Spring 2019 • Issue 8 Department of

From the Chair Professor Allan Sly Receives MacArthur Fellowship

Congratulations to Sly works on an area of probability retical computer science, where a key the Class of 2019 theory with applications from the goal often is to understand whether and all the finishing physics of magnetic materials to it is likely or unlikely that a large set graduate students. computer science and information of randomly imposed constraints on a Congratulations theory. His work investigates thresh- system can be satisfied. Sly has shown to the members of olds at which complex networks mathematically how such systems of- class of 2018 and suddenly change from having one ten reach a threshold at which solving new Ph. D.s who set of properties to another. Such a particular problem shifts from likely are reading Fine Letters for the first questions originally arose in phys- or unlikely. Sly has used a party invi- time as alumni. As we all know, the ics, where scientists observed such tation list as an analogy for the work: Math major is a great foundation for shifts in the magnetism of certain As you add interpersonal conflicts a diverse range of endeavors. This metal alloys. Sly provided a rigorous among a group of potential guests, it is exemplified by seventeen '18's who mathematical proof of the shift and can suddenly become effectively im- have gone to industry and seventeen a framework for identifying when possible to create a workable party. to grad school; ten to advanced study such shifts occur. in math and seven in CS, Economics, “How long until I can no longer find Philosophy or Biostatistics. His work has extended into theo- 40 people who will all be OK in the ...continued on page 7 This academic year we welcomed Instructors Kenneth Ascher, Clark Butler, Remy van Dobben de Bruyn, Theo Drivas, Jiequn Han *18, Casey Kelleher, Chao Li, Boyu Zhang, and Ian Zemke, and assistant professor Aleksandr Logunov. We are very happy to report that twelve instruc- tors and assistant professors have accepted offers to join our ranks in upcoming years, including six wom- en. In addition, Chenyang Xu *08 accepted our professorial offer starting Fall 2020.

