Princeton University Faculty

"Board member and co organizer of Addis Coder as well as Jam Coders."

"I am a co organizer of the Harvard Machine Learning Foundations seminar and a steering committee member and associate faculty of the Kempner Institute, for the Study of Natural and Artificial Intelligence."

"I have not discussed these issues at all."

"Starting in the Fall of 2025 I will be part time at Harvard as a Catalyst Professor and part time at OpenAI."

"I was a principal researcher at Microsoft Research New England."

"I am a theoretical computer scientist, and have worked on computational complexity, algorithms, cryptography, quantum computing."

"I have written numerous blog posts and opinion articles on this matter."

"I am a professor of Computer Science at Harvard University."

"In recent years my focus has been onfoundations of machine learning and safety of artificial intelligence systems."

"I’m an Israeli American tenured professor, and I felt it was my duty to stand up for Jewish and Israeli students."

"I helped organize an open letter denouncing antisemitism."

"I was on the steps of Harvard’s Widener Library, taking part in a vigil for the victims of Hamas’s terrorist attack."

"Modern civilization depends on science...James Smithson was well aware that knowledge should not be viewed as existing in isolated parts, but as a whole, each portion of which throws light on all the other, and that the tendency of all is to improve the human mind, and give it new sources of power and enjoyment...narrow minds think nothing of importance but their own favorite pursuit, but liberal views exclude no branch of science or literature, for they all contribute to sweeten, to adorn, and to embellish life...science is the pursuit above all which impresses us with the capacity of man for intellectual and moral progress and awakens the human intellect to aspiration for a higher condition of humanity."

"My parents separated after my father resigned his university position to focus on his inventions, and my mother then finished her education and became a school mathematics teacher. We moved to a very cosmopolitan area of London, which was like a new birth to me; it was there that my interest in mathematics really began."

"I cannot claim to find it easy to balance my ambitions in mathematical research with the desire to be a good parent, to be an inspiring teacher, or to effect positive social change in the world, I do feel very fortunate to be able to spend my life tackling these challenges, which are extremely interesting and important to me."

"I learned mathematics on my own from textbooks, which is perhaps strange given that both my parents were involved in the subject. At the same time, I spent a good deal of time studying art and wanted to follow a career in that direction until I was eventually convinced by my family that I should first work for a mathematics degree to ensure that I could earn a living."

"I went to Cambridge, which represented a second major change in my life. As I learned more mathematics, I saw that it is an entire world of its own which many people choose to live in, a world in many ways more real than the real world: it feels permanent, eternal, and offers a deep sense of security because nearly everyone who understands it agrees on what is truth. By the time I had finished at Cambridge, I was very involved with mathematics and did not consider other careers."

"While I was growing up, the elementary school I attended was extremely ethnically homogeneous. I was unable to escape from heavy issues concerning race, which my mother always explained in a political context."

"My research is in the field of spectral geometry, the study of how the shape of an object affects the modes in which it can resonate. A famous question in the field is, Can one hear the shape of a drum? Spectral geometry bridges different branches of science, including engineering and physics, as well as a number of different fields of mathematics. However, quite different sorts of questions are studied within each discipline. I am a mathematical analyst, which gives me an appreciation for the infinite and the infinitesimal. At the moment, one of the things I am working on understanding is the total wavelength of a surface like a sphere or something of greater complexity, such as the surface of a bagel or a pretzel. What is this total wavelength? If you strike a surface it can resonate at any one of a list of frequencies, and the wavelength of the sound produced by the vibration is inversely proportional to the frequency. In the mathematically idealized model, there are infinitely many possible wavelengths. The total wavelength should be the sum of all of these individual wavelengths except that this infinite sum equals infinity. Fortunately, a finite number can be assigned to it by a slightly elusive process called regularization. (This process is also used in mathematical physics to mysteriously obtain true answers from formulas which do not really make sense!) I first became interested in the total wavelength as a model related to a question which can be roughly stated as, can one hear the shape of the universe? However, the total wavelength shows up in many quite different areas of mathematics and I am finding these connections intriguing."

