mathematical physics Epic Effort to Ground Physics in Math Opens Up the Secrets of Time By mathematically proving how individual molecules create the complex motion of fluids, three mathematicians have illuminated why time can’t flow in reverse. 10 A woman surfing on a wave formed by circular objects Wei-An Jin/Quanta Magazine ByLeila Sloman Contributing Correspondent June 11, 2025 View PDF/Print Mode fluid dynamicsmathematical physicsmathematicsNavier-Stokes equationspartial differential equationsAll topics At the turn of the 20th century, the renowned mathematician David Hilbert had a grand ambition to bring a more rigorous, mathematical way of thinking into the world of physics. At the time, physicists were still plagued by debates about basic definitions — what is heat? how are molecules structured? — and Hilbert hoped that the formal logic of mathematics could provide guidance. On the morning of August 8, 1900, he delivered a list of 23 key math problems to the International Congress of Mathematicians. Number six: Produce airtight proofs of the laws of physics. The scope of Hilbert’s sixth problem was enormous. He asked “to treat in the same manner [as geometry], by means of axioms, those physical sciences in which mathematics plays an important part.” His challenge to axiomatize physics was “really a program,” said Dave Levermore(opens a new tab), a mathematician at the University of Maryland. “The way the sixth problem is actually stated, it’s never going to be solved.” But Hilbert provided a starting point. To study different properties of a gas — say, the speed of its molecules, or its average temperature — physicists use different equations. In particular, they use one set of equations to describe how individual molecules in a gas move, and another to describe the behavior of the gas as a whole. Was it possible, Hilbert wondered, to show that one set of equations implied the other — that these equations were, as physicists had assumed but hadn’t rigorously proved, simply different ways of modeling the same reality? For 125 years, even axiomatizing this small corner of physics seemed impossible. Mathematicians made partial progress, coming up with proofs that only worked when they considered the behavior of gases on extremely short timescales or in other contrived situations. But these fell short of the kind of result that Hilbert had imagined. Share this article (opens a new tab) Newsletter Get Quanta Magazine delivered to your inbox Subscribe now Recent newsletters (opens a new tab) Black-and-white photo of a bald man sitting in a wicker chair. In 1900, David Hilbert came up with a list of 23 problems to guide the next century of mathematical research. His sixth problem challenged mathematicians to axiomatize physics. University of Gottingen Now, three mathematicians have finally provided such a result. Their work not only represents a major advance in Hilbert’s program, but also taps into questions about the irreversible nature of time. “It’s a beautiful work,” said Gregory Falkovich(opens a new tab), a physicist at the Weizmann Institute of Science. “A tour de force.” Under the Mesoscope Consider a gas whose particles are very spread out. There are many ways a physicist might model it. At a microscopic level, the gas is composed of individual molecules that act like billiard balls, moving through space according to Isaac Newton’s 350-year-old laws of motion. This model of the gas’s behavior is called the hard-sphere particle system. Now zoom out a bit. At this new “mesoscopic” scale, your field of vision encompasses too many molecules to individually track. Instead, you’ll model the gas using an equation that the physicists James Clerk Maxwell and Ludwig Boltzmann developed in the late 19th century. Called the Boltzmann equation, it describes the likely behavior of the gas’s molecules, telling you how many particles you can expect to find at different locations moving at different speeds. This model of the gas lets physicists study how air moves at small scales — for instance, how it might flow around a space shuttle(opens a new tab). What mathematicians do to physicists is they wake us up. Gregory Falkovich Zoom out again, and you can no longer tell that the gas is made up of individual particles. It acts like one continuous substance. To model this macroscopic behavior — how dense the gas is and how fast it’s moving at any point in space — you’ll need yet another set of equations, called the Navier-Stokes equations. Physicists view these three different models of the gas’s behavior as compatible; they’re simply different lenses for understanding the same thing. But mathematicians hoping to contribute to Hilbert’s sixth problem wanted to prove that rigorously. They needed to show that Newton’s model of individual particles gives rise to Boltzmann’s statistical description, and that Boltzmann’s equation in turn gives rise to the Navier-Stokes equations. Mathematicians have had some success with the second step, proving that it’s possible to derive a macroscopic model of a gas from a mesoscopic one in various settings. But they couldn’t resolve the first step, leaving the chain of logic incomplete. Now that’s changed. In a series of papers, the mathematicians Yu Deng(opens a new tab), Zaher Hani(opens a new tab) and Xiao Ma(opens a new tab) proved the harder microscopic-to-mesoscopic step(opens a new tab) for a gas in one of these settings, completing