Two extraordinary members of our Courtesy of the John D. & Catherine T. MacArthur Foundation community passed away in December. The Department lost Professor Emer- itus Elias Stein, who made many deep Inside this issue: and influential research contribu- Faculty Memorials — Pages 4–5 tions. He was an unusually effective and popular mentor and teacher at all Honors and Awards — Pages 6–7 levels, as well as a master expositor Tips on Teaching — Page 8 whose books are classics. As a citizen, The Mathematics of Black Holes — Page 9 Professor Stein was an essential figure in all aspects of the Math department RTG Summer Programs — Page 10 ...continued on next page Mentoring — Page 11 From the Chair from the Chair ...continued from previous page for many years, including two terms as demics and non-academic employment Events Specialist, won the President's Chair. We also lost , a alike. Some of the independent research Achievement Award, the University's Professor at the Institute for Advanced leads to publications in excellent jour- highest recognition for a staff member. Study. An extraordinarily prolific and nals, or serves as an entrée to coveted Her activities far exceed her job title. creative , he had many positions in the business, non-profit, or Gale also mentors newer staff, offers ties to our Department. government sectors. her ear to students and faculty, greets visitors, and when the elevators go out The Department's Minerva Program From graduate students to postdocs to she dashes from the B-level to the 12th had its most active year with three junior faculty to long-time professors, floor helping Fine Hall members. Minerva Distinguished Visitors: we are all striving for essentially the Aaron Naber*09, Melanie Wood*09 same goals; solving the most interest- This academic year concludes my seven and Xinwen Zhu in the fall. Hugo ing and fundamental mathematical years as Chair of the Department. Duminil-Copin delivered our Minerva problems and/or finding new directions Being Chair has been an amazing Lecture Series in the spring. All were and connections. For some of these experience. Watching undergrads and very engaged with our students and efforts, our faculty have won major grad students run with opportunity, faculty and gave beautiful, well-received recognitions. These include Allan and working with extraordinary col- research lectures. Sly (MacArthur Fellowship), Michael leagues, has been incredible. Among Aizenman (Poincaré Prize), Weinan many other things, the Department has In addition to busily learning math- E (Henrici Prize), Aleksandr Logu- undergone generational change and ematics in the classroom, our under- nov (Salem Prize) and Francesco Lin the Common Room is once again an grads are learning to think and explore (Sloan Fellowship). We are happy to energetic center of activity. Whatever independently, first through their junior report that Visiting Senior Lecturer has been accomplished these last seven independent work and then through Karen Uhlenbeck won the 2019 Abel years is due to ever ready colleagues their senior theses. All juniors partici- Prize, *02 won a who step up when called, staff that go pate in a seminar, which sharpens the 2018 , and Visiting Asso- above and beyond, and our Nassau Hall ability to pursue topics independently ciate Research Scholar June Huh won a administration that has steadfastly sup- and to communicate their knowledge 2018 New Horizons Prize. ported our efforts. I am grateful to all effectively, important skills in both aca- of you! Igor Rodnianski will become We take teaching very seriously, and Chair as of July 1, 2019. Igor's out- Department Administration recognized Nicolas Boumal and Evita standing leadership as Acting Chair in Chair Nestoridi with Departmental Junior the 2015-16 academic year bodes well David Gabai Faculty teaching awards, and Maggie for the future. Miller with the Departmental Graduate Associate Chair Student teaching award. We are very Special thanks go to the Fernholz Foun- János Kollár (Fall) pleased that Hansheng Diao won his dation, the Class of 1971 Endowment Christopher Skinner (Spring) fourth consecutive Engineering Council and Wu-Chung Hsiang and Vicky Teaching Award and David Villalobos- Kwoh Ching for their generous support Departmental Representative Paz won one of seven Graduate School of the Department. János Kollár (Fall) Ana Menezes (Spring) Teaching Awards. Jennifer M. Johnson I hope that you can attend this year's Our extraordinary staff works behind Alumni Reception, 2:00 PM on Friday, Director of Graduate Studies the scenes to support the teaching May 31 in Fine Hall's Common Room. Javier Gómez Serrano and research mission of the Depart- Reunions provide an opportunity not Zoltán Szabó ment. I'd like to take this opportunity only for you to connect with former to thank Michelle Matel, who is now classmates and faculty, but to also meet Senior Advisors our Undergraduate Administrator, our current students and hear about Zeev Dvir for her six years of superb service as their projects. One of the highlights of John Pardon Assistant to the Chair and Department being chair is hearing from our alumni. Junior Advisor Manager. We welcome Kristie Dacey Please share your thoughts and stories! Mark McConnell who is taking on the Assistant role. We are delighted that Gale Sandor, our David Gabai *77 *80 PlacementPage 2 Officer [email protected] Ana Menezes Faculty Appointments 2018-19 Academic Year Kenneth Ascher, Instructor Chao Li, Instructor Algebraic & Kenneth Ascher received his Ph.D. from Brown University in 2017 after Chao Li completed his Ph.D. at Stan- completing his undergraduate work at ford University in 2018 after received Stony Brook. Before coming to Princ- his bachelors degree from Peking eton, Ascher was an NSF Postdoctoral University in 2012. Li received the Fellow at the Massachusetts Institute of Chinese National Scholarship in 2012 Technology during the 2017-18 aca- in addition to being named Outstand- demic year. ing Student of Peking University for three consecutive years. Clark Butler, Veblen Research Instructor Aleksandr Logunov, Assistant Dynamical Systems Professor Clark Butler completed his Ph.D. Univer- Analysis and PDEs sity of Chicago in 2018 and his B.S. at Joins our department after spending State University in 2012. While at the 2017-18 academic year as a mem- the Butler received ber at the Institute for Advancecd an NSF Graduate Research Fellowship as Study and working as a Postdoctoral well as that school's Harper Dissertation Fellow at Tel-Aviv University from Fellowship. 2015-17. Logunov received both his graduate and undergraduate degrees Remy van Dobben de Bruyn, Veblen from St. Petersburg State University. Research Instructor Algebraic Geometry Boyu Zhang, Instructor Remy van Dobben de Bruyn did his Symplectic Geometry & Low- undergraduate work at Universiteit dimensional Topology Leiden, followed by studies at the Boyu Zhang completed his Ph.D. and Uni- at in 2018 and versité Paris-Sud XI before moving his undergraduate degree at Peking to Columbia University, where he University in 2013. While at Harvard, completed his Ph.D. in 2018. Zhang received their Graduate School of Arts and Sciences Merit Fellowship. Theo Drivas, Instructor Analysis Theo Drivas received his Ph.D. in Ian Zemke, Instructor 2017 from Johns Hopkins University Symplectic Geometry & Low- after completing his undergraduate dimensional Topology work at the University of Chicago. Ian Zemke completed his Ph.D. in Drivas was an NSF Postdoctoral Fel- 2017 at the University of California, low in our department before being Los Angles after his undergraduate promoted to Instructor this year. studies at the University of Washing- ton, Seattle. Zemke joined our depart- ment in 2017 as an NSF Postdoctoral Jiequn Han, Instructor Fellow before being promoted to In- Scientific Computing, Machine structor this year. Learning Jiequn Han did his doctoral work at Upcoming Appointments Princeton's Program in Applied and Daniel Álvarez-Gavela, Instructor Computational Mathematics, receiv- ing his Ph.D. in 2018. Han completed Chiara Damiolini, Instructor his undergraduate work in math- ematics with a minor in economics at Duncan Dauvergne, Instructor Peking University. Bariş Kartal, Instructor Joaquin Moraga, Instructor Casey Kelleher, Instructor Evita Nestoridi, Assistant Professor Differential Geometry & Geometric Analysis Sarah Peluse, Veblen Research Instructor Casey Kelleher received her Ph.D. in 2017 from the University of Califor- Sophie Spirkl, Instructor nia, Irvine after her undergraduate Maxime van de Moortel, Instructor work at the California Polytechnic State University at San Luis Obispo. Hong Wang, Instructor Before being promoted to Instructor Jingwei Xiao, Veblen Research Instructor Kelleher was an NSF Postdoctoral Fellow in our department. Andrew Yarmola, Instructor Page 3 Faculty Memorials Elias M. Stein 1931—2018 by Emily Aronson, Office of Communications

Elias Menachem Stein, a Princeton ago still play an important role in current teacher, he was extremely popular with University mathematician for more than research, and he kept on proving new students. And as a citizen, he was an 50 years, died on Dec. 23 from complica- theorems almost to the end of his life.” essential figure in all aspects of the Math tions related to mantle cell lymphoma. He department for many years.” Before becoming his colleague in the was 87. Math department, Fefferman was one of Stein became an emeritus professor in Stein, the Albert Baldwin 2012, though he contin- Dod Professor of Math- ued to teach some classes. ematics Emeritus, was In his 70s, he helped a pioneer in the field of create a series of advanced harmonic analysis, an area undergraduate math of mathematics that has courses and co-wrote a applications throughout four-volume textbook to the sciences. accompany the courses.

