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What Math Can Teach Physics

Question: In what new ways could math be applied to solve the problems of physics?

Peter Woit: The thing that most fascinates me about this whole subject is that something that probably quickly get very technical, but there's an area of mathematics which is known as representation theory. One way of thinking about it is in terms of what physicists often call symmetries. So, for instance, one of the basic facts about the laws of nature is that there are symmetric laws of nature are the same at - if you move in any direction, or you move in time back and forth, the laws of physics don't change. If you rotate things around in three dimensions, the laws of physics don't change. And so, these so-called symmetries have very important physical implications. The fact that the laws of physics don't change as if you move in time has physical implications that there's this thing called energy and energy is conserved, and the same thing - and the fact that the laws of physics don't change of you move back and forth in different directions in space implies that there is something called momentum and momentum is conserve and doesn't change as you evolve in time.

So, the very, very fundamental facts about physics are kind of deeply grounded in the symmetries of nature. And so to a mathematician this question is a question about what we call groups and representation of groups. So, if you go into any math department and look at what they're doing, you'll see that a lot of people in different kinds of mathematics are studying different structures which are also called groups and they're often studying what is called representations of the groups. So, even people studying number theory, abstract things about prime numbers or something, they're also studying groups and certain representations of these groups. So, there's kind of a, to the extent that mathematics has a kind of unifying theme and a unifying principle which shows up in different areas of mathematics, it's about these representations, or representation theory. And the thing that most strikes me about physics, and what fascinates me about it is, if you look at quantum mechanics, quantum mechanics initially looks like a very odd structure. It's not something were we have any kind of intuitive understanding of it. It doesn't look like the way we're used to thinking about physics, based on every day experience. But if you look at the mathematical structure and the basic structure of quantum mechanics, they're exactly the structures that show up in this theory of representations. So, there's a kind of a deep relation between math and physics which is surrounding this whole notion of symmetries and representation of symmetries.

So, that's one thing that's always fascinated me, and my own research and my own interests is in developing - taking a lot that has been learned in mathematics. There's a lot that's been learned in mathematics over the years about how to think about representations and how to construct them and how to work with them. Some of it has made its way into physics and have been used in physics and was used in physics since the early days of quantum mechanics. So, for instance, one of the great ****, there's a kind of hero of this book I wrote about this, it's The Mathematician called Herman Vial, who was one of the first people to understand how quantum mechanics worked and to understand the relation to representations.

But anyway, I think there's still a lot to be learned in that way and very specifically the so-called standard model has this group of symmetries which is called the gage symmetry and it's an infinitial group and the standard kind of physics understanding of the representations of this group is that that should not be an interesting question. There should only be a trivial representation of this group. But anyway, my conjecture is that there is actually a more interesting question there and in pursuing this question of how do you deal with the gauged symmetry of this theory in terms of using ideas from representation theory that are more well-known in mathematics that hadn't been used in physics before that you can actually get somewhere. Well that's maybe too technical, but that's as good as I can do with this.

Recorded on December 16, 2009
Interviewed by Austin Allen

Peter Woit explains the "deep relation" between the two disciplines and the most mind-bending new ways in which that relation is being explored.

Does conscious AI deserve rights?

If machines develop consciousness, or if we manage to give it to them, the human-robot dynamic will forever be different.

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  • Does AI—and, more specifically, conscious AI—deserve moral rights? In this thought exploration, evolutionary biologist Richard Dawkins, ethics and tech professor Joanna Bryson, philosopher and cognitive scientist Susan Schneider, physicist Max Tegmark, philosopher Peter Singer, and bioethicist Glenn Cohen all weigh in on the question of AI rights.
  • Given the grave tragedy of slavery throughout human history, philosophers and technologists must answer this question ahead of technological development to avoid humanity creating a slave class of conscious beings.
  • One potential safeguard against that? Regulation. Once we define the context in which AI requires rights, the simplest solution may be to not build that thing.

A new hydrogel might be strong enough for knee replacements

Duke University researchers might have solved a half-century old problem.

Photo by Alexander Hassenstein/Getty Images
Technology & Innovation
  • Duke University researchers created a hydrogel that appears to be as strong and flexible as human cartilage.
  • The blend of three polymers provides enough flexibility and durability to mimic the knee.
  • The next step is to test this hydrogel in sheep; human use can take at least three years.
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Hints of the 4th dimension have been detected by physicists

What would it be like to experience the 4th dimension?