"I am a mathematical analyst, and most of my research is in the area of spectral geometry. Problems in spectral geometry are also studied by various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others. What is Spectral geometry? Spectral geometry most usually means the study of how the geometry of an object is related to the natural frequencies of the object. These are the frequencies at which the object can vibrate. A vibrating object often produces a sound, and the frequencies can be heard as the dominant tone and the overtones of the sound. The well-known question highlighting what spectral geometry is all about is the question "Can one hear the shape of a drum?" I am a mathematical analyst, and most of my research is in the area of spectral geometry. Problems in spectral geometry are also studied by various kinds of geometers, number theorists, applied mathematicians, mathematical physicists, and others. In mathematical terms, the natural frequencies of an object (or rather their squares) are the eigenvalues of a partial differential operator called the Laplacian. This Laplacian takes each function defined on the object and differentiates it twice to give a new function. The eigenvalues of the Laplacian form an infinite sequence of numbers tending to infinity. In spectral geometry we study how these numbers depends on the shape of the object. For people who like to know the full story, I should mention that many spectral geometers (including me) who work on the Laplacian on smooth manifolds study the whole sequence of eigenvalues of the Laplacian. Now the low eigenvalues can give accurate values for the frequencies at which a real-life object vibrates, but the very high eigenvalues do not correspond to genuine physical vibrations of the object because of molecular forces and damping. These effects are not included in the model where the vibration is driven by the Laplacian alone. This means that my research is rather different from that of an engineer who wishes to model precisely the vibrations of a real-life object. In actual fact the questions I work on are more closely related to mathematics arising in quantum physics and string theory. In addition, I don't always study the Laplacian, but also the eigenvalues of other operators, which might represent other physical quantities than the frequencies of vibration. I mostly study spectral geometry for nice smooth objects such as spheres and tori, but some people work on rough objects and even discrete objects like graphs. In the last eight years, I have worked mostly on the spectral zeta function, which is an infinite sum of powers of the eigenvalues. In particular, I have worked on the zeta-regularised determinant, which is used in topology, quantum field theory, and string theory. Recently, I have been very interested in the sum of squares of the wavelength of a surface, which is related to all kinds of different things including vortex theory."

"My mother is British, from a family with a trade union background and a central interest in class struggle; she met my father, who is Nigerian, while both were students of mathematics in London. My father was a very talented mathematician, and after my parents married, he went on to a position in the mathematics department of the University of East Anglia."

"David Todd Wilkinson died on 5 September 2002 in Princeton, New Jersey, after having battled cancer for 17 years. His role in the measurements of the thermal (CMB) was key to the completion of the program of cosmological tests that began with Edwin Hubble’s discovery of the expanding universe in 1929."

"The universe is filled with thermal radiation having a current temperature of 2.75 K. Originating in the very early universe, this radiation furnishes strong evidence that the Big Bang cosmology best describes our expanding universe from an incredibly hot, compacted early stage until now. The model can be used to extrapolate our physics backward in time to predict events whose effects might be observable in the 2.75 K radiation today. The spectrum and isotropy are being studied with sophisticated microwave radiometers on the ground, in balloons, and in satellites. The results are as predicted by the simple theory: the spectrum is that of a blackbody (to a few percent) and the radiation is isotropic (to 0.01 percent) except for a local effect due to our motion through the radiation."

"When a student picks a research topic, the decision is influenced by the research interests of the advisor, the state of the field in which the problem lives, the personality of the advisor, the chemistry between them ... who else in working on whichever problem. It's immensely complicated. And asking for a sort of simple prescription for how to assign research problems — it reminds a little of the question of how to decide who should marry whom."

"First of all, there are problems that no one knows how to solve. There are problems that have been studied but untouched, or problems on which there is partial progress. There are problems that sound compelling when formulated, but which no one has thought of yet. There are concepts which are very useful in solving problems .. or which perhaps ... sound very natural and compelling when formulated, but have not been formulated yet. And all of these things interact. So, by solving problems, one is led to concepts — and, by thinking about concepts, one is led to problems."

"Charles Fefferman (Charlie) is a mathematician of the first rank whose outstanding findings, both classical and revolutionary, have inspired further research by many others. He is one of the most accomplished and versatile mathematicians of all time, having so far contributed with fundamental results to harmonic analysis, s, , , quantum mechanics, fluid mechanics, and , together with more sporadic incursions into other subjects such as neural networks, financial mathematics, and crystallography."

"That geometric considerations enter in a decisive way in many questions of harmonic analysis is by now a well-known fact. In explicit form such ideas arose first in the estimation of the of surface-carried measure; they have since played a key role in averages over lower-dimensional varieties, restriction theorems, in connection with the study of s and Fourier integral operators, and in application to linear and non-linear s."

"(edited by Noga Alon, Jean Bourgain, Alain Connes, Misha Gromov, and Vital D. Milman; quote from p. 434)"

"What I like about mathematics is the interplay with other people. And that generates extra energy — because it really helps you think."

"Working with Eli Stein is amazing. There are good teachers and there are great teachers — and then there is Eli. He was simply the best teacher, I believe, of advanced math, I believe, in the whole world, by far. ... he conveyed a spirit of optimism. ... He has a wonderful style of explaining math — in which almost all of the effort of explanation goes into finding the right point of view and getting the essence of the problem."

