News from www.caltech.eduhttps://www.caltech.edu/about/news2024-07-15T17:33:12.394876+00:00The Office of Strategic Communicationswww@caltech.eduCopyright © 2024 California Institute of TechnologyTom Hutchcroft Receives European Mathematical Society Prize2024-07-15T17:33:12.394876+00:002024-07-15T17:33:12.329922+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/tom-hutchcroft-receives-european-mathematical-society-prize<p data-block-key="b51mk">Professor of Mathematics <a href="https://pma.caltech.edu/people/thomas-m-hutchcroft">Tom Hutchcroft</a> has been awarded a 2024 European Mathematical Society (EMS) prize "for his revolutionary contributions to probability theory and geometric group theory, in particular to percolation theory on general graphs, using tools from geometry, operator theory, group theory and functional analysis."</p><p data-block-key="d0hiq"></p><p data-block-key="2vlee">The award will be given in person at a ceremony in Seville, Spain, on July 15, 2024. Every four years, the <a href="https://euromathsoc.org/ems-prizes">EMS awards up to 10 prizes</a> to early career researchers "not older than 35 years at the time of nomination, of European nationality or working in Europe, in recognition of excellent contributions in mathematics," according to the society.</p><p data-block-key="70bkp"></p><p data-block-key="9s68b">Hutchcroft, who is from England, specializes in percolation theory, which is the mathematical study of what happens when liquids flow through a porous medium, such as water percolating through ground coffee beans to make an espresso. He studies what happens during a critical phase transition in percolation, the point where an abrupt qualitive change in the system occurs. During this phase transition, interesting fractal patterns (self-similar patterns seen at different scales) emerge. "You only change one factor in the system a tiny bit, and then you get a big change," said Hutchcroft when explaining these phase transitions in a Caltech interview.</p><p data-block-key="12jfp"></p><p data-block-key="63vbu">Hutchcroft is known in particular for his work exploring the behavior of percolation and other probabilistic processes in curved, non-Euclidean geometries. It is believed that additional phase transitions occur in these curved geometries that do not occur in ordinary, flat Euclidean space, and Hutchcroft has proved that this is so for several important cases. His work is noted for its interdisciplinary character, blurring the borders between probability theory and group theory, the latter of which is the abstract mathematical study of symmetry.</p><p data-block-key="3251j"></p><p data-block-key="3824q">Hutchcroft earned his bachelor's degree in mathematics from Cambridge University in 2013 and his PhD in mathematics from the University of British Columbia, Canada, in 2017. He held internships at Microsoft Research Theory Group during his graduate studies and later completed postdoctoral fellowships at the University of Cambridge from 2017 to 2021. He joined the Caltech faculty in 2021.</p><p data-block-key="aa09s"></p><p data-block-key="83ov2">Other Caltech-affiliated mathematicians who have received the EMS prize include Stanislav Smirnov (PhD '96) in 2004, former Caltech professor Alexei Borodin in 2008, and former Caltech postdoc Adrian Ioana in 2012.</p>Math Celebration at Caltech Builds Community2024-07-05T16:42:00+00:002024-07-11T20:55:47.515809+00:00Caltech Communicationswww@caltech.eduhttps://www.caltech.edu/about/news/math-celebration-at-caltech<p data-block-key="319i8">In the Richard N. Merkin Center for Pure and Applied Mathematics high in Caltech Hall, the American Institute of Mathematics (AIM) gathers mathematicians to solve confounding puzzles, often using only chalk. AIM carried that analog approach outside to celebrate its 30th birthday with a free public math party on the lawn beside Caltech Hall on June 29.</p><p data-block-key="5d99k">Hundreds of families and individuals turned out to play math games while sitting on blankets spread out under trees and at tables under a shaded arcade. Pythagoras would have felt at home.</p><p data-block-key="7qnbj">At more than a dozen stations, volunteers coached participants and explained the mathematical principles at work in games including SET, Prime Climb, Rubik's Cube, wooden 3D puzzles, unfair dice, and walking puzzles. Traditional paper games featured wolves and sheep, cats and dogs, and magic circles. One improvised paper game taught children to spot base systems, from binary's base 2 to the ancient Sumerian and Babylonian base 60.</p><p data-block-key="elasm">"We wouldn't have been able to pull off this milestone event without the incredible dedication and enthusiasm of everyone who volunteered to help," says Sergei Gukov, the John D. MacArthur Professor of Theoretical Physics and Mathematics and consulting director of AIM. The volunteers included <a href="https://aimath.org/">AIM</a> staff, postdoctoral scholars organized by the <a href="https://cpa.caltech.edu/">Caltech Postdoctoral Association</a>, Caltech professors and students, and math professors and teachers from around LA.</p><p data-block-key="f8gcc">"Today's activities opened my eyes to non-online games, ones families used to play around the dinner table. This is where a child's exponential growth starts," said visitor Ms. Y. M. Wong. She is homeschooling her son, Atti, who played beside her, and she relies on math circles and activities like the AIM fair. "I wasn't sure how homeschooling would go, but with this kind of community, I am happy."</p><p data-block-key="bg52a">By way of fuel and fun, children lined up for free churros and Sno-Kones, transformed their looks with free face painting, accessorized with balloon animals, and crowded the podium during raffle announcements for free games and books to take home. A taco truck did brisk business.</p><p data-block-key="4odak">AIM's birthday party drew hundreds of people who wanted to engage their curiosity, challenge their minds, and play together. That same sense of community is vital to progress in mathematics as the field transitions from an individualistic model to a collaborative one, says Gukov. The collaborative model was not widely accepted 30 years ago. Gukov credits its gradual adoption to the foresight and dedicated efforts of AIM's founders. Together with the <a href="https://merkincenter.caltech.edu/">Merkin Center</a> (which <a href="/about/news/New_Merkin_Center_Takes_in_AIM">took in AIM in 2023</a>), AIM continually works to connect people, from the wisest mathematicians to the newest.</p>Caltech Professors Win 2024 Sloan Fellowships2024-02-20T17:21:35.741172+00:002024-02-20T17:21:35.531604+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/caltech-professors-win-2024-sloan-fellowships<p data-block-key="sjv42">Four Caltech faculty members have been awarded the prestigious Sloan Research Fellowship for 2024: Katie Bouman, assistant professor of computing and mathematical sciences, electrical engineering and astronomy; Lee McCuller, assistant professor of physics; Vikram Ravi, assistant professor of astronomy; and Antoine Song, assistant professor of mathematics.</p><p data-block-key="99nha">The fellowships honor exceptional U.S. and Canadian researchers "whose creativity, innovation, and research accomplishments make them stand out as the next generation of leaders," according to the <a href="https://sloan.org/">Alfred P. Sloan Foundation</a>, which has been granting the awards annually since 1955. The Caltech professors are among 126 early-career scientists who have been selected to receive the fellowships, which come with $75,000 over two years to advance research projects.</p><p data-block-key="4l2id"><a href="/about/news/seeing-farther-and-deeper-interview-katie-bouman">Katie Bouman</a> is a computational imaging scientist whose methods combine ideas from signal processing, computer vision, machine learning, and physics to bring out hidden signals in scientific and technical data. She is a key member of the <a href="https://eventhorizontelescope.org/">Event Horizon Telescope</a> project, which made history in 2019 by unveiling the first image of a black hole, in this case, a supermassive black hole lying at the heart of the M87 galaxy. Using data acquired by a global network of radio telescopes, Bouman and her teammates developed a computational approach that transformed the black hole data into an image. Since then, her team has helped produce an <a href="/about/news/caltech-researchers-help-generate-first-image-of-black-hole-at-the-center-of-our-galaxy">image of the supermassive black hole at the heart of our Milky Way Galaxy</a>, called Sagittarius A*, as well as a new <a href="/about/news/new-data-same-great-appearance-for-m87">image of the M87 black hole made with enhanced data</a>. Bouman is also developing next-generation computational cameras for other imaging problems in astronomy, medicine, and seismology, where traditional cameras will not work.</p><p data-block-key="899b3"><a href="/about/news/at-the-edge-of-physics">Lee McCuller</a> is an expert at creating technologies that make the most precise measurements in the world. These quantum measurements are at the heart of the <a href="https://www.ligo.caltech.edu/">Laser Interferometer Gravitational-wave Observatory</a>(LIGO), which has been detecting gravitational waves from colliding black holes and neutron stars since it first detected ripples in space-time in 2015. McCuller helped lead the development of essential technology at LIGO based on a concept called quantum squeezing. This method helps reduce unwanted noise in LIGO's detectors—noise that bubbles up from the quantum realm—to make the facilities even more sensitive to gravitational waves. Recently, Lee and his LIGO collaborators <a href="/about/news/ligo-surpasses-the-quantum-limit">took the technology one step further</a> to make quantum squeezing work across the range of gravitational frequencies detected by LIGO. The research helped surpass limits imposed by quantum physics and made LIGO even more powerful.</p><p data-block-key="1kqt6"><a href="/about/news/build-it-and-they-will-come-fast-radio-bursts">Vikram Ravi</a> is an astronomer specializing in energetic dynamic events, such as fast radio bursts, or FRBs, which are powerful eruptions of radio waves that <a href="/about/news/fast-radio-burst-pinpointed-distant-galaxy">originate primarily from remote galaxies</a> and whose cause remains unknown. Ravi co-led the development of the Deep Synoptic Array-110, an array of radio dishes at Caltech's Owens Valley Radio Observatory, which has now identified and pinpointed over 60 FRBs to their galaxies of origin. Ravi plans to use this growing sample—a significant fraction of the fewer than 90 FRBs so far associated with galaxies—to better understand <a href="/about/news/cosmic-burst-probes-milky-ways-halo">how matter is distributed within and in between galaxies</a>. He is also embarking on a new program to identify tidal disruption events, or TDEs, which occur when a star wanders too close to a black hole and is devoured. Ravi and his colleagues will use data from the <a href="https://ztf.ipac.caltech.edu/">Zwicky Transient Facility</a> (ZTF) at Caltech's Palomar Observatory to search for more obscure TDEs, and help piece together the puzzles of how often these events occur as well as why they do not look the same when viewed at different wavelengths of light.