the chain(opens a new tab) for the first time. The result and the techniques that made it possible are “paradigm-shifting,” said Yan Guo(opens a new tab) of Brown University. A man standing by a chalkboard Yu Deng usually studies the behavior of systems of waves. But by applying his expertise to the realm of particles, he has now resolved a major open problem in mathematical physics. Courtesy of Yu Deng Declaration of Independence Boltzmann could already show that Newton’s laws of motion give rise to his mesoscopic equation, so long as one crucial assumption holds true: that the particles in the gas move more or less independently of each other. That is, it must be very rare for a particular pair of molecules to collide with each other multiple times. But Boltzmann could not definitively demonstrate that this assumption was true. “What he could not do, of course, is prove theorems about this,” said Sergio Simonella(opens a new tab) of Sapienza University in Rome. “There was no structure, there were no tools at the time.” A bespectacled man in a suit The physicist Ludwig Boltzmann studied the statistical properties of fluids. Creative Commons After all, there are infinitely many ways a collection of particles might collide and recollide. “You just get this huge explosion of possible directions that they can go,” Levermore said — making it a “nightmare” to actually prove that scenarios involving many recollisions are as rare as Boltzmann needed them to be. In 1975, a mathematician named Oscar Lanford managed to prove this(opens a new tab), but only for extremely short time periods. (The exact amount of time depends on the initial state of the gas, but it’s less than the blink of an eye, according to Simonella.) Then the proof broke down; before most of the particles got the chance to collide even once, Lanford could no longer guarantee that recollisions would remain a rare occurrence. In the decades since, many mathematicians tried to extend his result, to no avail. Then, in November of 2023, Deng, now at the University of Chicago, and Hani, of the University of Michigan, posted a preprint(opens a new tab) that teased the desired proof. A forthcoming paper, they wrote, would build off their latest result to investigate “the long-time extension of Lanford’s theorem.” Other mathematicians didn’t know what to make of the announcement. “I didn’t think it was possible,” said Pierre Germain(opens a new tab) of Imperial College London. Deng and Hani didn’t even usually work with particle systems; until that point, they’d mainly been studying systems made up of waves (like rays of light). So mathematicians eagerly awaited the promised proof. When Particles Collide Deng and Hani’s 2023 result involved an analysis of the transition from the microscopic scale to the mesoscopic scale in the context of waves. About a year before the mathematicians posted their paper online, Deng was at a conference, where he met with a graduate student at Princeton University named Xiao Ma. They ended up discussing Deng and Hani’s work, and how they might adapt the methods to particles. Doing so would allow them to extend Lanford’s result — to show that particle recollisions are rare even on longer timescales. It was an idea that Deng and Hani had already been considering. Impressed by Ma’s insights on the topic, Deng invited him to help them turn their intuition into a proof. The trio hoped to focus on a much-studied scenario where mathematicians had already proved the second, meso-to-macro step in Hilbert’s sixth problem. In this scenario, a dilute gas of spherical particles is trapped in a box. If a particle hits one of the box’s walls, it reappears on the opposite wall. But to prove the harder micro-to-meso step for this setting — thereby resolving Hilbert’s sixth problem — Deng, Hani and Ma had to port their wave-based techniques over to particles. So they started in a setting where that task would be a little bit easier. They worked with a gas whose particles are distributed randomly in an infinite amount of space; unlike the particles in the boxed gas, which keep bouncing off each other forever, these particles eventually disperse and stop colliding. “In the whole-space case, there is a shortcut,” Deng said. A woman surfing The three mathematicians first needed to tabulate the different patterns of collisions that might occur in their gas, and how likely each of those patterns was. They could easily rule out scenarios with particularly high rates of recollisions. This left them with a finite, though still massive, number of patterns to analyze — each involving a certain subset of particles colliding, in a certain order. Once they knew exactly what each pattern entailed, they could use that information to estimate its likelihood of occurring. But that often felt like an impossible task, because many of the patterns involved huge numbers of particles and intricate, indirect interactions between them. “The structure of these sets [of colliding particles] gets exceedingly complicated,” Deng said. In principle, the mathematicians would need to keep track of every one of these particles simultaneously to compute the probability estimates they needed. That’s where Deng and Hani’s previous work on waves gave them an important insight. In that result, they’d figured out ways to break up complicated patterns of interacting waves into simpler ones. They’d carefully crafted their technique so that, by