David Hilbert

A brief illustrated guide to 'scissors congruence' − an ancient geometric idea that’s still fueling cutting-edge mathematical research

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Wednesday, August 9, 2023
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Remember “base x height” for rectangles and “½ base x height” for triangles?

Key Points: 
  • Remember “base x height” for rectangles and “½ base x height” for triangles?
  • But if you were in math class in ancient Greece, you might have learned something very different.
  • Ancient Greek mathematicians, such as Euclid, thought of area as something geometric, not algebraic.

Euclid’s notion of area

    • So what does it mean to think of area as something geometric the way the ancient Greeks did?
    • Using your algebraic formulas for area, you could check that the area of your new shape is equal to the original area of the construction paper.
    • In other words, for polygons, Euclid’s notion of area is exactly the same as the modern one.
    • Since area is preserved if you cut the pentagon up into smaller triangles, you can instead find the area of these triangles (using “½ base x height”) and add them up to get the answer.

Hilbert’s third problem

    • The third problem on the list, and the first to be resolved, is about scissors congruence.
    • Dehn’s solution to the problem was very different from the two-dimensional case.
    • So, if Dehn could find two polyhedra with the same volume but different values of this invariant, that would prove the answer to Hilbert’s third problem is no – scissors congruence doesn’t precisely capture 3D volume.
    • In 1965, Jean-Pierre Sydler confirmed that the answer is yes, closing this chapter on scissors congruence.

Strange shapes and stranger connections

    • Mathematics is full of shapes living in higher dimensions – like 4D, 100D, 3,485D or any dimension you can imagine – which are impossible to visualize.
    • An active new research area called generalized scissors congruence seeks to uncover whether Hilbert’s question about scissors congruence can also be stated – and maybe even solved – for these strange shapes.
    • A recent research program pioneered by mathematicians Jonathan Campbell and Inna Zakharevich proposes a unifying framework for generalized scissors congruence.
    • With a little bit of adjustment, mathematicians can harness the machinery of K-theory and apply it to generalized scissors congruence problems.

NTT Study: Artificial Neural Networks for Recognizing Natural Sounds Exhibit Human-Like Responses

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Tuesday, August 1, 2023
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Nippon Telegraph and Telephone Corporation (headquartered in Chiyoda-ku, Tokyo; Akira Shimada, President & CEO; hereinafter "NTT") has discovered that artificial neural networks (NNs, *1) that recognize natural sounds (*2) show human-like responses to changes in sound amplitude.

Key Points: 
  • Nippon Telegraph and Telephone Corporation (headquartered in Chiyoda-ku, Tokyo; Akira Shimada, President & CEO; hereinafter "NTT") has discovered that artificial neural networks (NNs, *1) that recognize natural sounds (*2) show human-like responses to changes in sound amplitude.
  • NTT Laboratories has been conducting studies using artificial neural networks (NN, *1) to understand auditory AM processing.
  • AM sounds were fed to NNs trained to recognize natural sounds (*4) and their responses were examined.
  • The results suggest that the response to AM sound in animal brains might be a result of adaptation to recognize natural sounds.

Has a mathematician solved the 'invariant subspace problem'? And what does that even mean?

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Monday, June 12, 2023
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The paper is just 13 pages long and its list of references contains only a single entry.

Key Points: 
  • The paper is just 13 pages long and its list of references contains only a single entry.
  • The paper purports to contain the final piece of a jigsaw puzzle that mathematicians have been picking away at for more than half a century: the invariant subspace problem.

Per Enflo: mathematics, music, and a live goose

    • By solving the approximation problem, Enflo cracked an equivalent puzzle called Mazur’s goose problem.
    • Polish mathematician Stanisław Mazur had in 1936 promised a live goose to anyone who solved his problem – and in 1972 he kept his word, presenting the goose to Enflo.

What’s an invariant subspace?

    • But what about the invariant subspace problem itself?
    • Another way to think about this is to say that the matrix transforms the eigenvectors (and any lines parallel to them) back onto themselves: these lines are invariant for this matrix.
    • Taken together, we call these lines invariant subspaces of the matrix.

What about spaces with an infinite number of dimensions?

    • The invariant subspace problem is a little more complicated: it is about spaces with an infinite number of dimensions, and it asks whether every linear operator (the equivalent of a matrix) in those spaces must have an invariant subspace.
    • Mathematicians working on this problem narrowed their focus by restricting the problem to particular classes of spaces and operators.
    • He answered the problem in the negative, by constructing an operator on a Banach space without a non-trivial invariant subspace.

What’s new about this new proposed solution?