</p><p data-block-key="5fteo"><a href="/about/news/Geometry_of_Minimal_Surfaces">Antoine Song</a> specializes in differential geometry, the study of shapes using analysis and differential equations. His goal is to better understand minimal surfaces, geometrical shapes that minimize total surface area and total energy; for instance, a circular wire frame dipped in soapy water leads to a film with a minimal optimal shape. For his PhD thesis, Song showed that there are infinitely many minimal surfaces in any three-dimensional space. Recently, Song and his colleague Conghan Dong, a graduate student at Stony Brook University, helped prove an aspect of Albert Einstein's general theory of relativity, showing that "a sequence of curved spaces with smaller and smaller amounts of mass will eventually converge to a flat space with zero curvature," according to <a href="https://www.quantamagazine.org/a-century-later-new-math-smooths-out-general-relativity-20231130/">Quanta Magazine</a>.</p><p data-block-key="12ql6">Read more about the Sloan Research Fellowships at its <a href="https://sloan.org/fellowships/">website</a>.</p>A Two-Way Street Connecting Math and Machine Learning2023-12-06T22:12:00+00:002023-12-08T01:12:40.017617+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/a-two-way-street-connecting-math-and-machine-learning<p data-block-key="l68x4">Traditionally, mathematicians jot down their formulas using paper and pencil, seeking out what they call pure and elegant solutions. In the 1970s, they hesitantly began turning to computers to assist with some of their problems. Decades later, computers are often used to crack the hardest math puzzles. Now, in a similar vein, some mathematicians are turning to machine learning tools to aid in their numerical pursuits.</p><p data-block-key="4m45">"Mathematicians are beginning to embrace machine learning," says Sergei Gukov, the John D. MacArthur Professor of Theoretical Physics and Mathematics at Caltech, who put together the upcoming <a href="https://mathml2023.caltech.edu/">Mathematics and Machine Learning 2023</a> conference, which takes place at Caltech December 10–13.</p><p data-block-key="bfk8n">"There are some mathematicians who may still be skeptical about using the tools," Gukov says. "The tools are mischievous and not as pure as using paper and pencil, but they work."</p><p data-block-key="dtf6f"><a href="https://scienceexchange.caltech.edu/topics/artificial-intelligence-research/artificial-intelligence-vs-machine-learning">Machine learning is a subfield of AI</a>, or artificial intelligence, in which a computer program is trained on large datasets and learns to find new patterns and make predictions. The conference, the first put on by the new <a href="https://merkincenter.caltech.edu">Richard N. Merkin Center for Pure and Applied Mathematics</a>, will help bridge the gap between developers of machine learning tools (the data scientists) and the mathematicians. The goal is to discuss ways in which the two fields can complement each other.</p><p data-block-key="4s0ml">"It's a two-way street," says Gukov, who is the director of the new Merkin Center, which was <a href="/about/news/merkin-center-for-pure-and-applied-mathematics">established by Caltech Trustee Richard Merkin</a>.</p><p data-block-key="a7f0i">"Mathematicians can help come up with clever new algorithms for machine learning tools like the ones used in <a href="https://scienceexchange.caltech.edu/topics/artificial-intelligence-research/generative-ai?utm_medium=web&utm_campaign=cseai&utm_source=caltechcarousel&utm_content=&utm_term=">generative AI programs</a> like ChatGPT, while machine learning can help us crack difficult math problems."</p><p data-block-key="bdbvq">Yi Ni, a professor of mathematics at Caltech, plans to attend the conference, though he says he does not use machine learning in his own research, which involves the field of topology and, specifically, the study of mathematical knots in lower dimensions. "Some mathematicians are more familiar with these advanced tools than others," Ni says. "You need to know somebody who is an expert in machine learning and willing to help. Ultimately, I think AI for math will become a subfield of math."</p><p data-block-key="994l3">One tough problem that may unravel with the help of machine learning, according to Gukov, is known as the Riemann hypothesis. Named after the 19th-century mathematician Bernhard Riemann, this problem is one of seven Millennium Problems selected by the Clay Mathematics Institute; a $1 million prize will be awarded for the solution to each problem.</p><p data-block-key="4rfqd">The Riemann hypothesis centers around a formula known as the Riemann zeta function, which packages information about prime numbers. If proved true, the hypothesis would provide a new understanding of how prime numbers are distributed. Machine learning tools could help crack the problem by providing a new way to run through more possible iterations of the problem.</p><p data-block-key="emktd">"Machine learning tools are very good at recognizing patterns and analyzing very complex problems," Gukov says.</p><p data-block-key="cardk">Ni agrees that machine learning can serve as a helpful assistant. "Machine learning solutions may not be as beautiful, but they can find new connections," he says. "But you still need a mathematician to turn the questions into something computers can solve."</p><p data-block-key="ecpgs">Gukov has used machine learning himself to untangle problems in knot theory. Knot theory is the study of abstract knots, which are similar to the knots you might find on a shoestring, but the ends of the strings are closed into loops. These mathematical knots can be entwined in various ways, and mathematicians like Gukov want to understand their structures and how they relate to each other. The work has relationships to other fields of mathematics such as representation theory and quantum algebra, and even quantum physics.</p><p data-block-key="dvn8u">In particular, Gukov and his colleagues are working to solve what is called the smooth Poincaré conjecture in four dimensions. The original Poincaré conjecture, which is also a Millennium Problem, was proposed by mathematician Henri Poincaré early in the 20th century. It was ultimately solved from 2002 to 2003 by Grigori Perelman (who famously turned down his prize of $1 million). The problem involves comparing spheres to certain types of manifolds that look like spheres; manifolds are shapes that are projections of higher-dimensional objects onto lower dimensions. Gukov says the problem is like asking, "Are objects that look like spheres really spheres?"</p><p data-block-key="am2rc">The four-dimensional smooth Poincaré conjecture holds that, in four dimensions, all manifolds that look like spheres are indeed actually spheres. In an attempt to solve this conjecture, Gukov and his team develop a machine learning approach to evaluate so-called ribbon knots.</p><p data-block-key="b2eis">"Our brain cannot handle four dimensions, so we package shapes into knots," Gukov says. "A ribbon is where the string in a knot pierces through a different part of the string in three dimensions but doesn't pierce through anything in four dimensions. Machine learning lets us analyze the 'ribboness' of knots, a yes-or-no property of knots that has applications to the smooth Poincaré conjecture."</p><p data-block-key="3eo20">"This is where machine learning comes to the rescue," writes Gukov and his team in a preprint paper titled "<a href="https://arxiv.org/pdf/2304.09304.pdf">Searching for Ribbons with Machine Learning</a>." "It has the ability to quickly search through many potential solutions and, more importantly, to improve the search based on the successful 'games' it plays. We use the word 'games' since the same types of algorithms and architectures can be employed to play complex board games, such as Go or chess, where the goals and winning strategies are similar to those in math problems."</p><p data-block-key="48akc">On the flip side, math can help in developing machine learning algorithms, Gukov explains. A mathematical mindset, he says, can bring fresh ideas to the development of the algorithms behind AI tools. He cites Peter Shor as an example of a mathematician who brought insight to computer science problems. Shor, who graduated from Caltech with a bachelor's degree in mathematics in 1981, famously came up with what is known as Shor's algorithm, a set of rules that could allow quantum computers of the future to factor integers faster than typical computers, thereby <a href="https://scienceexchange.caltech.edu/topics/quantum-science-explained/quantum-cryptography">breaking digital encryption codes</a>.</p><p data-block-key="11leu">Today's machine learning algorithms are trained on large sets of data. They churn through mountains of data on language, images, and more to recognize patterns and come up with new connections. But data scientists don't always know how the programs reach their conclusions. The inner workings are hidden in a so-called "black box." A mathematical approach to developing the algorithms would reveal what's happening "under the hood," as Gukov says, leading to a deeper understanding of how the algorithms work and thus can be improved.</p><p data-block-key="6g56q">"Math," says Gukov, "is fertile ground for new ideas."</p><p data-block-key="eubf">The conference will take place at the Merkin Center on the eighth floor of Caltech Hall.</p>Gary A. Lorden (BS '62), (1941–2023)2023-11-11T00:27:00+00:002023-11-10T19:41:31.213758+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/gary-a-lorden-bs-62-19412023<p data-block-key="aznlh">Gary Allen Lorden (BS '62), professor of mathematics, emeritus, passed away on October 25, 2023. He was 82.</p><p data-block-key="em0b5">Lorden was born in Los Angeles, California, on June 10, 1941. He received a BS from Caltech in 1962 and a PhD from Cornell University in 1966. He rejoined the Caltech community as an assistant professor of mathematics in 1968, became associate professor in 1971, and professor in 1977. He retired in 2009.</p><p data-block-key="1psag">"In 1958, when Gary and I arrived at Caltech as a freshmen, he, in addition to being superb at mathematics, was also an outstanding pianist," recalls Kip S. Thorne (BS '62), the Richard P. Feynman Professor of Theoretical Physics, Emeritus. "He did unbelievably good imitations of Liberace, much to the enjoyment of all his classmates."</p><p data-block-key="25nsr">Lorden's mathematical research involved statistics. He was especially interested in applications to real-world problems and often served as an expert witness in trials. Over the years, Lorden developed connections to Hollywood, and he served as technical advisor to the 2005 crime drama show <i>NUMB3RS</i>, consulting on 99 of the 118 episodes. He co-wrote a book inspired by the show, titled "<a href="https://www.amazon.com/Numbers-Behind-NUMB3RS-Solving-Mathematics-ebook/dp/B000UG78MW"><i>The Numbers behind</i> <i>NUMB3RS: Solving Crime with Mathematics</i></a><i>,"</i> which explains real-life math techniques used by the FBI and other law enforcement agencies.</p><p data-block-key="3075s">"Through <i>NUMB3RS</i> and his accompanying book, Gary aroused public appreciation for mathematics in a manner that inspired me in my subsequent effort to arouse people about the beauty of astrophysics via <i>Interstellar</i> and <i>The Science of Interstellar</i>, " Thorne says. "Gary was a wonderful friend and inspiration."