The most ambitious project in math seeks to answer fundamental questions by connecting disparate branches of the field, like calculus and geometry. This effort, known as the Langlands program, recently received a rare gift that has vastly expanded its potential. (Thread)
The Langlands program seeks to assemble a mathematical dictionary that uses objects from calculus to investigate polynomials. An adjacent effort seeks to do something similar in geometric terms.
Questions about numbers can often be answered more quickly when they are translated into geometric problems. Now, two mathematicians have found the first shape whose properties communicate directly with the Langlands program’s main concerns.
The object is called the Fargues-Fontaine curve. Originally developed in 2010 by the mathematicians Laurent Fargues and the late Jean-Marc Fontaine, the curve was only initially useful for a narrow technical purpose.
In 2014, Laurent Fargues heard Peter Scholze, a prodigious mathematician, give a lecture on a nascent geometric theory he had developed. Inspired, Fargues later pitched Scholze that the curve, joined with this new theory, could advance the geometric Langlands program.
Both Scholze’s theory and the Fargues-Fontaine curve concern special number systems called the 𝘱-adics that help mathematicians investigate prime numbers. @kelseyahe explains 𝘱-adics here:…
@kelseyahe After seven years of collaboration, Fargues and Scholze finished equipping the Fargues-Fontaine curve with the features necessary to fulfill the needs of the geometric Langlands program, as described in a 350-page paper published this February.
The work is one of the biggest advances so far on the Langlands program — and also the most tangible evidence yet that earlier mathematicians weren’t foolish to attempt the Langlands program by geometric means.
Read @KSHartnett full article on the Fargues-Fontaine reinvention:…
@KSHartnett To read more about the Langlands program, and for ongoing reporting on mathematics, computer science, biology and physics, head to (/Thread)

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More from @QuantaMagazine

15 Jul
In 1873, the mathematician Georg Cantor shook math to the core when he discovered that infinity comes in many sizes. Setting out to climb the tower of infinities that he created, a mystery stopped him in his tracks. (Thread)
There exist an infinite amount of “natural” numbers, like 1, 2 and 3, but Cantor proved that there are even more “real” numbers that sit between the natural numbers on the number line — most with never-ending digits, like 3.14159…. In other words, a larger infinity.
The mystery, called the continuum hypothesis, concerns how the size — or “cardinality,” designated with the aleph symbol, ℵ — of these infinities progresses. Cantor theorized that the cardinality of the real numbers is exactly ℵ₁, one level up from the natural numbers.
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29 Jan
THREAD: The researchers developing epidemiological models are often expected to provide answers where there are none, with a certainty they can’t guarantee. Here is how these models are made, and where their uncertainties lie. (Reporting by @jordanacep)
To understand pandemics and other disease outbreaks, scientists turn to two well-established approaches to quantitative epidemiological modeling. Today’s models usually fall on a spectrum between the two.
On one end of the spectrum are models that split a population into “compartments” based on whether they are susceptible, infected or recovered (S, I or R). Because COVID-19 has such a long incubation period, its models also need to include an exposed (E) compartment.
Read 14 tweets
4 Nov 20
In the 1970s, Steven Hawking proposed that information that falls into a black hole gets destroyed, never to be retrieved. A series of breakthrough papers have now shown that’s not correct. Here’s a thread about the famous black hole information paradox:
According to Einstein’s general theory of relativity, the gravity of a black hole is so intense that nothing can escape it. Hawking’s “semiclassical” approach brought together quantum mechanics and relativity and predicted the famous information paradox.
Now, theoretical physicists have demonstrated that additional semiclassical effects emerge in older black holes, allowing information to spill out.
Read 10 tweets
28 Oct 20
Here’s a thread about the fundamental constituents of the universe, as explained by @nattyover, with graphics by Samuel Velasco and @LucyIkkanda.
In the 1970s, physicists formed a framework that encapsulates our best understanding of nature’s fundamental order. Yet most visualizations of the Standard Model of particle physics are too simple, ignore important interconnections or are overwhelming.
The most common visualization of the Standard Model shows a periodic table of particles, but doesn’t offer insight into the relationships between them. It also leaves out key properties like “color.” Image
Read 21 tweets
8 May 20
Time has fascinated the human mind for millennia. Through the vantage points of culture, physics, timekeeping and biology, we have compiled a special timeline organizing some of the efforts that humans have made to understand time. (thread)…
Western culture tends to emphasize a linear conception of time, but the ancestors of today’s Australian aboriginal peoples embraced a timeless view of nature. In Asia, followers of Hinduism and Buddhism adopted a cyclic view.
Some of the best evidence for how ancient cultures viewed time can be found in artifacts of timekeeping mechanisms, like Egyptian sundials and circular Mayan calendars.
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30 Mar 20
The revered condensed matter physicist Philip Anderson passed away yesterday at the age of 97. Here is a sampling of some of Anderson’s ideas that have propelled modern physics. (Thread)
Anderson won the 1977 Nobel Prize in Physics for his discovery of what is now called Anderson localization, a phenomenon in which some waves stay within a given “local” region rather than advancing freely.
Quanta covered new advances in the understanding of Anderson localization in 2017 in an article that later became an episode of the Quanta Science Podcast:…
Read 6 tweets

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