working with only a few waves at a time, they could still get a good estimate for the likelihood of the more complicated complete wave pattern. They hoped the same idea would work in the particle setting. But after a collision, particles behave very differently from waves. For instance, particles, unlike waves, bounce off each other, greatly affecting the resulting pattern of collisions and its probability of happening. Deng, Hani and Ma needed to rework the details of their strategy from the beginning. A smiling man Zaher Hani studies solutions to equations that arise in oceanography, plasma physics and quantum mechanics. Courtesy of Zaher Hani First, they tackled the simplest cases, in which each particle collides just a few times over a very short time span, with no recollisions. They then gradually moved on to harder and harder cases — longer amounts of time, with more collisions and recollisions. It was as much an art as a science. “The intuition was developed gradually, starting with some unsuccessful attempts,” Deng said. They had to get a sense for how to slice up large, complicated patterns of particle collisions in a way that would simplify their calculations while keeping their estimates highly accurate. Famous Fluid Equations Are Incomplete mathematical physics Famous Fluid Equations Are Incomplete July 21, 2015 “This is a process that takes months,” Hani said. “We would be stuck constantly.” Nearly every day, they jumped on a Zoom meeting to talk things through. “Much to the dismay of my wife, some of them happened very late at night, or very early in the morning,” Hani said. “I would put my daughter to sleep, and then we would have two or three hours of Zoom meetings.” Finally, by the spring of 2024, the trio was sure they had covered everything. Their proof, which they posted online(opens a new tab) that summer, confirmed that recollisions had to be very, very uncommon. They’d shown, as they’d hoped to, that in their infinite-space setting, Boltzmann’s description of the gas could be derived from Newton’s. The microscopic and mesoscopic scales fell under a single rigorous mathematical framework. “I think it’s outstanding work,” said Alexandru Ionescu(opens a new tab), a mathematician at Princeton who was also Deng and Ma’s doctoral adviser. “These are some of the most significant advances in many, many years.” They were now ready to return to the gas-in-a-box setting, where they could finally solve Hilbert’s sixth problem. The Completed Chain It didn’t take long for them to extend their result from the infinite-space setting to the boxed one. “Eighty percent of the proof is still the same in the whole-space case,” Deng said. In March, they posted a new paper(opens a new tab) that combined their proof with the earlier results connecting the Boltzmann equation to the Navier-Stokes equations. The logical chain was complete: They’d shown that, for a realistic model of a gas, a microscopic description of individual particles does indeed ultimately give rise to a macroscopic description of the gas’s large-scale behavior. The work didn’t just mark the resolution of a major case of Hilbert’s sixth problem. It also provided a rigorous mathematical resolution of an old paradox. At the microscopic scale, where particles act like billiard balls, time is reversible. Newton’s equations predict both where a particle comes from and where it’s going. The future is not fundamentally different from the past. But at the mesoscopic and macroscopic levels, there is no going back in time. “We know very well that, going forward in time, one ages but does not rejuvenate; heat does not spontaneously pass from a cold body to a warm body; a drop of ink in a glass of water spreads, darkening the liquid, but does not spontaneously return to the small, round shape it originally had,” Simonella wrote(opens a new tab). Neither the Boltzmann equation nor the Navier-Stokes equations are time-reversible; if you try to run time backward, the results will be nonsensical. To Boltzmann’s contemporaries, this was perplexing. How could a time-irreversible equation be derived from a time-reversible system? But Boltzmann argued that there was no paradox: Even if each particle can be modeled in a time-reversible way, almost every collision pattern ends up with a gas dispersing. The chance of, say, a gas suddenly contracting is essentially zero. Related: The Trouble With Turbulence New ‘Superdiffusion’ Proof Probes the Mysterious Math of Turbulence New Proofs Probe the Limits of Mathematical Truth Lanford had confirmed this intuition mathematically for his very short time frame. Now Deng, Hani and Ma’s result confirms it for more realistic situations. Going forward, mathematicians — who are still poring over the details of the new proof — want to test whether similar techniques might be useful in other, even more realistic contexts. These might include gases made up of particles of different shapes, or particles that interact in more complicated ways. Meanwhile, Falkovich said, these sorts of rigorous proofs can help physicists understand why a gas behaves a certain way at various scales, and why different models might be more or less effective in different scenarios. “What mathematicians do to physicists,” he said, “is they wake us up.” Editor’s Note: Deng and Hani’s work on the system of waves was funded in part by the Simons Foundation, which also funds this editorially independent magazine.