</p><p data-block-key="771hg">"He loved solving problems and had a passion for math," says Barry Simon, the International Business Machines Professor of Mathematics and Theoretical Physics, Emeritus. "The math department is a very friendly place, but I would say that Gary was one of the sweetest people there."</p><p data-block-key="1k39k">Richard (Rick) Wilson, professor of mathematics, emeritus, says he once took over teaching one of Lorden's classes in statistics, which is not Wilson's primary field. "Gary used to say that mathematical statistics was important for students to learn because it uses a different way of thinking. After I taught his class, I wholeheartedly agreed."</p><p data-block-key="4sdoi">Lorden also served the Caltech community through leadership roles. He was dean of students from 1984 to 1988, vice president for student affairs from 1989 to 1998, and acting vice president for student affairs in 2002. He was executive officer for mathematics from 2003 to 2006. In addition, he served as chair of the Athenaeum's Board of Governors from 2010 until his passing.</p><p data-block-key="6gqev">"I came to know and deeply respect Gary through our shared services in Student Affairs at Caltech, roles in which we served for over 30 years," says Caltech's Christopher Brennen, the Richard L. and Dorothy M. Hayman Professor of Mechanical Engineering, Emeritus. "Through those connections and the various challenges they involved, I came to know and deeply respect Gary as person and a dear friend. Throughout those years he was, for me, the epitome of empathy, integrity, reliability, and kindness. He was, indeed, a dear, dear friend who also cared deeply for the Caltech students and the Caltech community."</p><p data-block-key="fi150">According to Brennen, Lorden was understanding and sympathetic with students due in part to his own undergraduate experiences at Caltech. "He knew what it was like to be a student at Caltech, and the pressures they can feel."</p><p data-block-key="ai5b1">Lorden liked to act and regularly participated in the <a href="http://www.its.caltech.edu/~plays/">Caltech Playreaders</a> series, in which members of the Caltech and JPL communities put on semi-staged readings of plays. He was also a member of the Caltech Athenaeum's wine committee, which selects wines for the faculty club. "I remember Gary as a very enthusiastic chair of the Athenaeum Wine Committee," says Anneila Sargent, the Ira S. Bowen Professor of Astronomy, Emeritus.</p><p data-block-key="druec">"And I remember his devotion to his wife Louise," Sargent adds. "And the pleasure they took, until her untimely death, in showing the students how well they danced together."</p><p data-block-key="8fqui">Lorden is survived by his children Lisa and Diana. His wife Louise passed away in 2015.</p>What Do a Jellyfish, a Cat, a Snake, and an Astronaut Have in Common? Math.2023-08-09T20:29:00+00:002023-08-09T20:37:27.495357+00:00Emily Velascoevelasco@caltech.eduhttps://www.caltech.edu/about/news/what-do-a-jellyfish-a-cat-a-snake-and-an-astronaut-have-in-common-math<p data-block-key="0pko4">Across the animal kingdom there are creatures that move through their environments not by walking or running or climbing but instead by simply changing the shape of their bodies. This kind of locomotion is found in snakes as they slither, in stingrays as a they swim, and even in cats as they twist themselves to land on their feet as they fall.</p><p data-block-key="41ev7">The contortions of a falling cat and a sidewinding snake might not appear to have much in common, but a team of researchers led by Caltech's <a href="https://www.eas.caltech.edu/people/ps">Peter Schröder</a> say they have discovered a single algorithm that describes both kinds of motion and the motions of many other animals that get around simply by changing shape.</p><p data-block-key="bij72"></p><p data-block-key="de4ga"></p><p data-block-key="7oaqt"></p><p data-block-key="ah4md">A paper describing their work appears in the August issue of <i>ACM Transactions on Graphics</i> and is being presented at this year's SIGGRAPH (Special Interest Group on Computer Graphics and Interactive Techniques) conference, which is being held August 6–10 at the Los Angeles Convention Center.</p><p data-block-key="97amg">"You have all kinds of creatures who have in common that they move about by changing their shape," says Schröder, the Shaler Arthur Hanisch Professor of Computer Science and Applied and Computational Mathematics. "One classic example is a single-cell organism. How does it move? It doesn't have legs. It doesn't have wings to fly with. The only thing it can really do is change its shape.</p><p data-block-key="dvum1"></p><embed alt="A microscope image of a white blood cell swimming through its environment." embedtype="image" format="MiddleAlignMedium" id="9944"/><p data-block-key="372nf"></p><p data-block-key="4oks8">"Once you understand that basic observation, you see there are all kinds of creatures who move by changing their shape. A snake makes an undulating motion and yet manages to move forward. Astronauts can turn in zero gravity by doing a dance-like motion that manages to turn them without the need to push off of a surface."</p><p data-block-key="425sa"></p><embed alt="An animated gif showing a jellyfish swimming, followed by a snake slithering, and then followed by an aardvark astronaut using a motion to turn in zero gravity." embedtype="image" format="MiddleAlignMedium" id="9942"/><p data-block-key="6cn1f"></p><p data-block-key="282kf">Schröder says all of these types of motion can be explained by the principle of least dissipation, a concept positing that natural systems will always try to be as efficient as possible. As an example, he cites an ice skate, which can easily slide forward or backward on ice but which has a great deal of difficulty sliding side to side. If a person is wearing ice skates and wants to skate forward, they push their skates sideways away from the center line of their body. Since it's difficult for a skate to skid sideways, the skates (and the person wearing them) will move forward, since forward motion is easier and more efficient. The system consisting of the skater, the skates, and the ice favors forward motion because it wastes the least energy.</p><p data-block-key="d4c4l">The same principle is at work when a snake undulates its way across the sandy ground of a desert. A snake, being long and skinny, can slide forward and backward much more easily than it can slide sideways. Since that undulation causes its body to slide sideways in a back-and-forth motion, much energy would be lost due to friction—unless the snake is moving forward while undulating. Since motion along the length of the snake encounters less friction, the system favors it, and the snake slithers along its merry, scaly way.</p><p data-block-key="f6e5m"></p><embed alt="A computer generated animated gif of a snake slithering." embedtype="image" format="MiddleAlignMedium" id="9943"/><p data-block-key="29cnh"></p><p data-block-key="7ovqr">All these kinds of locomotion were modeled on computers using the principle of least dissipation as a starting point. The animals were rendered as sets of flexible nodes connected by rigid bars and allowed the researchers to examine how the creatures move in a simulated space and compare it with real life data.<br/></p><p data-block-key="eb39s">Guided by the principle of least dissipation (and other math), these animal models showed movement remarkably like that seen in their real-world counterparts.</p><p data-block-key="8uf6l">"It's just beautiful that you can identify a fairly simple governing principle of a whole class of different kinds of motion," Schröder says. "It's not 100 percent accurate, but shows remarkable agreement with motion observed in real life, suggesting that it captures a major part of what happens in nature.</p><p data-block-key="1j3bt">"There's a certain mathematical beauty when you have a very simple principle that can explain a whole bunch of things at once. That's what gets me up in the morning."</p><p data-block-key="7jts">The paper describing the research is titled "<a href="https://dl.acm.org/doi/10.1145/3592417">Motion from Shape Change</a>." Additional co-authors Yousuf Soliman, graduate student in applied and computational math; and Oliver Gross, Marcel Padilla, Felix Knöppel, and Ulrich Pinkall of the Technical University of Berlin.</p>Caltech Mourns the Passing of David B. Wales (1939–2023)2023-07-20T20:38:00+00:002023-07-26T17:54:28.816478+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/caltech-mourns-the-passing-of-david-wales-19392023<p data-block-key="qd6y4"></p><p data-block-key="dmp2g">Caltech's mathematics professor David B. Wales, who spent more than 50 years at the Institute, passed away on July 17. In addition to serving on the faculty in the Division of Physics, Mathematics and Astronomy, he was also Caltech's associate dean of students from 1976 to 1980, dean of students from 1980 to 1984, executive officer for mathematics from 1985 to 1991, and the master of student houses from 1991 to 1997. Wales retired in 2008 but remained active in math research and attended talks at Caltech.</p><p data-block-key="48k7b">Wales was an expert in group theory, algebraic combinatorics, and representation theory. He spent the most time on finite group theory, searching for and studying what are called simple groups. In the same way that prime numbers can be thought of as the building blocks of integers, simple groups are the building blocks of finite groups, which are those groups with a finite number of elements.</p><p data-block-key="fsd5g">In an <a href="https://heritageproject.caltech.edu/interviews-updates/david-wales">interview with the Caltech Heritage Project</a>, Wales explained that "groups are algebraic constructions, which are sets with a multiplication between elements, which gives another element of the set. The multiplication has properties much like multiplication of ordinary numbers but with different rules. Certain groups are called simple. In a sense, they are the building blocks of all groups."</p><p data-block-key="4vs2n">"I always enjoyed talking to David over the years, and he always answered patiently a lot of questions in group theory, which is a neighboring discipline of my field of number theory," says Dinakar Ramakrishnan, the Taussky-Todd-Lonergan Professor, Emeritus, at Caltech. "I know many people liked to talk to David to see if his work could be useful to them. David was always quite helpful to others."</p><p data-block-key="dd6cm">Gary Lorden (BS '62), professor of mathematics, emeritus, and a longtime colleague and friend of Wales described him as a "master mentor." Wales, he said, had remarkable ability to distill complex concepts and information into the most accessible and understandable format. He did this in his research, publishing "beautiful proofs," in his teaching, and in his administrative roles and management. "He gave everybody eureka moments," Lorden said.</p><p data-block-key="dkh3e">Wales was born in 1939 in Vancouver, Canada. His father was a high school physics and math teacher, and later a school principal. Wales said that his mother, who had a degree in library science, spent her time raising him and his two younger brothers, Terry and Keith.