Two different experiments show hints of a 4th spatial dimension. Credit: Zilberberg Group / ETH Zürich
Technology & Innovation

Physicists have understood at least theoretically, that there may be higher dimensions, besides our normal three. The first clue came in 1905 when Einstein developed his theory of special relativity. Of course, by dimensions we’re talking about length, width, and height. Generally speaking, when we talk about a fourth dimension, it’s considered space-time. But here, physicists mean a spatial dimension beyond the normal three, not a parallel universe, as such dimensions are mistaken for in popular sci-fi shows.

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Predicting PTSD symptoms becomes possible with a new test

An algorithm may allow doctors to assess PTSD candidates for early intervention after traumatic ER visits.

Image source: camillo jimenez/Unsplash
Technology & Innovation
  • 10-15% of people visiting emergency rooms eventually develop symptoms of long-lasting PTSD.
  • Early treatment is available but there's been no way to tell who needs it.
  • Using clinical data already being collected, machine learning can identify who's at risk.

The psychological scars a traumatic experience can leave behind may have a more profound effect on a person than the original traumatic experience. Long after an acute emergency is resolved, victims of post-traumatic stress disorder (PTSD) continue to suffer its consequences.

In the U.S. some 30 million patients are annually treated in emergency departments (EDs) for a range of traumatic injuries. Add to that urgent admissions to the ED with the onset of COVID-19 symptoms. Health experts predict that some 10 percent to 15 percent of these people will develop long-lasting PTSD within a year of the initial incident. While there are interventions that can help individuals avoid PTSD, there's been no reliable way to identify those most likely to need it.

That may now have changed. A multi-disciplinary team of researchers has developed a method for predicting who is most likely to develop PTSD after a traumatic emergency-room experience. Their study is published in the journal Nature Medicine.

70 data points and machine learning

nurse wrapping patient's arm

Image source: Creators Collective/Unsplash

Study lead author Katharina Schultebraucks of Columbia University's Department Vagelos College of Physicians and Surgeons says:

"For many trauma patients, the ED visit is often their sole contact with the health care system. The time immediately after a traumatic injury is a critical window for identifying people at risk for PTSD and arranging appropriate follow-up treatment. The earlier we can treat those at risk, the better the likely outcomes."

The new PTSD test uses machine learning and 70 clinical data points plus a clinical stress-level assessment to develop a PTSD score for an individual that identifies their risk of acquiring the condition.

Among the 70 data points are stress hormone levels, inflammatory signals, high blood pressure, and an anxiety-level assessment. Says Schultebraucks, "We selected measures that are routinely collected in the ED and logged in the electronic medical record, plus answers to a few short questions about the psychological stress response. The idea was to create a tool that would be universally available and would add little burden to ED personnel."

Researchers used data from adult trauma survivors in Atlanta, Georgia (377 individuals) and New York City (221 individuals) to test their system.

Of this cohort, 90 percent of those predicted to be at high risk developed long-lasting PTSD symptoms within a year of the initial traumatic event — just 5 percent of people who never developed PTSD symptoms had been erroneously identified as being at risk.

On the other side of the coin, 29 percent of individuals were 'false negatives," tagged by the algorithm as not being at risk of PTSD, but then developing symptoms.

Going forward

person leaning their head on another's shoulder

Image source: Külli Kittus/Unsplash

Schultebraucks looks forward to more testing as the researchers continue to refine their algorithm and to instill confidence in the approach among ED clinicians: "Because previous models for predicting PTSD risk have not been validated in independent samples like our model, they haven't been adopted in clinical practice." She expects that, "Testing and validation of our model in larger samples will be necessary for the algorithm to be ready-to-use in the general population."

"Currently only 7% of level-1 trauma centers routinely screen for PTSD," notes Schultebraucks. "We hope that the algorithm will provide ED clinicians with a rapid, automatic readout that they could use for discharge planning and the prevention of PTSD." She envisions the algorithm being implemented in the future as a feature of electronic medical records.

The researchers also plan to test their algorithm at predicting PTSD in people whose traumatic experiences come in the form of health events such as heart attacks and strokes, as opposed to visits to the emergency department.

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