Even with a bullet proof vest the hate still hurts.

72% of American Jews hold a favorable view of Israel. While American Neo Nazi's hold a 0% percent favor for Israel and prefer its's destruction. What do you think?

Pro-Israel EU Members:Germany, Italy, Austria, Hungary, Czech Republic, Croatia: These countries blocked an anti-Israel initiative led by Spain, Slovenia, and Ireland in July 2025, indicating support or at least reluctance to criticize Israel harshly.

Our Additional Commitments to Combatting Antisemitism July 15, 2025 Alma Mater looks over Low Plaza. Dear members of the Columbia community, In recent weeks, in anticipation of the fall semester, my focus has been on the work we can do to support a civil, tolerant, flourishing community at Columbia. A place where all feel welcome, where different viewpoints, different ideas, can be shared respectfully, and where, therefore, deep intellectual vibrancy is possible. Many efforts are underway, and you will be hearing more about all of them in the coming weeks. Our work toward an agreement with the federal government has put a harsh spotlight on many of the difficult issues regarding discrimination and harassment we’ve seen on our campuses. The fact that we’ve faced pressure from the government does not make the problems on our campuses any less real; a significant part of our community has been deeply affected in negative ways. In my view, any government agreement we reach is only a starting point for change. Committing to reform on our own is a more powerful path. It will better enable us to recognize our shortcomings and create lasting change. Today, I write to you, specifically, about our ongoing efforts to combat antisemitism. There is no place for intimidation, hateful language, or targeting of Jews or Israelis at Columbia, and we have zero tolerance for this behavior. Over the last year, guided by our principles of academic freedom, inclusion, and respect, and the important work of our Antisemitism Task Force, we have enacted meaningful reforms, understanding that more would be needed. After deep consultation with our Jewish community, and many friends and experts outside of our institution, we are taking some important additional steps. International Holocaust Remembrance Alliance (IHRA) Definition of Antisemitism As part of our March 21st commitments, Columbia announced we would incorporate the definition of antisemitism, as recommended by our Antisemitism Task Force in August 2024, into our anti-discrimination policies. We felt then, as we do now, that it is important to use a definition of antisemitism that reflects the experiences of many within Columbia’s Jewish community. Our Task Force had recommended that definition for use in education and pedagogy. While we remain committed to that carefully constructed definition, we are today also formally incorporating the IHRA definition of antisemitism into the work of our Office of Institutional Equity (OIE), housed under the Office of the Provost. While OIE has operated in a manner consistent with applicable regulations and guidance from the Department of Education’s Office for Civil Rights (OCR), including OCR’s 2021 and 2024 guidance, the formal incorporation of this definition will strengthen our response to and our community’s understanding of modern antisemitism. That guidance directs schools to consider the IHRA definition of antisemitism and its accompanying examples to the extent that any such examples might be useful as evidence of discriminatory intent. The IHRA definition is similarly used by many universities and colleges across the country. Columbia is committed to taking all possible steps to combat antisemitism and the University remains dedicated to ensuring that complaints of discrimination and harassment of all types, including complaints based on Jewish and Israeli identity, are treated in the same manner. Formally adding the consideration of the IHRA definition into our existing anti-discrimination policies strengthens our approach to combating antisemitism. Appoint Title VI and Title VII Coordinators Columbia will appoint Title VI and Title VII Coordinators to review and respond to allegations under University policies implementing the requirements of Title VI and Title VII, ensuring compliance with the laws’ prohibition of discrimination, based on race, color, religion, sex, or national origin, as well as its prohibition on retaliation. The coordinators will be part of the OIE and will have both advising and enforcement responsibilities. We believe the addition of these positions will ensure swift attention and action on complaints about violations of these policies. Additionally, these coordinators will contribute to a publicly available annual report to University leadership and the Board of Trustees, which will be part of a broader OIE annual report and will include reporting on Title VI and Title VII complaints, investigations, and outcomes. We hope this expanded annual report will give our community confidence that our systems addressing discrimination and harassment on