</p><p data-block-key="eegrt">Wales grew up during World War II and, in his Heritage Project interview, recalled how his family had to pull down blackout blinds over the windows every night. "People were afraid that the Japanese would bomb Vancouver the way they did Pearl Harbor," he said. When the war ended in Europe, he and his family drove around town in his mother's Model-T Ford. "There were all kinds of people in the street waving flags and shouting and cheering, so it was exciting."</p><p data-block-key="2v3o">Wales earned his bachelor's and master's degrees from the University of British Columbia in 1961 and 1962, respectively. He earned his PhD from Harvard in 1967, studying under Richard Brauer, who was known for creating representations of finite groups. Wales joined Caltech as a Bateman Research Fellow in 1967; he became an assistant professor in 1968 and a full professor in 1977.</p><p data-block-key="1cq73">Wales said that one of the reasons he chose to come to Caltech was to work with the late Marshall Hall, a Caltech professor whom he called "a famous group theorist." Wales was also drawn to the small size of Caltech and the students. "The students are extremely good, very good all around, but the very good mathematics students are excellent," he said.</p><p data-block-key="b41l9">"David was a joyful person, always positive and affirming, which was very comforting for students going through a tough time," says Kitty Cahalan (PhD '00), assistant director for educational outreach. "I don't think I ever heard him say an unkind thing to or about anyone. After he retired, he and his wife Kathy were always planning their next adventure to some far-flung place in the world, neither of them the type to let the grass grow under their feet. Family was always very important to him, and Caltechers were very lucky to have him consider all of us as part of his family."</p><p data-block-key="4s3ed">In his early years at Caltech, Wales worked on several projects involving the representation theory of finite groups, explains Michael Aschbacher, the Shaler Arthur Hanisch Professor of Mathematics, Emeritus. "He determined the finite groups with a faithful representation of small degree. He and his students determined the simple groups whose order is divisible by just three primes."</p><p data-block-key="69epd">Aschbacher notes that Wales worked with Hall on a simple group known as the Hall-Janko group, sometimes referred to as HJ. "Wales proved the uniqueness, subject to suitable constraints, of HJ, using a pretty argument applicable to other <a href="https://mathworld.wolfram.com/SporadicGroup.html">sporadic groups</a>," Aschbacher says. Wales also proved the existence of another sporadic simple group, the <a href="https://mathworld.wolfram.com/RudvalisGroup.html">Rudvalis group</a>, together with the late mathematician John Conway.</p><p data-block-key="bk16r">In his Heritage Project interview, Wales said of his time as associate dean and dean of students: "Although things were sometimes stressful, especially the calls during the middle of the night, I really had fun with the people who I was working with and enjoyed the contact with students." Later, as a master of student houses (a position that no longer exists), Wales was responsible for hosting social gatherings for students at Steele House.</p><p data-block-key="6ldk3">"Steele House was a wonderful place for hosting students," he said. "Up to around 24 people could be comfortably seated in the dining room and adjoining sun porch. We had a pool table put in the library. The biggest attraction was the real pipe organ in Steele House. There was a large room at the side of the house that held the 3,000 pipes. The organ was later removed when Steele House was converted to other uses."</p><p data-block-key="fpjam">Christopher Brennen, the Richard L. and Dorothy M. Hayman Professor of Mechanical Engineering, Emeritus, who worked closely with Wales for more than a decade in overlapping faculty administration roles in Student Affairs, said Wales was a "model of good judgment, caring, and empathy" in his interactions with students and in his efforts to ensure that all undergraduates were happy, healthy, and ultimately successful in their studies and future careers.</p><p data-block-key="8ame4">"I think he loved the students, just as I did, and he loved seeing them prosper and seeing them be successful," Brennen said. "He had a very genuine humanity about him but, even more, a real desire to contribute to the students and by doing so contribute to the Institute."</p><p data-block-key="eascf">In his spare time, Wales and Brennen liked to hike in the San Gabriel and Sierra Nevada mountains, and they often brought Caltech's students along on many of their weekend adventures.</p><p data-block-key="3k4g1">"We clambered down untraveled canyons, rappelled down great waterfalls, and swam through canyon-wide pools with joy in the wilderness and our fellow canyoneers," Brennen recounted. "Those adventures, the all-day adventures, were deeply appreciated by all of us."</p><p data-block-key="3bmn1">Tom Mannion, senior director of student activities and programs at Caltech, recalled some of his interactions with Wales, beginning in 1993 when Mannion came to Caltech as director of student housing. "I went on many retreats with David and the RAs, and was always struck with how much he cared about our students. When David saw students, he would always ask how they were doing, and he was always ready to help."</p><p data-block-key="19bko">Even after his retirement from Caltech in 2008, Wales continued to contribute to his field and to students' education. For instance, since 2015, he had been helping a Dutch colleague, Arjeh Cohen, translate mathematical educational materials to English.</p><p data-block-key="8ts3v">Wales is survived by his wife, Kathy TeStrake Wales; his children, Jonathan Wales and Jennifer Wales Singleton; and his grandsons Matthew and Joshua Singleton.</p>Counting Curves2023-01-13T21:55:00+00:002023-01-19T19:36:04.478552+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/counting-curves<p data-block-key="w1hgr">While counting is the first skill a child learns in mathematics, it's also something studied at the highest levels of the discipline, albeit in a more exciting way. New professor of mathematics Tony Yue Yu's research involves counting curves in a geometric space, which places his work in the field of <i>enumerative geometry</i>. One of the earliest examples of enumerative geometry is the Problem of Apollonius, named after a mathematician in ancient Greece. In this problem, one counts the number of circles that are tangent to three given circles in a plane (black in the illustration). There are in general eight such tangent circles; one is shown in pink.</p><p data-block-key="sst2"></p><embed alt="The Problem of Apollonius -- diagram showing tangent circles" embedtype="image" format="RightAlignSmall" id="9402"/><p data-block-key="chqmh">Such questions are not only intuitively appealing, but also practically important because counting geometric objects with certain constraints is the same type of problem as counting the number of solutions to a system of equations.</p><p data-block-key="9cmh5">Yu is developing a new theory of curve counting via what is called non-Archimedean geometry. Typically, when you take two numbers such as 1 and 100, in which one number is less than the other, you can add the lesser number to itself again and again to eventually surpass the greater number (100). This concept stems from work by Archimedes, an ancient Greek mathematician. However, in non-Archimedean geometry, you can keep adding up smaller numbers but you will <i>never</i> surpass the larger number. Yu explains that these "exotic, non-intuitive" numbers are at the heart of his work.</p><p data-block-key="9idk3">Yu, who grew up in Ningbo, China, completed his undergraduate studies in mathematics at Peking University in 2010, and later completed his graduate studies at the Ecole normale supérieure in Paris. He was a permanent researcher in the French National Centre for Scientific Research until joining the Caltech faculty in 2021.</p><p data-block-key="88oon">We met with Yu over Zoom to learn more about his research and how it relates to a concept familiar to physicists known as mirror symmetry.</p><h4 data-block-key="5jvup">When did you first become interested in math?</h4><p data-block-key="8hio4">I have been fascinated by math and science since childhood. Growing up in Ningbo, there were plenty of science activities and competitions for kids, and I enjoyed all of them. I went to Peking University, majoring in mathematics, because in high school I was able to read university textbooks on many other science subjects but couldn't understand much from the math textbooks. I became very curious about modern mathematics.</p><p data-block-key="ghpf">Then I went to Ecole normale supérieure in Paris for graduate school. Paris is the birthplace of modern algebraic geometry, founded by Alexander Grothendieck in the 1960s. People like to say that Paris is the center of the fashion world, but it is also a center of mathematical research in many areas. Once in Paris, surrounded by so many mathematicians, it was natural for me to pursue mathematical research.</p><h4 data-block-key="avmlt">Can you tell us more about non-Archimedean geometry?</h4><p data-block-key="atreo">The Archimedean property says that given any two positive numbers A <i><</i> B, if we add A to itself sufficiently many times, the sum A+ ⋯ +A will eventually exceed B.</p><embed alt="Illustration of the Archimedean property" embedtype="image" format="MiddleAlignMedium" id="9403"/><p data-block-key="d297q"></p><p data-block-key="39rc2">You will say that this is obvious because this is how length behaves in our daily life. However, in modern mathematics, there is a great interest to study quantities and geometric spaces where the Archimedean property fails. We call them non-Archimedean numbers and non-Archimedean spaces. In this realm, numbers do not fall on a number line nor represent any notion of distance. They are exotic and don't match our intuition.</p><p data-block-key="cjthp">Non-Archimedean geometry is a branch of algebraic geometry, where we study geometric shapes defined over non-Archimedean numbers. Since we do not live in a non-Archimedean world, it is hard to study non-Archimedean spaces, and many research mathematicians considered it to be a difficult and abstract field.</p><p data-block-key="dvv68">I became interested in the field while I was a graduate student in Paris. One day I asked my advisor what a non-Archimedean space is. He replied that it is some very "hairy" space. I used to think of mathematical objects as austere and solemn, and I couldn't believe how a geometric space can be hairy like an animal! I became very fascinated by the subject afterward.</p><h4 data-block-key="64vsv">Can you tell us more about enumerative geometry?</h4><p data-block-key="f89b3">Enumerative geometry is about counting geometric objects, such as the Problem of Apollonius, in which one counts circles in a plane. While it is fun and important to count and compute the precise numbers, the thrill of the field is to discover deeper structural relations behind these numbers. One of the most mysterious relations is described by so-called mirror symmetry, which is a duality of shapes first discovered by theoretical physicists studying string theory, a mathematical theory that aims to describe the fundamental particles and forces in nature.