For his contributions to Upon his transfer to the field, Stein received emeritus status, col- the 2002 National Medal leagues wrote in tribute: of Science. Other awards “Eli’s combined influence include the Wolf Prize, as a researcher, collabora- one of the highest honors tor, teacher and expositor in mathematics; the is unmatched. …His Schock Prize, which is lectures are characterized given by the Royal Swed- by perfect clarity, con- ish Academy of Sciences; and a Lifetime centration on essentials and impeccable Stein’s Ph.D. students. Achievement Award from the American taste. In his interaction with students and Mathematical Society in recognition of “Eli’s mathematical descendants — his co-workers he has managed to convey the Stein’s fundamental contributions to dif- Ph.D. students, their Ph.D. students and strong sense of optimism that is essential ferent branches of . so on — number over 600,” Fefferman for mathematical discovery. He has been said. “He inspired generations of stu- a major influence on many lives.” Stein joined Princeton’s Department of dents and researchers. He will be sorely Mathematics in 1963 and twice served as Stein held a life-long view of mathematics missed.” department chair. as a brilliant balance of imagination and Stein was equally passionate about his rational investigation, said his daughter, “His death is a huge loss to the depart- research as he was about teaching gradu- Karen Stein. ment, to mathematics and to Princeton,” ate and undergraduate students. said Department Chair David Gabai, the He pioneered work in harmonic analysis, Hughes-Rogers Professor of Mathemat- “By so many different measures Elias a field of mathematics originally devel- ics. Stein was truly an incredible and extraor- oped to understand the flow of heat and dinary mathematician,” Gabai said. “As the vibrations of a string. He also pio- , the Herbert E. Jones, a research mathematician, he did incred- neered applications of harmonic analysis Jr. ’43 University Professor of Math- ible and influential work. He was an in other fields of mathematics. These ad- ematics, said Stein was “a great math- extraordinary mentor; many of his Ph.D. vances led, in turn, to deeper understand- ematician, a great teacher and a great students have become extraordinary lead- ing of a broad range of subjects including human being. It sounds like hype, but ers in mathematics. As an undergraduate the stock market, gravitational waves and it is true. His discoveries from 50 years

Page 4 Faculty Memorials sound recordings. including “Singular Integrals and Dif- Stein is survived by Elly, his wife of 59 ferentiability Properties of Functions” years; a brother, Daniel; a son, Jeremy, Born in Belgium in 1931, Stein and his Princeton Class of 1983, the Moise Y. family fled the country after the German Safra Professor of Economics at Harvard invasion during World War II. They He has managed to convey University and a member of the Board eventually immigrated to the United the strong sense of opti- of Governors of the U.S. Federal Reserve States and settled in , mism that is essential for from 2012-14; a daughter Karen, Princ- where Stein served as captain of the math mathematical discovery. eton Class of 1984, an architecture critic team at Stuyvesant High School. and former member of the jury of the He earned his undergraduate and and “Harmonic Analysis: Real Variable Pritzker Architecture Prize; a daughter- graduate degrees from the University of Methods, Orthogonality, and Oscillatory in-law, Anne; and grandchildren Carolyn, Chicago, and held positions there and at Integrals.” He also served on the editorial Alison and Jason. the Massachusetts Institute of Technol- boards of the premiere journal Annals of A conference honoring Stein's work will ogy before coming to Princeton. Mathematics and several of Princeton’s be held next year. book series. Stein was the author of several books,

Jean Bourgain 1954—2018 Honoring the work of Jean Bourgain, IBM von Neumann as "a giant in the field of math- John N. Mather Professor in the School of Mathemat- ematical analysis, which he has The department hosted a confer- ics at the Institute for Advanced applied broadly and to great effect. ence October 1-3, 2018 to honor the Study and a Visit- In many instances, he memory and the lasting contributions ing Lecturer with provided foundations of Professor John N. Mather. Speak- Rank of Professor at for entirely new areas ers at the conference were: Princeton, revered of study and in other James Damon for the exceptional instances he gave University of North Carolina range, depth, and mathematics new Albert Fathi power of his math- tools and techniques." Georgia Institute of Technology ematical work, A widely celebrated Charles Fefferman passed away in Bon- mathematician whose Princeton University heiden, Belgium, on influence was ac- Jacques Féjoz December 22, at the knowledged with University of Paris, Dauphine age of 64. many awards, including the 2017 Giovanni Forni University of Maryland Breakthrough Prize in Mathemat- Bourgain joined the Institute's Mark Goresky ics, Bourgain produced important School of Mathematics as a Professor Institute for Advanced Study work with impact across theoreti- in 1994, the same year he received Vadim Kaloshin cal computer science, group theory, a Fields Medal, and served as IBM University of Maryland spectral theory, number theory, par- von Neumann Professor at the time Leonid Polterovich tial differential equations, harmonic University of Tel Aviv of his death. In awarding the 2018 analysis, and functional analysis. Alfonso Sorrentino Steele Prize for Lifetime Achieve- University of Rome Tor Vergata ment to Bourgain, the American Mathematical Society recognized him Stony Brook University Page 5 Honors and Awards