"I think he would have made a great actor. His lectures were polished: He would finish at the right moment and march off the scene. A very lively individual with many interests: music, astronomy, chemistry, history.... He loved to teach. I had a feeling that he loved to teach anybody anything. Being his student was a wonderful experience; I couldn’t have had a better start to my mathematical career. It was a remarkable accident. My favorite theorem, which I had first learned from Bell’s book, was Gauss’s law of , and there, entirely by chance, I found myself at the same university as the man who had discovered the . It was just amazing."

"The essential point in the definition of an algebra is that it is a vector space of finite dimension over a field. This fact allows us to conclude that of subalgebras will terminate. After the great success that Emmy Noether had in her ideal theory in rings with ascending chain condition, it seemed reasonable to expect that in rings where the ascending and the descending chain condition holds for left ideals one should obtain results similar to those of . As one of the papers written from this point of view we mention E. Artin, Zur Theorie der hyperkomplexen Zahlen (Abh. Math. Sem. Hamburgischen Univ. vol. 5 (1926)). In 1939 showed (Rings with minimal condition f or left ideals, Ann. of Math. vol. 40) that the descending chain condition suffices."

"The majority opinion in Trump v. United States, the most sweeping judicial reconstruction of the American presidency in history, secures the monumental historic disgrace of the John Roberts Court. Since last winter, the Supreme Court has intervened directly in the 2024 presidential campaign by effectively shielding Donald Trump from being tried on major federal charges before the November election. No previous Supreme Court has protected a political candidate in this way. Far more ominously, in March the Court in Trump v. Anderson openly nullified the section of the Fourteenth Amendment that bars insurrectionists from holding federal or state office, discarding basic lessons about threats to American democracy dating back to the Civil War. Now, in Trump v. United States, handed down on the last day of its 2023–2024 term, the Court has seized the opportunity to invent, with no textual basis, "at least presumptive" and quite possibly "absolute" presidential criminal immunity for official acts, a decision so broad that it essentially places the presidency above the law."

"Over the past forty years the doctrine of originalism (along with its sibling, textualism) has been the cornerstone of the jurisprudence of the conservative majority that now dominates the [[w:Supreme Court of the United States|[Supreme] Court]]. Concocted in the 1980s to roll back the constitutional precedents of the New Deal and Great Society eras, supposedly in the name of judicial restraint, originalism purports to divine the original intentions of the framers by presenting tendentious renderings of the past as a kind of scripture. This bad-faith invocation of the framers has become a ploy to justify overturning Roe v. Wade, gutting the Voting Rights Act of 1965, eliminating commonsense gun regulation, and more. But now this originalist petard is exploding in the majority’s face. No degree of cherry-picking or obfuscation can deny the historical record of the Fourteenth Amendment, which is unequivocal: if Donald Trump engaged, in any way, in the insurrection of January 6, he is automatically barred from holding any public office, federal or state."

"War is hell. War is unpredictable, difficult, dangerous—and usually (though not always) worse than its initiators expect when conflict begins. War has features that remain fairly similar from century to century, once you take a reflective perspective and look back on things."

"[American] Civil War was effectively our first industrial-scale war. It led to mass mobilization at the human and economic/industrial levels. It also involved long-distance movements and communications, by train and telegraph, in ways that arguably make it the first modern war. It produced as many American casualties as all the rest of our wars combined, so can hardly be ignored in a book about major U.S. wars. It also illustrated warfare at the campaign or theater level of analysis perhaps more clearly and cleanly than any other conflict."

"We are at once the most powerful nation in the history of the human race, yet because of nuclear weapons, we are vulnerable to rapid destruction by foreign powers in a way we cannot prevent. That is a paradox. We also have lopsided advantages over most other countries (perhaps not China, but definitely Russia). Yet those advantages do not guarantee victory in war in the modern era (we struggled against the Vietcong, the Taliban, Iraqi insurgents and militias—the list goes on)."

"...she is enormously strong. Pagels not only survived two tragedies in the space of fifteen months but since then has written another book, reared her children, taught her many students. She is, by all reports, a good colleague, a devoted friend. Her mind is quick and generous. At fifty-two, she has a mild, earnest appearance (a rounded, friendly face, windblown blond hair), and yet in conversation she is absolutely fierce, focussed, picking apart the careless question, delighted by the unexpected one. When she delivers her lectures for an undergraduate course on the New Testament—Monday and Wednesday mornings at ten—she does not so much pace the room as prowl it. Pagels radiates so much intensity that you somehow imagine a fast-burning cigarette in her hand. There is none. She does not smoke. She smolders."