</p><p data-block-key="9qokj">Regardless of whether string theory can be proven by experiments, it has made a great impact on mathematical research. In mirror symmetry, the numbers of curves in one space can be related to solutions of differential equations on the mirror space. The full extent of this phenomenon, as well as its underlying mathematical mechanism, are still largely unknown.</p><h4 data-block-key="alaqu">What problems are you working on?</h4><p data-block-key="do0st">I initiated the study of enumerative geometry using non-Archimedean methods, in particular with the aim of solving conjectures in the field of mirror symmetry. In fact, non-Archimedean spaces appear naturally in the study of mirror symmetry via a process known as degeneration of spaces. One can think of degeneration as crashing a big complicated space into smaller, simpler broken pieces. The parameter for the degeneration becomes a non-Archimedean number, and the degeneration process gives rise to a non-Archimedean space. However, most researchers were not keen on applying non-Archimedean methods to curve counting problems because non-Archimedean geometry was considered to be exotic and difficult. My research has been exploring this direction in the last few years, and it has turned out to be a rewarding experience.</p><p data-block-key="5m0ej"></p><embed alt="Diagram showing non-Archimedean curves in mirror symmetry" embedtype="image" format="LeftAlignSmall" id="9404"/><p data-block-key="3iv6b">I aim to further develop this non-Archimedean approach and hopefully make new contributions to the mathematical foundation of mirror symmetry. It is also important to compare this work with methods studied by other researchers. While mirror symmetry completely revolutionized the field of enumerative geometry, I also look forward to exploring applications of mirror symmetry to broader areas of algebraic geometry, such as moduli theory and birational geometry. Both concern the classification of spaces, and classification has always been a central theme across different areas of mathematics.</p><h4 data-block-key="cd3i7">Is there anything else you would like to add?</h4><p data-block-key="94ija">In addition to being the indispensable tool for science and technology, math is also an art. Sometimes you hear a piece of music and you like it. Math has a lot of surprises but requires years of training to fully appreciate.</p>2022 Year in Review2022-12-19T19:49:00+00:002022-12-23T16:16:02.688609+00:00Caltech Media Relationsmr@caltech.eduhttps://www.caltech.edu/about/news/2022-year-in-review<p data-block-key="makzi">As the end of 2022 quickly approaches, Caltech News looks back at our coverage of the research, discoveries, events, and experiences that shaped the Institute. Here are some highlights.</p><h2 data-block-key="7pr79">Revealing the Secrets of the Red Planet</h2><p data-block-key="bb3av">Caltech researchers used data gathered both in space by the Mars Reconnaissance Orbiter (MRO) and on the ground by the Mars Perseverance Rover to continue to probe the Red Planet's past and any potential signs it was previously hospitable to life. In January, MRO survey data revealed that liquid water was on Mars about one billion years earlier than suspected. Meanwhile, Perseverance made a beeline across the floor of Jezero Crater during spring 2022, <a href="/about/news/as-mars-perseverance-rover-rolls-along-the-delta-scientists-at-caltech-roll-up-their-sleeves">arriving at an ancient river delta</a> in April. The delta is thought to be one of the best possible places to search for past signs of life; there, Perseveranc<i>e</i> found <a href="/about/news/rock-samples-from-the-floor-of-jezero-crater-show-significant-contact-with-water-together-with-possible-organic-compounds">signs of past water along with evidence of possible organic compounds</a> in the igneous rocks on the crater floor. After a few months at the delta, Perseverance project scientist <a href="https://www.gps.caltech.edu/people/kenneth-a-farley">Ken Farley</a> announced in September the discovery of a class of <a href="https://www.nasa.gov/press-release/nasa-s-perseverance-rover-investigates-geologically-rich-mars-terrain">organic molecules in two samples of mudstone rock</a> collected from a feature called Wildcat Ridge. While these organic molecules can be produced through nonliving chemical processes, some of the molecules themselves are among the building blocks of life.</p><h2 data-block-key="a57fq">Surveying the Cosmos and Our Interaction with It</h2><p data-block-key="9afbb">Not all eyes aimed toward space are set on Mars, however. New instruments and surveys provided insights related to other celestial bodies in our Milky Way galaxy, such as asteroids, and helped discover an abundance of planets outside of our solar system.</p><p data-block-key="b66m5">In March, the <a href="https://exoplanetarchive.ipac.caltech.edu/">NASA Exoplanet Archive</a>, an official catalog for exoplanets—planets that circle other stars beyond our sun—housed at Caltech's IPAC astronomy center, officially <a href="/about/news/exoplanet-count-tops-5000">hit a new milestone</a>: 5,000 exoplanets.</p><p data-block-key="a6013">Looking even farther out into the universe from planet Earth, Caltech researchers made several discoveries, including a tight-knit pair of supermassive black holes locked in an <a href="/about/news/colossal-black-holes-locked-in-dance-at-heart-of-galaxy">epic waltz</a>, and a new "<a href="/about/news/black-widow-star-devours-its-rapidly-circling-companion">black widow</a>" star system, spotted by the Zwicky Transient Facility (ZTF), in which a rapidly spinning dead star called a pulsar is slowly evaporating its companion.</p><p data-block-key="6rk5i">Caltech's ZTF sky survey instrument, based at Palomar Observatory, had <a href="/about/news/first-asteroid-found-inside-orbit-venus">previously discovered the first known asteroid to circle entirely within the orbit of Venus</a>. To honor the Pauma band of Indigenous peoples whose ancestral lands include Palomar Mountain, the ZTF team asked the band to <a href="/about/news/native-americans-name-asteroid-ayl%C3%B3chaxnim-or-venus-girl">name the asteroid</a>. They chose <a href="https://caltech.box.com/s/21v4d2vw0qcmw9g1eqdjmlmwgii42637">'Ayló'chaxnim</a>, which means "Venus girl" in their native language of Luiseño.</p><p data-block-key="8gbm2">And far closer to home, new faculty member and historian <a href="https://www.hss.caltech.edu/people/lisa-ruth-rand">Lisa Ruth Rand</a> set her sights on the debris we have left in Earth's orbit (and beyond), and what it can tell us about humanity and our evolving relationship with space.</p><h2 data-block-key="9rjgo">Building Better Ways to See the Universe</h2><p data-block-key="bagcn">Caltech astronomers continue to lead the way in the development of ever more powerful instruments for answering fundamental questions about our place in the universe. The <a href="/about/news/keck-observatorys-newest-planet-hunter-puts-its-eye-on-the-sky">new Keck Planet Finder</a>, led by astronomer <a href="/about/news/keck-observatorys-newest-planet-hunter-puts-its-eye-on-the-sky">Andrew Howard</a>, will take advantage of the W. M. Keck Observatory's giant telescopes to search for and characterize hundreds, and ultimately, thousands of exoplanets, including Earth-size planets that may harbor conditions suitable for life.</p><p data-block-key="1tge2">NASA has also <a href="/about/news/nasa-selects-uvex-mission-proposal-for-further-study">selected the UltraViolet EXplorer (UVEX)</a> proposal, led by astronomer <a href="https://pma.caltech.edu/people/fiona-a-harrison">Fiona Harrison</a>, for further study. If selected to become a mission, UVEX would conduct a deep survey of the whole sky in ultraviolet light to provide new insights into galaxy evolution and the life cycle of stars. Harrison's current NASA mission, NuSTAR (Nuclear Spectroscopic Telescope Array), an X-ray telescope that hunts black holes, <a href="/about/news/nustar-celebrates-10-years-in-space">celebrated 10 years in space</a>. Meanwhile, the development of NASA's SPHEREx (Spectro-Photometer for the History of the Universe, Epoch of Reionization and Ices Explorer), led by astronomer <a href="https://pma.caltech.edu/people/james-j-jamie-bock">Jamie Bock</a>, is forging ahead with a customized test chamber <a href="/about/news/a-test-chamber-for-nasas-new-cosmic-mapmaker-makes-a-dramatic-entrance">delivered this year to Caltech</a>.</p><p data-block-key="9niks">As new telescopes continue to come together, a venerable Caltech telescope is being taken apart atop Maunakea in Hawai‘i. The Caltech Submillimeter Observatory (CSO) <a href="/about/news/caltech-submillimeter-observatory-decommissioning-receives-final-permits-and-selects-contractors">received the final permits</a> to begin its decommissioning process. Scientists plan to ultimately repurpose the telescope and put it back together in Chile.</p><h2 data-block-key="69uv1">Improving Human Health</h2><p data-block-key="5ge0n">Caltech's fundamental quest for understanding life and our origins also inspires many research efforts and innovations with the potential to improve human health and well-being.</p><p data-block-key="78dfs">Continuing work that began with the COVID-19 pandemic, Pamela Björkman and colleagues developed a <a href="/about/news/sars-coronavirus-variant-vaccine-bjorkman">new type of vaccine</a> that protects against the virus that causes COVID-19 and closely related viruses, while Sarkis Mazmanian has shown how an imbalance of gut microbes <a href="/about/news/gut-microbes-influence-binge-eating-of-sweet-treats-in-mice">can cause binge eating</a>. Meanwhile, other researchers made real what would have seemed like science fiction only a few years ago: Caltech medical engineer Wei Gao created an <a href="/about/news/artificial-skin-gives-robots-sense-of-touch-and-beyond">artificial skin for robots</a> that interfaces with human skin and allows a human operator to "feel" what the robot is sensing; chemical engineer Mikhail Shapiro engineered a strain of <a href="/about/news/fighting-cancer-with-sound-controlled-bacteria">remote-controlled bacteria</a> that seek out tumors inside the human body to deliver targeted drugs on command; and neuroscientist Richard Andersen and colleagues developed a brain–machine interface that can <a href="/about/news/brain-machine-interface-device-predicts-internal-speech">read a person's brain activity</a> and translate it into the words the person was thinking— technology that may one day allow people with full-body paralysis to speak. Additionally, Caltech researchers created a <a href="/about/news/synthetic-mouse-embryo-with-brain-and-beating-heart-grown-from-stem-cells">"synthetic" mouse embryo</a>, complete with brain and beating heart; completed a 20-year quest to decode one of <a href="/about/news/decoding-a-key-part-of-the-cell-atom-by-atom">the most complex and important pieces of machinery in our cells</a>; and discovered how fruit flies' <a href="/about/news/how-fruit-flies-sniff-out-their-environments">extremely sensitive noses help them find food</a>.