Michael Aizenman June Huh was named the Visiting Pro- ICM 2018 Akshay Venkatesh *02 Aisenstadt Chair fessor at the was one of four by the Centre de Institute for Recherches recipients of the Advanced Study Mathématiques 2018 Fields Medal and Visiting As- and received the at the Interna- sociate Research Henri Poincaré tional Congress of Prize at the Scholar at Princ- International eton shares the 2019 New Horizons in Rio de Janeiro. Congree of . Prize in Mathematics with Karim Venkatesh, who Adiprasito completed his Ph.D. at Princeton in Sun-Yung Alice Chang Franesco Lin 2002 under the supervision of Peter was selected for the has been se- Sarnak, was cited for his synthesis of 2019 class of lected to received analytic number theory, homogeneous AWM Fellows by a 2019 Sloan dynamics, topology, and representa- the Association for tion theory, which has resolved long- Women in Math- Research Fellow- standing problems in areas such as the ematics "for ship. equidistribution of arithmetic objects. shattering the glass ceiling and inspir- ing women mathematicians to follow her Aleksandr Logunov David Donoho '78 lead... and her extraordinary record of has been awarded Princeton alum groundbreaking research in geometric the 2018 Salem David Donoho '78, analysis that has had worldwide impact." Prize for his work Stanford's Anne Weinan E on the volumes T. and Robert M. was awarded the of the zero sets of Bass Professor in 2019 Peter Henrici Laplacian eigen- the School of Hu- Prize at the 9th functions. manities and Sci- International Con- ences, was award- gress on Indus- Karen Uhlenbeck ed the 2018 Carl Friedrich Gauss Professor Emerita trial and Applied Prize at the International Congress of of Mathematics at Mathematics for Mathematicians in Rio de Janeiro for the University of his "breakthrough his fundamental contributions to the contributions in various fields of applied Texas at Austin mathematical, statistical and computa- mathematics and scientific computing..." and a Visiting tional analysis of important problems Senior Research in signal processing. Charles Fefferman Scholar at Princ- received an Hon- eton, will receive the 2019 orary Doctor of National Academy “for her pioneering achievements in Science degree Congratulations to Jennifer Chayes geometric partial differential equa- from the University *83, Barry Simon *70 and Karen tions, and integrable of Warwick at their Smith '87 for their elections to the summer graduate systems, and for the fundamental National Academy of Sciences in 2019. ceremonies July impact of her work on analysis, geom- 17-25. etry and mathematical physics.”

Page 6 Honors and Awards

MacArthur Fellowship ...continued from front page same room?” Sly said. insights, Sly is reshaping our under- area at the intersection of many differ- standing of fundamental probabilistic ent fields,” said Sly. “It’s very exciting Assaf Naor, a professor in the De- phenomena,” Naor said. to take ideas or predictions from one partment of Mathematics at Princ- area and apply them to problems from eton, described Sly’s work in this In addition to interacting with different fields.” area as a “tour-de-force” in rigor- physicists, Sly actively collaborates ously pinpointing the location of a with computer scientists and informa- “Allan Sly is a brilliant mathemati- satisfiability threshold of random tion theorists. In related work, Sly cian whose extraordinary work has formulas, “thus answering a ques- currently is co-advising a graduate resolved challenging problems in the tion that resisted the efforts of many student with Emmanuel Abbe, associ- fields of probability and statistics,” mathematicians, computer scientists, ate professor of Electrical Engineer- said President Eisgruber. “We are statisticians and physicists.” ing and Applied and Computational fortunate that he is at Princeton, and Mathematics. we are delighted by the news of his “Through his originality, spectacular selection as a MacArthur Fellow.” analytic powers, and the depth of his “Probability is a really fascinating

DEPARTMENT TEACHING AWARDS Graduate Student and Junior Faculty they are following. This is invaluable in was called an "amazing instructor" who The Department annual teaching a STEM class and I had not had such is "super great in helping to understand awards were announced at a special an engaged instructor at Princeton yet.” the material" by her students. They also tea on Monday, October 15. This year's praised her responsiveness to questions Junior Faculty Teaching Awards went Miller, a fourth year graduate student, and her mastery of colored chalk. to Assistant Professor Nicolas Boumal and Instructor Evita Nestoridi, while the Graduate Student Teaching Award went to Maggie Miller.

The awards are given by the senior faculty and are based largely on stu- dent evaluations. Boumal's students have noted he was "very receptive to questions, and genuinely cared that everyone understood."

Nestoridi's students also noted her ap- proachable demeanor and commitment to ensuring every student understood the material, with one student writing “Evita was very careful about explain- ing the material coherently and in depth, continuously taking the initia- Left to right: Evita Nestoridi, Maggie Miller, and Nicolas Boumal tive to check with students whether