our campuses are working. Additional Training on Antisemitism In addition to the existing Title VI antidiscrimination training—which specifically addresses antisemitism and is required of all students, faculty, and staff—we believe that supporting a deeper education on antisemitism is critical. Over the last six months, after extraordinarily thoughtful and often painful conversations with leaders from our own Jewish community, and nationally and internationally, it’s become clear that ongoing educational work will be the most critical and consequential of all our efforts to combat antisemitism. Implementing meaningful educational programs that engage not only our students, but also our faculty and staff, will have the most lasting impact, and they will also take the most time and effort. To begin, we are initiating programming and training partnerships with several national Jewish organizations including Project Shema, the Anti-Defamation League (ADL), the Foundation to Combat Antisemitism, and Kalaniyot; we are also exploring a cooperation with Yad Vashem. We hope to engage our community, in many ways, in an ongoing dialogue on the roots of antisemitism and its modern forms. These partnerships are intended to bring our community together and foster a greater understanding of these issues using tools we value at Columbia—education and respectful conversation. We are also collaborating with Interfaith America, the Constructive Dialogue Institute, and StoryCorps' One Small Step program to build programs that will go beyond traditional trainings, and the focus on antisemitism, to build bridges more broadly, to create constructive dialogue, and to deepen our understanding of each other. Affirmation of Zero Tolerance I want to reiterate that the University has zero tolerance for discrimination and harassment based on protected traits, including Jewish and Israeli identity. And while our University Rules and Policies are well-defined on this matter, beginning this 2025-26 academic year, Columbia will make clear our “Zero Tolerance for Antisemitism and Hate” in regular community messages. As part of this initiative, Columbia has not, and will not, recognize or meet with the group that calls itself “Columbia University Apartheid Divest” (CUAD), its representatives, or any of its affiliated organizations. Organizations that promote violence or encourage disruptions of our academic mission are not welcome on our campuses and the University will not engage with them. ***** I am so grateful to the many community members who have given so generously of their time to offer their deep concerns and constructive and critical ideas. I would also add that making these announcements in no way suggests we are finished with the work. In a recent discussion, a faculty member and I agreed that antisemitism at this institution has existed, perhaps less overtly, for a long while, and the work of dismantling it, especially through education and understanding, will take time. It will likely require more reform. But I’m hopeful that in doing this work, as we consider and even debate it, we will start to promote healing and to chart our path forward. Sincerely, Claire Shipman Acting President, Columbia University in the City of New York

did england help start the UN Absolutely — England played a major role in the founding of the United Nations, though technically it was the United Kingdom as a whole (which includes England, Scotland, Wales, and Northern Ireland). 🇬🇧 Here's how the UK helped shape the UN: Early Visionaries: British Prime Minister Winston Churchill and U.S. President Franklin D. Roosevelt co-authored the Atlantic Charter in 1941, which laid the groundwork for the UN’s principles. Key Conferences: The UK was deeply involved in pivotal wartime meetings like the Tehran, Yalta, and San Francisco conferences, where the structure and goals of the UN were hammered out. Founding Member: The UK was one of the original 51 countries that signed the UN Charter in 1945 and is a permanent member of the UN Security Council. First UN Meetings: The very first sessions of the UN General Assembly and Security Council were held in London in 1946 — a symbolic nod to Britain’s influence. So yes, England — as part of the UK — was absolutely central to the birth of the UN.

Germany learns from History and look!

The Rapture!

UN is an Iranian Proxy: Samantha Ettus @samanthaettus · 23h PLEASE WATCH: There is NO famine, there is an intentional aid blockage. TODAY 950 food trucks inspected and approved by Israel are rotting on the Gaza side of the Kerem Shalom crossing because the UN refuses to distribute them. Why? Because they want to control the entry of food so that they can get it to go directly into the hands of Hamas. The Humanitarian Aid Fund (GHF) today offered the UN to distribute the aid on its behalf free of charge - and the UN refused. Together, the UN and Hamas are preventing the entry of aid to the Gaza Strip while simultaneously promoting the false campaign of famine so that they can turn the world against Israel while continuing the war.