</p><h2 data-block-key="ae3i">Advancing Sustainability Solutions</h2><p data-block-key="l8li">In 2022, Caltech paid tribute to its long history of advances in sustainability and then looked forward to pioneering new initiatives and technologies that will reduce humanity's footprint on Earth's fragile environment. Through the <a href="https://magazine.caltech.edu/post/caltech-heritage-projects-oral-history">newly launched Caltech Heritage Project</a>, a series of oral histories published this year captured the pivotal role Caltech alumni played in <a href="/about/news/todays-electric-vehicles-owe-debt-to-caltech-alumni">the electric car revolution</a>. Meanwhile, in April, <a href="/about/news/caltech-energy-10-ce10-aims-to-develop-the-roadmap-toward-a-50-percent-reduction-in-us-global-warming-gas-emissions-by-2032">Caltech hosted the Caltech Energy 10 (CE10) conference</a>, bringing thought leaders to campus to chart a path toward achieving the Biden administration's stated goal to cut U.S. global warming gas emissions by 50 percent within the next 10 years.</p><p data-block-key="5hh40">Caltech researchers continue to contribute to research to generate cleaner energy, ranging from work in the laboratory of John Dabiri (MS '03, PhD '05) to <a href="/about/news/tweaking-turbine-angles-squeezes-more-power-out-of-wind-farms">optimize wind farms</a> to efforts to create and commercialize technology for capturing carbon already released into the atmosphere (which earned <a href="/about/news/startup-from-caltech-nabs-xprize-award">a Caltech-based startup an XPrize Award</a>).</p><p data-block-key="cd9kn">On campus, Caltech began construction of the <a href="https://magazine.caltech.edu/post/caltechs-green-gateway-the-resnick-sustainability-center">Resnick Sustainability Center</a>, scheduled to open in 2024, which will bring together talent from across campus to tackle issues related to climate change and other human impacts on the natural environment. And as the year wraps up, the Space-based Solar Power Project is <a href="/about/news/space-solar-power-atwater-hajimiri-pellegrino">preparing to launch a demonstration</a> into space to test three key elements of its ambitious plan to harvest solar energy in space—where there are no cloudy days—and beam it wirelessly down to Earth.</p><h2 data-block-key="919l0">Harnessing the Power of Data to Advance Science</h2><p data-block-key="3r4bo">As the <a href="https://www.ai4science.caltech.edu/">AI4Science Initiative</a> continually demonstrates and the <a href="https://scienceexchange.caltech.edu/topics/artificial-intelligence-research?utm_source=caltechnews&utm_medium=web&utm_campaign=cseai">Caltech Science Exchange recently highlighted</a>, artificial intelligence (AI) and machine learning (ML) have applications that reach every corner of campus. In 2022, AI was used to generate <a href="/about/news/caltech-researchers-help-generate-first-image-of-black-hole-at-the-center-of-our-galaxy">the first-ever picture of the black hole at the center of our own galaxy</a> (only the second image of a black hole ever created), to pave the way to <a href="/about/news/improving-aircraft-design-with-machine-learning-and-a-more-efficient-model-of-turbulent-airflows">improve aircraft design</a>, to <a href="/about/news/rapid-adaptation-of-deep-learning-teaches-drones-to-survive-any-weather">help drones fly autonomously</a> in real-weather conditions, and to <a href="/about/news/researchers-tackle-covid-19-with-ai">fight COVID-19</a>. This election year, researchers from Caltech discussed how machine learning can both <a href="https://scienceexchange.caltech.edu/topics/artificial-intelligence-research/artificial-intelligence-experts/ai-science-alvarez-anandkumar?utm_source=caltechnews&utm_medium=web&utm_campaign=cseai">combat misinformation</a> and fight online bullying.</p><h2 data-block-key="e1lla">Forging Quantum Frontiers</h2><p data-block-key="futes">Caltech continues its role as a major hub of quantum research. The newly announced <a href="/about/news/ginsburgs-give-to-create-new-quantum-center-and-building-at-caltech">Dr. Allen and Charlotte Ginsburg Center for Quantum Precision Measurement</a> will unite a diverse community of theorists and experimentalists devoted to understanding quantum systems and their potential uses (see a <a href="https://youtu.be/zqRezABOAdY">video about the new center</a>). The 25th annual <a href="https://qipconference.org/">Conference on Quantum Information Processing</a>, or QIP, the world's largest gathering of researchers in the field of quantum information, a discipline that unites quantum physics and computer science, <a href="/about/news/caltech-hosts-largest-quantum-information-conference">was held in Pasadena</a> for the first time and represented the first major collaboration between Caltech and the new <a href="/about/news/caltech-and-amazon-partner-to-create-new-hub-of-quantum-computing">AWS Center for Quantum Computing</a> on campus.</p><p data-block-key="51e4a">Fundamental research in the quantum sciences charged ahead, with findings that included a quantum computer-based experiment to <a href="/about/news/physicists-observe-wormhole-dynamics-using-a-quantum-computer">test theoretical wormholes</a> and new demonstrations showing how graphene can be used in <a href="/about/news/graphene-boosts-flexible-and-wearable-electronics">flexible and wearable electronics</a>.</p><h2 data-block-key="17b89">Pioneering People</h2><p data-block-key="daf2">This year, members of the Caltech community received recognition for expanding the boundaries of scientific knowledge, but also for humanitarian endeavors and for blazing new educational and occupational paths for others to follow.</p><p data-block-key="d0309">In March, Roman Korol, a Caltech graduate student, <a href="/about/news/organizing-aid-to-his-native-ukraine">launched a project</a> to collect and distribute humanitarian aid for families affected by the war in Ukraine.</p><p data-block-key="88vng">In April, Jessica Watkins, who worked on the Mars Curiosity rover mission while a postdoc at Caltech, made history as the <a href="/about/news/former-caltech-postdoc-launches-into-space">first Black woman on the International Space Station</a>. From space, she <a href="/about/news/nasa-astronaut-jessica-watkins-holds-a-qa-from-space">hosted a live Q&A</a> for Caltech students and faculty in Ramo Auditorium and <a href="/about/news/wind-drives-geology-on-mars-these-days">reviewed a paper</a> describing how geology on Mars works in dramatically different ways than on Earth.</p><p data-block-key="27l2o">In May, alumna Laurie Leshin (MS '89, PhD '95) <a href="/about/news/caltech-names-laurie-leshin-ms-89-phd-95-director-of-jpl">assumed leadership of JPL</a>, becoming its first female director.</p><p data-block-key="83gle">In June, Carver Mead (BS '56, MS '57, PhD '60), one of the fathers of modern computing, <a href="/about/news/carver-mead-awarded-kyoto-prize-by-inamori-foundation">received the 2022 Kyoto Prize</a> for leading contributions to the establishment of the guiding principles for very large-scale integration systems design, which enables the basis for integrated computer circuits.</p><p data-block-key="99gae">In October, Caltech alumnus John Clauser (BS '64) <a href="/about/news/caltech-alum-wins-nobel-prize-in-physics">shared the 2022 Nobel Prize in Physics</a> "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science." The same month, <a href="https://www.jpl.nasa.gov/news/edward-stone-retires-after-50-years-as-nasa-voyagers-project-scientist">Edward Stone retired</a> as the project scientist for NASA's Voyager mission a half-century after taking on the role. Under his guidance, the Voyager probes explored the solar system's four gas-giant planets and became the first human-made objects to reach interstellar space, the region between stars containing material generated by the death of nearby stars. Also, Tracy Dennison <a href="/about/news/tracy-dennison-begins-tenure-as-chair-of-the-division-of-the-humanities-and-social-sciences">began her term</a> as the new Ronald and Maxine Linde Leadership Chair of the Division of the Humanities and Social Sciences.</p><p data-block-key="1i42u">In November, 50 years after they entered Caltech as the Institute's <a href="https://magazine.caltech.edu/post/reflections-on-72">first Black female students</a>, Karen Maples, MD (BS '76); Deanna Hunt (BS '76); and Lauretta Carroll (BS '77) reflected on the challenges and successes they experienced then and in the years that followed.</p><h2 data-block-key="f2mn6">Institute News</h2><p data-block-key="88d5a">Throughout the year, the Institute took steps to implement new programs and bolster existing ones that underscore Caltech's guiding values, such as supporting students and postdoctoral scholars, creating a more inclusive environment, and celebrating and accounting for its history.</p><p data-block-key="5jsko">To create more opportunities for students and increase interdisciplinary research, Caltech <a href="/about/news/new-graduate-track-to-combine-study-of-medical-and-electrical-engineering">created a new graduate education track</a> that combines medical engineering and electrical engineering. To further boost interdisciplinary research and expand Caltech's prominence as a hub for mathematics, the Institute became the <a href="/about/news/american-institute-of-mathematics-moves-to-caltech">new home of the American Institute of Mathematics</a>, an independent nonprofit organization funded in part by the National Science Foundation.</p><p data-block-key="9fosq">The Institute, which this year kicked off a partnership with the Carnegie Institution for Science, also <a href="/about/news/caltech-joins-sea-change-as-charter-member">became a charter member of SEA Change</a>, an initiative of the American Association for the Advancement of Science that supports educational institutions as they systemically transform to improve diversity, equity, accessibility, and inclusion in science, technology, engineering, mathematics, and medicine.</p><p data-block-key="9en86">The Institute expanded its <a href="/about/news/presidential-postdoctoral-fellows">Presidential Postdoctoral Fellowship</a>, which supports efforts to diversify academia by recruiting and supporting promising postdoctoral scholars from underrepresented communities.</p><p data-block-key="1bi6e">On campus, Caltech marked the <a href="/about/news/grant-d-venerable-house-dedication">dedication of the Grant D. Venerable House</a>, honoring its namesake alumnus, who was the first Black undergraduate student to graduate from Caltech and an active student leader and athlete during his time on campus. It also celebrated the <a href="/about/news/caltech-celebrates-dedication-of-the-lee-f-browne-dining-hall">dedication of the Lee F. Browne Dining Hall</a>, honoring the late Lee Franke Browne, a former Caltech employee and lecturer who dedicated his life and career to efforts that expanded students' access to STEM and who advanced human rights.</p><h2 data-block-key="e33ik">In the Community</h2><p data-block-key="8itgo">With the return of in-person events, the Institute was able to reestablish and strengthen ties to the local community through educational programs for area students, and through cultural events and lectures whose online components often reached even broader audiences across the world.