Page 7 Excellence in Teaching Teaching tips from Hansheng Diao

Hansheng Diao, an instructor in our math course at Princeton. And it will can easily survive on the Wall Street”. department since 2015, has received be the best math class you’re gonna Understanding math in the ways of our departmental teaching award as take!” I also try to indicate how excited teaching well as being awarded the Excellence in I am, conveying the impression that Teaching Award by the Under- Being able to solve problems as graduate and Graduate Engi- a mathematician is not the same neering Student Council for four as being able to present those consecutive years. He has taught problems with a variety of strate- both introductory and advanced gies so that they are accessible courses in the department, and to all students. This is about here he shares his advice on con- understanding math in the ways necting with students to convey of teaching. complex ideas. Allow me to demonstrate using Here are two teaching tips for the notion of “parameterized the first-time calculus teachers. curve” in vector calculus. Dur- Both tips are more or less cliché, ing the lecture, I give out three but in reality, easier said than different approaches to the same done. For this reason, I also in- concept. After two relatively clude a couple of real classroom formal definitions, I wrap up examples from my own teaching with a third one: a parameter- experiences. ized curve can be viewed as the Hansheng Diao (right) receiving his fourth consecu- trajectory of a “flying bug”, and Show your enthusiasm tive teaching award from the Undergraduate and then emphasize how this third, Conveying your passion and en- Graduate Engineering Student Council more dynamical, approach is thusiasm for the subject, as well the “correct” way to think. This as your willingness to provide help, is way, the bug becomes the “avatar” there is literally nowhere else I’d rather the key to a successful semester. One of an otherwise challenging concept. be. can never emphasize this enough. Be Consequently, the students, who are physically active and animated, oc- Here is another example. In MAT203 experiencing the concept for the first casionally walk around in front of the there was an entire week dedicated to time, at least grasp the key idea from blackboard, use your body language multivariable extreme value problems, the lecture, and hence ready to digest and voice to reflect your great fascina- debatably the most important week for the other two abstract approaches on tion with calculus. Try to make eye engineering and applied math ma- their own time. contact as you lecture and try to make jors. By that time some students were Surely, there is some danger in relying eye contact with each student equally. already stressed out after weeks of dif- too much on such avatars. The most When the students see their instruc- ficult material. At such moments, some noticeable of which is losing math- tor’s passion and confidence, they reassuring messages go a long way. ematical accuracy. There is no clear become more engaged and want to For instance, I started the week with: rule here. Depending on the course participate. “This week is the min/max week, and goal and your personal teaching style, you will LOVE it! Because, as you will On the first day of the semester, I a decision needs to be made about the see, it’s exactly like the single-variable always open with welcoming messages balance between “flying bugs” and Credit: Maryam Mirzakhani Maryam Credit: case.” You may choose to follow by, “If like: “Vector calculus is my favorite formal math. you master this week’s material, you Page 8 Faculty Research Mathematicians Disprove Conjecture Made to Save Black Holes by Kevin Hartnett for Quanta Magazine Nearly 40 years after it was proposed, falling into the black hole yourself. First boundary of space-time established at mathematicians have settled one of the you cross the event horizon, the point of the Cauchy horizon is less singular than most profound questions in the study of no return (though to you it looks just like Penrose had imagined. general relativity. In a paper posted online ordinary space). Here Einstein’s equa- last fall, Mihalis Dafermos, a mathemati- tions still work as they should, providing To Save a Black Hole cian at Princeton University, and Jona- a single, deterministic forecast for how Dafermos and Luk, a mathematician at than Luk have proven that the strong space-time will evolve into the future. , proved that the situ- cosmic censorship conjecture, which ation at the Cauchy horizon is not quite so concerns the strange inner workings of But as you continue to travel into the simple. Their work is subtle — a refuta- black holes, is false. black hole, eventually you pass another tion of Penrose’s original statement of the horizon, known as the Cauchy horizon. strong cosmic censorship conjecture, but “I personally view this work as a tremen- Here things get screwy. Einstein’s equa- not a complete denial of its spirit. dous achievement — a qualitative jump in tions start to report that many different our understanding of general relativity,” configurations of space-time could unfold. Building on methods established a emailed Igor Rodnianski, a mathemati- They’re all different, yet they all satisfy decade ago by Christodoulou, who was cian at Princeton University. the equations. The theory cannot tell Dafermos’s adviser in graduate school, us which option is true. For a physical the pair showed that the Cauchy horizon Relativity’s Cardinal Sin theory, it’s a cardinal sin. can indeed form a singularity, but not the In classical physics, the universe is pre- kind Penrose anticipated. The singularity dictable: If you know the laws that govern Roger Penrose proposed the strong in Dafermos and Luk’s work is milder a physical system and you know its initial cosmic censorship conjecture to restore than Penrose’s — they find a weak “light- state, you should be able to track its predictability to Einstein’s equations. The like” singularity where he had expected evolution indefinitely far into the future. conjecture says that the Cauchy horizon a strong “space-like” singularity. This The dictum holds true whether you’re is a figment of mathematical thought. It weaker form of singularity exerts a pull on using Newton’s laws to predict the future might exist in an idealized scenario where the fabric of space-time but doesn’t sunder position of a billiard ball, Maxwell’s equa- the universe contains nothing but a single it. “Our theorem implies that observers tions to describe an electromagnetic field, rotating black hole, but it can’t exist in crossing the Cauchy horizon are not torn or Einstein’s theory of general relativity any real sense. apart by tidal forces. They may feel a to predict the evolution of the shape of pinch, but they are not torn apart,” said space-time. “This is the basic principle The reason, Penrose argued, is that the Dafermos in an email. of all classical physics going back to Cauchy horizon is unstable. He said that Newtonian mechanics,” said Demetrios any passing gravitational waves should Dafermos and Luk prove that space-time Christodoulou, a mathematician at ETH collapse the Cauchy horizon into a singu- extends beyond the Cauchy horizon. They Zurich and a leading figure in the study of larity — a region of infinite density that also prove that from the same starting Einstein’s equations. “You can determine rips space-time apart. Because the actual point, it can extend in any number of evolution from initial data.” universe is rippled with these waves, a ways: Past the horizon “there are many Cauchy horizon should never occur in the such extensions that one could entertain, But in the 1960s mathematicians found wild. and there is no good reason to prefer one a physical scenario in which Einstein’s to the other,” said Dafermos. field equations — which form the core of As a result, it’s nonsensical to ask what his theory of general relativity — cease to happens to space-time beyond the Cauchy Dafermos and Luk indicate that while describe a predictable universe. Mathema- horizon because space-time, as it’s regard- space-time exists beyond the Cauchy hori- ticians and physicists noticed that some- ed within the theory of general relativity, zon, this extended space-time isn’t smooth thing went wrong when they modeled the no longer exists. “This gives one a way enough to actually satisfy Einstein’s equa- evolution of space-time inside a rotating out of this philosophical conundrum,” said tions. Thus, even with the strong cosmic black hole. Dafermos. censorship proven false, the equations are

...continued on page 13 To understand what went wrong, imagine This new work shows, however, that the Page 9 RTG Summer Programs