"Yet the gnostic Gospel of Thomas relates that as soon as Thomas recognizes him, Jesus says to Thomas that they have both received their being from the same source . . . Does not such teaching—the identity of the divine and human, the concern with illusion and enlightenment, the founder who is presented not as Lord, but as spiritual guide—sound more Eastern than Western? Some scholars have suggested that if the names were changed, the 'living Buddha' appropriately could say what the Gospel of Thomas attributes to the living Jesus. Could Hindu or Buddhist tradition have influenced gnosticism?"

"Could the title of the Gospel of Thomas—named after the disciple who, tradition tells us, went to India—suggest the influence of Indian tradition?"

"I try to point out in several of my works, including Wisdom Transformed, and in this book, that in some sense the Buddhism that we now believe in is the colonial product. It is a product of Western theories and translators of Buddhist text, who came in the nineteenth century. I'm not saying that's necessarily bad, but it led to a transformation of ethics, a transformation of the way we look at the world in Sri Lanka, and it is foolish to deny this colonial impact."

"Sri Lankan monks and educated laypersons found the Western interpretation of Buddhism especially appealing in their fight against the Protestant and Catholic missions. Soon the indigenous scholarship, strongly influenced by Western critical methods, carried on into the present day a rational view of Buddhism, treating the mythic, cultic, devotional elements as inessential to the religion, as accretions or interpolations superadded to a pristine, pure form of Buddhism. Concomitantly, the folk beliefs of ordinary peasants were viewed as animism, or superstitions, unworthy of the rational theosophy of old religion. 25"

"At one point in my life, I had to make a decision...I had to decide whether to have too little or too much in my life...Study, children, friends, travel—all of it. That was an easy one. I chose too much.”"

"Satan is a way of perceiving opponents. You may not believe the mythology of such a universe, but it’s in you, a background perception.”"

"There are certain ecological structures in any marriage—some with a traditional gender bias and some not. Simply, people take on certain roles. In a way, I had to do everything. But, most of all, I also wanted to take on the challenge of not giving up, of not despairing. Because Heinz was on the side of life. He loved life. He was full of explorative excitement, interest, passion. I realized that it would be no honor to him to say, ‘I can’t take this, I can’t go on, it’s too hard.’ Somehow, I wanted to take on something of what I’d learned from him, the way he embraced life, with all its dangers and difficulties. I was challenged to do that. I can’t say I’ve done it, but that’s what I wanted to do."

"From the very beginning of quantum mechanics, the notion of the position of a particle has been much discussed. In the nonrelativistic case, the proof of the equivalence of matrix and wave mechanics, the discovery of the uncertainty relations, and the development of the statistical interpretation of the theory led to an understanding which, within the inevitable limitations of the nonrelativistic theory, may be regarded as completely satisfactory."

"The family of mathematical problems discussed here has emerged in recent years as a result of efforts to put a small chapter of quantum field theory, the so-called external field problem, on a sound mathematical footing. The external field problems is special because the partial differential equations for the unknown field is linear, but the coefficients are allowed to vary in space and time and that gives rise to some surprises, which seem to be of general interest. There is a vast and in large part turgid mathematical physics literature on the subject. To make the general wisdom which has accumulated there more readily available to a mathematical audience I have, in the following, tried to place the problems in their physical context, and still to bring out the essential mathematical questions many of which remain to be answered."

"... when I was a graduate student in the 1950s, I had to learn a lot about cosmic ray physics — because that's where all the information was coming from about new particles. And I remember how surprised I was when a professor — at Princeton where I was a student — Arthur Wightman, told me that pretty soon physicists would no longer be worried about cosmic rays. They would be getting the information about particles from new kinds of s — which would accelerate known particles like s, which are the nuclei of hydrogen atoms, or s to very high energy where they would collide with each other or with stationary targets. And in that collision new matter would be formed."

"Vacuum expectation values of products of neutral scalar field operators are discussed. The properties of these distributions arising from , the absence of negative energy states and the positive definiteness of the scalar product are determined. The vacuum expectation values are shown to be boundary values of analytic functions. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved."

"… there are other things wrong with these models but the fundamental trouble is the non-uniqueness of the vacuum as was first shown by , , and STEINMANN … Actually, … has shown that the cluster decomposition property is not only necessary but sufficient for the uniqueness of the vacuum, if there is at least one cyclic vacuum. … HEPP, K., JOST, R., RUELLE, D. and STEINMANN, O., Necessary condition on Wightman functions, Helv. Phys. Acta 34 (1961) 542. BORCHERS, H.J., On the structure of the algebra of field observables, Nuovo Cim. 24 (1962) 214"