</p><p data-block-key="1t506">This year, the Institute celebrated the <a href="/about/news/caltechs-seismo-lab-celebrates-100-years-at-the-forefront-of-earthquake-science">centennial of the Caltech Seismological Laboratory</a>, marking an unparalleled century at the forefront of earthquake science and geophysics.</p><p data-block-key="4si1b">Caltech also celebrated the <a href="https://magazine.caltech.edu/post/watson-lectures-100-years">100th anniversary of the Watson Lectures</a>, which launched in 1922 as a way to benefit the public through education and outreach. Continuing that tradition, Caltech partnered with local schools to bring high school students to campus to see the lectures and engaged young students through other educational outreach programs, including the new <a href="/about/news/caltech-earthquake-fellows">Caltech Earthquake Fellows program</a> and the <a href="/about/news/outreach-program-engages-public-high-school-students-in-the-discovery-of-exoplanets">Caltech Planet Finder Academy</a>, both of which launched this year. Other programs designed to bolster science education for young students included <a href="/about/news/high-school-students-research-at-caltech">Summer Research Connection</a>, a program that invites high school students and teachers from Pasadena Unified School District and other nearby schools into Caltech laboratories, and the <a href="/about/news/caltech-virtual-host-national-science-olympiad-2022">National Science Olympiad Tournament</a>, which Caltech hosted this year for the first time and whose students played the main role in conducting the event.</p><p data-block-key="5fcm8">For the campus community, <a href="/about/news/techfest-2022-start-fall-term">TechFest</a> returned to campus for the first time since the start of the COVID-19 pandemic, welcoming students with an in-person block party on Beckman Mall complete with games and fireworks.</p><p data-block-key="2s02g"><a href="https://events.caltech.edu/">Caltech's Public Programming</a> was able to re-engage with the community through in-person events, including <a href="https://events.caltech.edu/series/caltechlive-performing-arts">CaltechLive!</a> events such as the performance of Nobuntu, a female a cappella quintet from Zimbabwe; and lectures from the <a href="https://events.caltech.edu/series/science-journeys"><i>Science Journeys</i></a>, <a href="https://events.caltech.edu/series/movies-that-matter"><i>Movies that Matter</i></a> and <a href="https://events.caltech.edu/series/behind_the_book"><i>Behind the Book</i></a> series that showcased such varied topics as a journey to the center of Jupiter, a discussion of the science of cooking, and how climate migration will reshape the world.</p>Pushing the Boundaries of Fluid Equations2022-11-22T19:13:38.149353+00:002022-11-22T19:13:38.074343+00:00Emily Velascoevelasco@caltech.eduhttps://www.caltech.edu/about/news/pushing-the-boundaries-of-fluid-equations<p data-block-key="th0wc">The motion of fluids in nature, including the flow of water in our oceans, the formation of tornadoes in our atmosphere, and the flux of air surrounding airplanes, have long been described and simulated by what are known as the Navier–Stokes equations.</p><p data-block-key="8g0s0">Yet, mathematicians do not have a complete understanding of these equations. While they are a useful tool for predicting the flow of fluids, we still do not know if they accurately describe fluids in <i>all</i> possible scenarios. This led the Clay Mathematics Institute of New Hampshire to label the Navier–Stokes equations as one of its seven Millennium Problems: the seven most pressing unsolved problems in all of mathematics.</p><p data-block-key="fbcum">The <a href="https://www.claymath.org/sites/default/files/navierstokes.pdf">Navier–Stokes Equation Millennium Problem</a> asks mathematicians to prove whether "smooth" solutions always exist for the Navier–Stokes equations. Put simply, smoothness refers to whether equations of this type behave in a predictable way that makes sense. Imagine a simulation in which a foot presses the gas pedal of a car, and the car accelerates to 10 miles per hour (mph), then to 20 mph, then to 30 mph, and then to 40 mph. However, if the foot presses the gas pedal and the car accelerates to 50 mph, then to 60 mph, then instantly to an infinite number of miles per hour, you would say there is something wrong with the simulation.</p><p data-block-key="efnmb">This is what mathematicians hope to determine for the Navier–Stokes equations. Do they always simulate fluids in a way that makes sense, or are there some situations in which they break down?</p><p data-block-key="bod9m"><i>For an in-depth explanation of this topic, see the blog post "</i><a href="https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/"><i>Why global regularity for Navier-Stokes is hard</i></a><i>" by Australian mathematician Terence Tao.</i></p><p data-block-key="6b5ph">In a paper published on the preprint site arXiv on October 19, Caltech's <a href="https://www.cms.caltech.edu/people/hou">Thomas Hou</a>, the Charles Lee Powell Professor of Applied and Computational Mathematics, and Jiajie Chen (PhD '22) of New York University's Courant Institute, provide a proof that resolves a longstanding open problem for the so-called 3D Euler singularity. The 3D Euler equation is a simplification of the Navier–Stokes equations, and a singularity is the point where an equation starts to break down or "blow up," meaning it can suddenly become chaotic without warning (like the simulated car accelerating to an infinite number of miles per hour). The proof is based on a scenario first proposed by Hou and his former postdoc, Guo Luo, in 2014.</p><p data-block-key="91trh">Hou's computation with Luo in 2014 discovered a new scenario that showed the first convincing numerical evidence for a 3D Euler blowup, whereas previous attempts to discover a 3D Euler blowup were either inconclusive or not reproduceable.</p><p data-block-key="fudja">In the latest paper, Hou and Chen show definitive and irrefutable proof of Hou and Luo's work involving 3D Euler equation blowup. "It starts from something that behaves nicely, but then somehow evolves in a way where it becomes catastrophic," Hou says.</p><p data-block-key="8psvs"></p><embed alt="A portrait of Thomas Hou. He wears as suit and glasses and smiles at the camera." embedtype="image" format="RightAlignSmall" id="9313"/><p data-block-key="bhh0h"></p><p data-block-key="7vu65">"For the first ten years of my work, I believed there was no Euler blowup," says Hou. After more than a decade of research since, Hou has not only proved his former self wrong, he's settled a centuries-old mathematics mystery.</p><p data-block-key="1l50c"></p><p data-block-key="uqwv3"></p><p data-block-key="2acoi"></p><p data-block-key="5ptgl">"This breakthrough is a testament to Dr. Hou's tenacity in addressing the Euler problem and the intellectual environment that Caltech nutures," says Harry A. Atwater, Otis Booth Leadership Chair of the Division of Engineering and Applied Science, Howard Hughes Professor of Applied Physics and Materials Science, and director of the <a href="https://www.liquidsunlightalliance.org/">Liquid Sunlight Alliance</a>. "Caltech empowers researchers to apply sustained creative effort on complex problems – even over decades – to achieve extraordinary results."</p><p data-block-key="qerg"></p><embed alt="A portrait of Jiajie Chen. He wears a polo shirt and glasses and smiles at the camera." embedtype="image" format="LeftAlignSmall" id="9314"/><p data-block-key="389sc"></p><p data-block-key="3ida6">Hou and colleagues' combined effort in proving the existence of blowup with the 3D Euler equation is a major breakthrough in its own right, but also represents a huge leap forward in tackling the Navier-Stokes Millennium Problem. If the Navier–Stokes equations could also blow up, it would mean something is awry with one of the most fundamental equations used to describe nature.</p><p data-block-key="43o49">"The whole framework that we set up for this analysis would be tremendously helpful for Navier–Stokes," Hou says. "I have recently identified a promising blowup candidate for Navier-Stokes. We just need to find the right formulation to prove the blowup of the Navier-Stokes ."</p><p data-block-key="5ov2t">The paper detailing the proof is titled <a href="https://arxiv.org/abs/2210.07191">"Stable Nearly Self-Similar Blowup of the 2D Boussinesq and 3D Euler Equations with Smooth Data."</a></p><p data-block-key="9vlqm">Funding for the research was provided by the National Science Foundation and by the Choi Family Postdoctoral Fund, Choi Family Gift Fund, and the Choi Family Graduate Fellowship Fund.</p>Caltech Mathematicians Solve 19th Century Number Riddle2022-10-31T21:45:00+00:002023-05-23T22:26:06.784375+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/caltech-mathematicians-solve-19th-century-number-riddle<p data-block-key="7blvg">For the past 175 years, a perplexing feature of numbers first stumbled upon by German mathematician Ernst Kummer has confounded researchers. At one point in the 1950s, this quirky aspect of number theory was thought to have been wrong, but then, decades later, mathematicians found hints that it was in fact true. Now, after several twists and turns, two Caltech mathematicians have at last found proof that Kummer was right all along.</p><p data-block-key="58so">"We had several ‘aha' moments, but then you have to roll up your sleeves and figure this out," explains Alexander (Alex) Dunn, a postdoc at Caltech and the Olga Taussky and John Todd Instructor in Mathematics, who wrote the proof with his advisor, professor of mathematics <a href="https://pma.caltech.edu/people/maksym-radziwill">Maksym Radziwill</a>, and posted it <a href="https://arxiv.org/pdf/2109.07463.pdf">online</a> in September 2021.</p><p data-block-key="a7fjg">The math problem has to do with Gauss sums, which are named after the 18th-century prolific mathematician Carl Friedrich Gauss. When Gauss was young, he amazed his classmates by <a href="https://www.youtube.com/watch?v=arf8wDP_MJE&t=272s">quickly developing a formula for adding up the numbers 1 to 100</a>. Later, he developed a complex concept known as Gauss sums, which readily map the distribution of solutions to equations. He then looked at the distribution of what are called square Gauss sums for nontrivial prime numbers (primes that have a remainder of 1 when you divide by 3) and found a "beautiful structure," according to Radziwill.</p><p data-block-key="3vfi2">This summing activity involves a type of math known as <a href="https://www.youtube.com/watch?v=-zEcHLdABfo">modular arithmetic</a>. An easy way to understand modular arithmetic is to think of a clock and its face divided into 12 hours. When noon or midnight rolls around, the numbers are reset and go back to 1. This "modulo 12" system simplifies timekeeping, since we do not have to keep counting up hours forever.</p><p data-block-key="48v8m">In the case of Gauss sums, the same idea is at play but the base "clock face" is divided up into <i>p</i> hours, where <i>p</i> is a prime number. "Modulo p math is a way of stripping out information and making impossibly complicated equations simpler," Radziwill says.