In 2015 members of our department ing each of the latter two weeks of the the middle of the program. received a Research and Teaching program we hosted small conferences, Grant (RTG) from the National Sci- each bringing about ten outside speak- In addition to three days of talks on ence Foundation. The main purposes ers to share their research. current research, one day of this sum- of this grant were to draw in students mer's conference be a festival focusing with interest but less background to This upcoming June we will host a on diversity in mathematics. the pursuit of mathematics, introduc- similar program on topics in geometric ing them to exciting new realms with- analysis, organized by Sun-Yung Alice A third summer school and confer- in the areas of topology and geometry. Chang, Otis Chodosh, Casey Kelleher, ence are being planned for the future, Ana Menezes, Stephen McKeown, and the department is grateful to the This goal is being realized through Rafael Montezuma, and Paul Yang. NSF for supporting these valuable three summer programs, one last sum- Again the format will be a three-week programs. mer (2018), one this summer (2019), intensive summer school, with a con- and one being planned for the future. ference featuring invited speakers in These programs consist of a three- week intensive summer school for undergraduate and first-year graduate students—approximately 25 students, mostly from other Universities—along with conferences during these summer schools to introduce current research.

The 2018 summer school focused on low-dimensional topolgy and its connections to symplectic geometry, organized by David Gabai, Jonathan Hanselman, Henry Horton, Francesco Lin, Peter Ozsváth, John Pardon, and Zoltán Szabó. The organizers each taught an excellerated course, and dur- Participants in the 2018 Summer School on Low-Dimensional Topology and Symplectic Geometry.

Highlighting Diversity in Mathematics

As part of this summer's workshop highlighting diversity in mathematics, the department is producing a series of inspir- ing videos about senior mathematicians from underrepresented groups.

The first of these videos features our own Professor Sun-Yung Alice Chang sharing her story on the importance of community and encouraging young women to pursue studies in mathematics and other sciences.

View her story at: https://math.princeton.edu/chang-video

Page 10 Mentoring

Closing the Gender Gap by Jenny Kaufmann '19 students in my year who would have The 2018 class where some students enter already liked to major in math, but didn't of math majors having a background in proof-based because they didn't believe they were was 1/3 women, math. Freshmen who do not have this smart enough. a department background may compare themselves all-time record. to the ones that do, and ascribe the dif- To close the gender gap, we have to This year, the ference to talent instead of to a tempo- close the confidence gap. We need fraction is 1/5. rary head start. supportive mentors and female role What causes this models for the younger students. We gender gap? Underconfidence doesn't exclusively need female math majors and grad afflict women. However, research has students and professors to talk to One of the biggest factors—and one found that it does so disproportion- freshmen, and say, "I've struggled too. that isn't often discussed—is under- ately: on average, women consider Everyone does. It's not just you." confidence. Math has a pernicious themselves lower-performing than men Accomplishing this is hard. But the reputation for being accessible only at the same performance level. This Princeton Noetherian Ring has risen to geniuses. If you're a student in a lines up with my experience. I have to the challenge. challenging math class, it is easy to get heard from more than a few female intimidated and think, "Man, this is re- math majors—no less talented than A year ago, math major Aria Wong ally hard, I must not be smart enough." their male counterparts—who at some spearheaded an initiative to encour- It is especially easy to fall victim to this point doubted their ability to succeed age more women to major in math. genius myth at places like Princeton, in math. I also know of two female ...continued on next page

The Noetherian Ring by Maggie Miller The department's student & faculty of speakers, who are asked to pres- gave the first lecture on March 6th, mentoring program, the Noetherian ent their work at a general graduate speaking about her research that cer- Ring, began several years ago as a way level so that the whole department can tain density-zero sets of prime num- to bring together female undergradu- participate. bers are infinite. The series continued ate students, graduate students, and on April 19th with Yaim Cooper from faculty to create a community and Princeton Instructor Yunqing Tang the IAS. structure for mentoring in the depart- ment.

What started as the occasional lunch gathering has expanded, and this semseter the group started a lecture series to feature the research of local and visiting women/nonbinary math- ematicians. The seminar is organized by graduate students Dev Dabke (PACM), Mari Kawakatsu (PACM), Maggie Miller, and Boya Wen.

The overarching goal of the series is In addition to being an organizer of the Noetherian Ring, Maggie Miller spoke to highlight the work of a diverse set at the department's Bring Your Child to Work Day, giving a demonstration on slicing Möbius strips Page 11 Distinguished Visitors The Minerva Programs This year was one of the busiest yet While the Minerva Distinguished for our Minerva Programs: Visitorship carries no official the Minerva Distinguished obligations, Naber, Wood, Visitorship and the Minerva and Zhu each delivered a se- Lecture Series. ries of lectures on their recent research, which were all well In the fall we hosted three Mi- attended by the department. nerva Distinguished Visitors: Aaron Naber *09, Professor of In the spring term Hugo Mathematics at Northwestern Duminil-Copin, a Profes- University; Melanie Matchett Aaron Naber Xinwen Zhu sor at both the Institut des Wood *09, Vilas Distinguished Hautes Études Scientifiques Achievement Professor of and the Université de Ge- Mathematics at the Univer- nève, delivered our Minerva sity of Wisconsin-Madison; Lecture Series. Working in and Xinwen Zhu, Professor of areas related to probability Mathematics at the California and mathematical physics, Institute of Technology. Duminil-Copin gave three talks on some intriguing aspects of the mathematical Melanie Matchett Wood Hugo Duminil-Copin study of phase transitions.

