem)*, Madhu Sudan (Harvard University)*, Corinna Ulcigrai (University of Zurich)*, Claire Voisin (Collège de France)

*new members

29 Fellows 2014–2020

Senior Fellows

Terry Hwa, UC San Diego 02.2014 – 09.2014 and 04.2015 – 08.2015 Gilles Brassard, Université de Montréal 06.2014 – 12.2014 and 04.2019 – 06.2019 Henryk Iwaniec, Rutgers University 08.2014 – 05.2015 and 06.2016 – 07.2016 Dmitry Chelkak, St. Petersburg 09.2014 – 08.2015 Alex Lubotzky, Hebrew University 02.2015 – 07.2015 and 02.2016 – 08.2016 Adi Shamir, Weizmann Institute 02.2015 – 07.2015 and 02.2016 – 07.2016 Eugene Demler, Harvard University 05.2015 – 06.2015 and 09.2015 – 12.2015 Robert Brandenberger, McGill University 08.2015 – 07.2016 Walter Schachermayer, University of Vienna 08.2015 – 08.2016 Riccardo Barbieri, SNS Pisa 11.2015 – 10.2016 Alexander Balatsky, Nordita and LANL 02.2016 – 04.2016 and 07.2016 – 03.2017 Eitan Tadmor, University of Maryland 08.2016 – 07.2017 Vadim Kaloshin, University of Maryland 09.2016 – 08.2017 Jean-Michel Coron, Université Pierre et Marie Curie 01.2017 – 12.2017 Claire Voisin, Collège de France 01.2017 – 12.2017 Gerhard Huisken, MFO 02.2017 – 09.2017 Leonid Glazman, Yale University 05.2017 – 12.2017 and 05.2018 – 08.2018 Sandu Popescu, University of Bristol 10.2017 – 12.2017 and 03.2018 – 12.2018 Yakov Eliashberg, Stanford University 06.2019 – 09.2019 Ivan Cherednik, University of North Carolina at Chapel Hill 07.2019 – 06.2020 Nicolai Reshetikhin, UC Berkeley 07.2019 – 07.2020 Kannan Soundararajan, Stanford University 09.2019 – 03.2020 Ming Yuan, Columbia University 01.2020 – 08.2020

Junior Fellows (with current affiliation of former Junior Fellows)

Emily Clader, University of San Francisco 09.2014 – 07.2016 Zur Luria, Azrieli College of Engineering 09.2014 – 09.2017 Alessandro Carlotto, ETH Zurich 09.2015 – 08.2016 Maria Colombo, EPF Lausanne 09.2015 – 08.2018 Lavinia Heisenberg, ETH Zurich 09.2015 – 09.2018 Titus Lupu, CNRS, U. Paris 11, Orsay 09.2015 – 12.2017 Aline Ramires, ICTP-SAIFR, Sao Paulo 09.2015 – 08.2018 Ran Tessler, Weizmann Institute 09.2015 – 10.2018 Shoham Letzter, University of Cambridge 09.2016 – 12.2019 William Sawin, Columbia University 09.2016 – 07.2018 Ulrike Rieß 09.2017 – 08.2020 Johannes Noller, University of Cambridge 09.2017 – 03.2020 Pierrick Bousseau 09.2018 – 08.2021 Nina Holden 09.2018 – 08.2021 Fanny Yang, ETH Zurich 09.2018 – 09.2019 Yi-Jun Chang 07.2019 – 07.2022 Stefan Glock 09.2019 – 08.2022 Dominik Schröder 09.2019 – 08.2022 Christoph Kehle 09.2020 – 08.2021 and 09.2022 – 08.2024 Alexandru Gheorghiu 09.2020 – 08.2023

30 Contact

ETH Zurich ETH Institute for Theoretical Studies Clausiusstrasse 47 8092 Zurich www.ethz.ch/eth-its

Publisher: ETH Institute for Theoretical Studies Editor: Rahul Pandharipande Layout: grafikvonfrauschubert Print: ETH Druckzentrum

© ETH Zürich, September 2020

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