</p><p data-block-key="fo8d5">In the 19th century, Kummer was interested in looking at the distribution of cubic Gauss sums for nontrivial primes, or in a modulo p system. He did this by hand for the first 45 nontrivial primes, and plotted the answers one by one on a number line (to do this, he had to normalize the answers first so that they fell between -1 and 1). The result was unexpected: the solutions were not random but tended to cluster toward the positive end of the line.</p><p data-block-key="d4472">"When dealing with the distribution of natural objects in number theory, the naive expectation is that one has an equal distribution, and if not, there should be a very convincing reason," Dunn says. "That is why it was so shocking that Kummer claimed that this wasn't the case for cubes."</p><p data-block-key="4vjnf">Later, in the 1950s, researchers led by the late Hedvig Selberg of the Institute for Advanced Study used a computer to calculate the cubic Gauss sums for all the nontrivial primes less than 10,000 (about 500 primes). When the solutions were plotted on the number line, the bias seen by Kummer disappeared. The solutions seemed to have a random distribution.</p><p data-block-key="bce5f">Then came mathematician Samuel Patterson who proposed a solution to the mix-up in 1978, now referred to as Patterson's conjecture. Patterson, who was a graduate student at the University of Cambridge at the time, recognized that the bias in the distribution of the solutions could be overwhelmed as the sample size gets bigger and bigger. That meant Kummer was right—something funny was going on with his sums for 45 primes. But proving why this is the case would have to wait until last year when Dunn and Radziwill finally figured it out.</p><p data-block-key="2c11i">"The bias seen with a few numbers is like having a physically impossible coin that is slightly weighted toward heads, but becomes less and less so the more often you flip it," Radziwill explains.</p><p data-block-key="9b8um">The two Caltech researchers decided to work together to try to crack the problem of Patterson's conjecture about two years ago. They had not spent much time together on campus due to the pandemic, but they bumped into each other in a parking lot in Pasadena and got to talking. They decided to meet in parks to work on the problem, where they would jot down their mathematical proofs on sheets of paper.</p><p data-block-key="3r68s">"I had just come to Caltech and didn't know many people," Dunn says. "So it was really great to run into Maks and be able to work together on the problem in person."</p><p data-block-key="6375g">Their solution was based on work by Roger Heath-Brown of the University of Oxford, who had seen a talk by Patterson at the University of Cambridge in the late 1970s. Heath-Brown and Patterson teamed up to work on the problem, and then, in 2000, Heath-Brown developed a tool known as a cubic large sieve to help prove Patterson's conjecture. He got close but the complete solution remained out of reach.</p><p data-block-key="c6vjs">Dunn and Radziwill cracked the problem when they realized that the sieve wasn't working properly, or had a "barrier" that they were able to remove.</p><p data-block-key="55tcm">"We were able to recalibrate our approach. In math, you can get trapped into a certain line of thinking, and we were able to escape this," Dunn says. "I remember when I had one of the ‘aha' moments, I was so excited that I ran to find Maks at the Red Door [a café at Caltech] and asked him to come to my office. Then we began the hard work of figuring this all out."</p><p data-block-key="72bo4">For a more in-depth read about this research, see <a href="https://www.quantamagazine.org/a-numerical-mystery-from-the-19th-century-finally-gets-solved-20220815/">Quanta Magazine</a>.</p>Mathematically Percolating2022-08-05T16:00:00.363895+00:002022-08-05T16:00:00.222954+00:00Whitney Clavinwclavin@caltech.eduhttps://www.caltech.edu/about/news/mathematically-percolating<p data-block-key="rf7em">When water flows through a bed of ground espresso beans, ultimately resulting in a delicious latte, the water is undergoing a process called percolation. The water slowly meanders through the coffee at just the right rate to extract the rich coffee flavors. In general, percolation refers to liquids filtering through a porous medium. The process can describe not only the generation of lattes, but also a host of other phenomena, such as how diseases spread and even physics concepts such as magnetism.</p><p data-block-key="90vim">For mathematicians like Tom Hutchcroft, who joined the Caltech faculty last year as a professor of mathematics, the most interesting aspect of percolation is what happens during a phase transition, the point where an abrupt qualitive change in the system occurs. "You only change one factor in the system a tiny bit, and then you get a big change," he says. A classical phase transition occurs when water freezes.</p><p data-block-key="7jonu">In mathematical percolation models, phase transitions can result in "really interesting mathematical behavior," according to Hutchcroft, including fractal patterns; fractal refers to self-similar patterns seen at different scales.</p><p data-block-key="dt4ma">Hutchcroft, who was born and raised in England, earned his bachelor's degree in mathematics from Cambridge University in 2013 and his PhD in mathematics from the University of British Columbia, Canada, in 2017. He held internships at Microsoft Research Theory Group during his graduate studies, and later completed postdoctoral fellowships at University of Cambridge from 2017 to 2021.</p><p data-block-key="ec25b">We met with Hutchcroft over Zoom to learn more about the math of percolation and what he is enjoying about Caltech so far.</p><h4 data-block-key="5vr3g">What does percolation have to do with the spread of a disease?</h4><p data-block-key="5939e">Percolation theory is a way of describing clustered components in random networks and can be applied to complex things like the spread of an infection through a population. Systems like these have phase transitions. With epidemics, there's a critical point or phase transition called "R nought," or R0. This value depends on the average number of people that an infected person infects. When R nought is below 1, the epidemic will die out; when it's above 1, it will grow exponentially. When R is exactly 1, the infection will die out but very slowly. When you look at this point in models, you tend to have a lot of mathematically interesting behavior. And when you use branching, tree-like models for epidemics, which are a form of a percolation model, you'll get some interesting fractal geometry in the tree. This has been well understood since the 1990s. But even though you are doing something very simple, you get really mathematically rich objects coming out at the end.</p><h4 data-block-key="9u1pr">How do mathematicians study these percolation models?</h4><p data-block-key="av2s7">While physicists and other scientists may study the statistical physics or statistical mechanics of similar systems as a means to explain the behavior of the components, we mathematicians are interested in the pure math, which can be very complex and interesting. In general, we draw out grids, with edges and nodes, where the edges connect the nodes. These are the percolation models that explain how liquid can flow through a porous media. Imagine that for each edge of this grid, you flip a coin that has a probability "p" of being heads. If the coin comes out heads, you keep the edge, and if it comes out tails, you delete the edge.</p><p data-block-key="9177r">When p is small, or the probability of keeping an edge is small, you will end up with small clusters of connections that are like small islands that don't connect to anything else. When p is greater than the critical parameter at which a phase transition occurs, called pc [pronounced pee-cee], you will get one big, connected cluster. When p is exactly equal to pc, we expect to get large, fractal-like clusters that permeate across the grid but do so in a zero-density way, with extremely long, tortuous paths.</p><p data-block-key="d6b3m">So, if this model were explaining coffee percolation, then when p is less than pc, the water would not get through the coffee—it would get stuck in the islands of small clusters. When p is larger than pc, the water would readily flow through. When p is equal to pc, at the phase transition, the water would slowly meander through the grinds, which is what you would want for a good cup of espresso.</p><p data-block-key="4u7fo">If you look at the connections between the nodes at this phase transition, and under different scales, you will start to see the fractal and winding math.</p><h4 data-block-key="ek181">What problems are you working on in this field?</h4><p data-block-key="5eoun">The two-dimensional models are very well understood, and even models with 100 dimensions are easier to understand. But the three-, four-, and five-dimensional cases are extremely hard to study. One thing I'm working on is trying to crack the three-dimensional problem. The most basic question is to figure out if the phase transition occurs with a jump, what we call discontinuous, or more smoothy, what we call continuous. This problem has been open for a really long time and needs to be solved before we can move on to understanding all the cool fractal stuff that should be happening at the phase transition. I'm also working on other related problems, such as long-range percolation where the probability of two nodes having an edge between them depends on the distance between the nodes. Changing how this probability falls off with the distance has a surprisingly similar effect to changing the dimension of the grid and lets us treat the dimension like a continuous parameter.</p><h4 data-block-key="198mi">What do you love about working on these math problems?</h4><p data-block-key="cbm5t">A lot of the appeal for me is that it's fun. When you get a good problem, you get hooked on it. Math is like the king of all puzzle games, but it goes beyond puzzles in that you the solution is very insightful. You not only solve the problem, but you build new conceptual frameworks for understanding other math problems.</p><h4 data-block-key="ej671">How do you like Caltech so far?</h4><p data-block-key="aa5ot">One of the things that drew me to Caltech was the small class size. It feels less like lecturing and more like doing a seminar where you get to interact with everyone individually. I like the tight community here. Of course, the small size of Caltech can mean less interaction with mathematical peers, but a lot of that is going to be balanced out by the fact that the American Institute of Mathematics is moving its headquarters to Caltech. That's going to bring a lot more activity here, and people will be passing through regularly.</p><p data-block-key="9tlcv">I also like the mountains in the Pasadena area, which we don't have back home in the UK. We've been going out hiking. You can look at a mountain on campus and then get in the car and be there in 15 minutes.</p><h4 data-block-key="b5l18">Anything else you'd like to add?</h4><p data-block-key="7tnro">Since coming to Caltech, I've also set up the <a href="https://sites.google.com/view/la-probability-forum/home">LA Probability Forum</a>, a monthly mini-conference for the LA-area probability community, so that I get to regularly interact with my colleagues at UCLA and USC, and anyone else who would like to be involved. This has been really enriching for me both scientifically and socially.</p>