Gender Gap ...continued from previous page

She made a Noetherian Ring website through. Two of the then-seniors, who It is too soon to tell how successful our with resources and advice from older became friends during their first proof- efforts will be at encouraging more students, and she sent personal wel- based math class, said they would not female math majors. But with serious come emails to every single one of the have gotten through that class without thought and persistent effort, I believe dozens of incoming freshman women each other. (They're both now getting we can bring about a time when, if the who indicated an interest in math on Ph.D.s at top-15 math grad schools.) gender ratio in a graduating class of their applications. math majors is 1/3, then that’s a local In Fall 2018, inspired by Aria, I minimum—not a global maximum. On top of that, Aria held a dinner for started running casual weekly study freshman women considering major- sessions for female math students. For more guidance on supporting ing in math. She invited all the female These sessions help younger female female students, see the "Guidelines math majors, along with a handful students form a support network. We on Best Practices" published by an of grad students and professors; we also started hosting dinners around NSF-funded initiative called WATCH were there to give advice and answer course selection time, where students US: https://www.womendomath.org/ questions for the freshmen. At one give and receive advice about choos- watch-us/ point during the dinner, we all went ing courses, applying to REUs and around one by one and talked about internships, and so on. With the help This article is excerpted from a post on our experiences: the struggles we of other math majors, we sent out wel- Jenny Kaufmann's blog, Math Within had faced, the friends we had found come emails to this year’s prospective Reach. You may read the full article in the department who had helped us female math students. And our group at: http://mathwithinreach.blogspot. has more ideas in the works. com/2019/04/closing-gender-gap.html

Page 12 Undergraduate Program Updates from the Math Club The Princeton Math Club has had a the Princeton University Math Compe- Lastly, there are some exciting new great start to 2019, and we are looking tition, which draws more than 500 high aspects of the Math Club that are forward to continuing to spread our school competitors to take a contest en- forthcoming. We will be starting a passion for mathematics throughout the tirely organized by our own undergrad- series of talks by undergraduates about undergraduate community. uate volunteers. Monthly board game their independent research or other nights on Saturday evenings in the Fine mathematical topics of interest. We are This semester, we have continued to Hall Common Room have been a great also working to revamp the Guide for hold academic events, including our place for taking a break from problem Math Students on the club website, longstanding colloquium series, which sets to sit back and enjoy board games which will give comprehensive informa- has featured several talks from our dis- and the company of other Math Club tion about all aspects of life as a student tinguished faculty at Princeton as well members. We also recently held our of mathematics at Princeton. We hope as special guests to the university Euge- annual Pi Day celebration, complete that this will prove to be an invalu- nia Malinnikova and Tadashi Tokieda. with pie (the sweet kind and the pizza able resource for anyone and everyone Additionally, we have continued to run kind!) and a pi-based trivia contest. It remotely interested in math on campus. the Mentoring Möbius program, in was a fun time for all who attended! which graduate students meet monthly That's all from the Math Club; we're with a small group of undergradu- Later in the semester, we will also be looking forward to what the rest of the ates to discuss math, life as a graduate holding Meet Your Professor dinners, year has in store! student, and much more. Our outreach as well as our annual end-of-year ban- to the community continues through quet in the Professor's Lounge of Fine Zachary Stier MathReach, a weekly program where Hall in May. We're looking forward to 2019 Math Club President undergraduate volunteers teach math to these events as well! students at a nearby charter school, and

Ryan Chen '19 awarded Churchill Scholarship Black Holes ...continued from page 9 Princeton Univer- III of the Mathematical Tripos, which still spared the ignominy of outputting sity senior Ryan confers a Master of Advanced Study in nonunique solutions. Chen, a mathemat- mathematics. ics major who is You could think of this outcome as a planning to pursue “In light of the vastness and diversity disappointing compromise: Even though a Ph.D. in math- of mathematics involved in modern you can extend space-time beyond the ematics with the number theory, I am eager to strength- Cauchy horizon, Einstein’s equations goal of entering en my foundations and knowledge as can’t be solved. But it’s precisely the fact that this middle ground seems to exist academia, has been much as possible,” Chen said. that makes Dafermos and Luk’s work so selected as one of interesting. this year’s Churchill Scholars. “My experience with math thus far has impressed upon me the diversity “This is people really discovering a new Chen is one of 16 scholarship winners of tools involved,” he said. “Connec- phenomenon in the Einstein equations,” who will spend a year studying at the tions between different techniques and said Rodnianski. University of Cambridge while living subfields offer beautiful theorems and This article was excerpted from Quanta at Churchill College, the only col- conjectures.” Magazine. For the full article, visit lege at Cambridge focused on STEM https://www.quantamagazine.org/ subjects. He plans to complete Part mathematicians-disprove-conjecture- made-to-save-black-holes-20180517/ Page 13 Graduate Program

Graduate Profile: Daniel Vitek Hi, this is Daniel! I'm a fifth-year As I leave Fine Hall, some memories the faculty-student soccer games, sup- student here in Fine Hall. These days stand out as highlights of everyday port (and the freedom to buy our own I work somewhere in the intersection life here. Department athletic events, supplies!) for Friday Socials, or orga- of stable homotopy theory and low-di- whether the F.C. Fine intramural nization for the recital. I've been here mensional topology. I definitely took a soccer team, the Open Discs intra- long enough to see some turnover in roundabout path to this project. I was mural ultimate Frisbee team, or the the administration, but the consistently fortunate enough to have my advisor, faculty-student soccer games (not to high standard that they provide is an Peter Ozsváth, suggest several prob- mention various pickup games), were underrated reason for why Fine Hall lems during my second and third year always a blast, and I'm glad that the functions as well as it does. that were nicely aligned with my inter- faculty-student soccer games are now ests during my undergraduate studies happening more often than they used at , though I didn't re- to. The graduate student community ally get anywhere worthwhile on these within Fine has offered some unique problems. Despite all this, it wasn't social events, from the intermittent until a conference the summer after movie nights to the now-very-successful my third year when I heard about the tradition of Friday Socials. Finally, the problems I've been thinking about (and department recital has always been a ever-so-slightly solving) since. While fantastic display of student and faculty I won't be continuing in academia— talents, and I'm very glad I managed to I'll be working at a think tank in town participate this spring. in the near term—I've tremendously enjoyed thinking about these problems One common thread among these very and am very glad that I both chose to distinct experiences is the support, go to grad school and chose to come to often unnoticed, of the department Princeton. administration, whether that's food for Daniel Vitek at this spring's recital

Javier Gómez Serrano: Director of Graduate Studies This is my second I strongly feel that our students enjoy rounds: first a committee formed out of year as one of their time here. seven faculty members from different the Directors of research areas and of varied seniority Graduate Stud- We continue to work with the Graduate met to review the 325 applications we ies (DGS) and it School and their representatives in the recieved this year. Then the commit- has been a great area of diversity and inclusion to guide tee presented its recommendations to experience. Work- us in the admissions process. There are the full faculty for final decisions. The ing with fellow always improvements that can be made. Open House for admitted students DGS Zoltán Szabó We will also be asking the graduate was held March 7-8 and provided our has been easy, and students to nominate representatives guests the opportunity to talk to current he has been very from each class year to serve as mem- faculty and students, attend talks, and effective at solving all the problems that bers of the Graduate Student Commit- also get a sense of how the town and arise on a day to day basis, both from tee, making diversity and inclusion a life in Princeton. In the end, the incom- the perspective of the students as well priority, and addressing the concerns ing class will consist of 12 graduate as the Graduate School. Our graduate as well as any suggestions our students students and we look forward to them administrator, Jill LeClair, puts in an may have. joining the department this September! enormous effort to keep our students on track and happy, which sometimes may Admissions is one of the most challeng- Finally, on behalf of my fellow DGS be a challenging task, helping them ing and at the same time rewarding and the Department, congratulations to navigate through any issues that arise. parts of the job. It consists of several the our 2019 Ph.D.s!

Page 14 Most recent Ph .D.s Name/ Field Undergrad Advisor Thesis Title Original Placement Matthew de Courcy- Fine-scale properties of random Universit of Lusanne /Post- Ireland McGill University Sarnak functions doctoral Researcher Analytic Number Theory Kathleen Emerson Cambridge Comparisons of different definitions Heap, Inc./Software Engi- Morel Number Theory University of pseudocharacter neer Ross Granowski Pursuing entrepreneurship in Shock formation and illposedness Nonlinear Wave Equa- Georgia Tech Klainerman property trading/ quantita- of quasilinear wave equations tions tive finance field Artem Kotelskiy Moscow State Bordered invariants in low-dimen- Indiana University/Zorn Low-dimensional Szabó University sional topology Postdoctoral Fellow Topology Daniel Kriz A new p-adic Maass-Shimura Princeton MIT/NSF Postdoctoral Re- Arithmetic Geometry/ Zhang operator and supersingular Rankin- University search Fellow Number Theory Selberg p-adic L-functions Lucia Mocz Hausdorff Center of Math- A new Northcott property for Falt- Arithmetic Geometry/ Harvard University Zhang ematics, University of Bonn/ ings height Number Theory Postdoctoral Research Fellow Georgios Moschidis National Technical Two instability results in general UC Berkeley/Miller Research Dafermos General Relativity University of Athens relativity Fellow

Lue Pan Fontaina-Mazur conjecture in the University of Chicago/ L.E. Peking University Taylor Number Theory residual reducible case Dickson Instructor Sobolev-type embedding on Cauchy- Xiaojie Shi National University Yang Riemann manifolds and nonlinear sub- Industry job/Chicago Geometric Analysis of Singapore elliptic PDE on Heisenberg group Charlie Stibitz University of Topics in Fano varieties and Northwestern University/ Kollár Algebraic Geometry Michigan singularities Boas Assistant Professor Courant Institute, New York Yury Ustinovskiy Moscow State Hermitian curvature flow and curva- Tian University/Courant Complex Geometry University ture positivity conditions Instructor Anibal Velozo Catholic University Yale University/Gibbs As- Tian The ergodic theory of geodesic flows Ergodic Theory of Chile sistant Professor Yao Wang Xi’an Jiaotong Estimation error for regression and Orsus Research, NY/ Re- Theoretical Machine E, Weinan University optimal convergence rate searcher Learning Junho Peter Whang Diophantine analysis on moduli of MIT/C.L.E. Moore Instruc- McGill University Sarnak Number Theory local systems tor Yuhou (Susan) Xia Irreducibility of Automorphic Galois Valassis Digital, NC/Data Algebraic Number Bryn Mawr College Taylor Representations of Low Dimensions Scientist Theory Cisinski (U. University of Regensburg, Masoud Zargar University of Regensburg), Voevodsky motives, stable homotopy Germany/Postdoctoral Re- Algebraic Geometry Toronto Weibel (Rut- theory, and integration search Assistant gers), Ozsváth

Page 15 Department of Mathematics Fine Hall — Washington Rd. Princeton, NJ 08544 math.princeton.edu [email protected]

Alumni , faculty, students, Annual friends, Alumni connect Reception with us, write to us at Friday, May 31, 2019• 2:00-3:30pm 3rd Floor Common Room, Fine Hall

The department's faculty-student soccer games started some years ago as a casual affair, and have grown to a twice-per-year event complete with an after-game BBQ and team t-shirts.