    • Resolving the invariant subspace problem for operators on Hilbert spaces has been stubbornly difficult, and it is this which Enflo claims to have achieved.
    • This time Enflo answers in the affirmative: his paper argues that every bounded linear operator on a Hilbert space does have an invariant subspace.

Expert review is still to come

    • Peer review of Enflo’s earlier proof, for Banach spaces in general, took several years.
    • However, that paper ran to more than 100 pages, so a review of the 13 pages of the new paper should be much speedier.

ColdQuanta Announces $110 Million Series B to Continue Commercializing Quantum Technology Products Across the Global Ecosystem

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Tuesday, November 1, 2022
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BOULDER, Colo., Nov. 1, 2022 /PRNewswire/ -- ColdQuanta, the global quantum ecosystem leader, today announced a $110 million Series B round of funding to continue commercializing the company's product portfolio, including quantum computing, quantum algorithms and applications, atomic clocks, sensors, and components.

Key Points: 
  • Breakthrough Victoria has participated in this round to help establish an Asia-Pacific quantum computing and technology facility at Swinburne University of Technology headquarters, known as the ColdQuanta Swinburne Quantum Technology Centre.
  • "The quantum ecosystem is experiencing a global wave of innovation, having our market leadership recognized by investors around the world validates our unique approach," said ColdQuanta CEO Scott Faris.
  • "Customers are already adopting ColdQuanta for quantum RF sensors, quantum atomic clocks, and quantum software.
  • ColdQuanta also offers components for quantum systems and applications, and quantum research and security solutions including high-precision clocks, navigation, radio frequency receivers, and quantum networking and communications, which are deployed throughout the quantum ecosystem today.

ColdQuanta Expands Leadership Team with Promotion of Chris Wood to Chief Technology Officer and Addition of New Top Talent

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Tuesday, October 4, 2022
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BOULDER, Colo., Oct. 4, 2022 /PRNewswire/ -- ColdQuanta, the global quantum ecosystem leader, today announced it has expanded its leadership team with a stable of quantum experts including: Dr. William Clark as Vice President of Quantum Development, Laura Hale as Vice President of Government Programs, and Steve Matthews, as Vice President of Sales & Business Development for Quantum Information Platforms. VP of Photonic Technologies, Dr. Chris Wood, has been promoted to Chief Technology Officer (CTO). Acting Chief Technology Officer, Dr. Dana Z. Anderson, will serve as Chief Strategy Officer (CSO), guiding the company's long-term quantum strategy. 

Key Points: 
  • VP of Photonic Technologies, Dr. Chris Wood, has been promoted to Chief Technology Officer (CTO).
  • Acting Chief Technology Officer, Dr. Dana Z. Anderson, will serve as Chief Strategy Officer (CSO), guiding the company's long-term quantum strategy.
  • Additionally, ColdQuanta announced the commercial beta launch of Hilbert , the world's first cold atom quantum computer.
  • The name ColdQuanta and the ColdQuanta logo are both registered trademarks of ColdQuanta, Inc.
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Sonic Foundry’s Global Learning Exchange™ Announces Partnerships with the University of North Alabama, Hilbert College, and SkillUp Online

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Wednesday, August 17, 2022
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We look forward to our collaborations in providing convenient, reliable, life-changing learning opportunities for traditional university students and working adults around the world.

Key Points: 
  • We look forward to our collaborations in providing convenient, reliable, life-changing learning opportunities for traditional university students and working adults around the world.
  • Hilbert President Michael Brophy, Ph.D. added, We are thrilled about the opportunity to join the Global Learning Exchange community.
  • On July 2, Global Learning Exchange celebrated the Grand Opening of the Hub facility that will anchor its inaugural program in the Bahamas.
  • Trusted by thousands of educational institutions, corporations, health organizations and government entities in over 65 countries, Sonic Foundrys solutions include Mediasite, Mediasite Connect, Vidable and Global Learning Exchange.

ColdQuanta Accelerates Momentum in First Half of 2022

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Wednesday, July 13, 2022
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BOULDER, Colo., July 13, 2022 /PRNewswire/ -- ColdQuanta, the global quantum ecosystem leader, today announced significant momentum for the first half of 2022 highlighted by business growth, technical milestones, leadership expansion and more. These accomplishments further demonstrate the company's commitment to driving maturation of its products and its ability to leverage game-changing talent and experience.

Key Points: 
  • BOULDER, Colo., July 13, 2022 /PRNewswire/ -- ColdQuanta , the global quantum ecosystem leader, today announced significant momentum for the first half of 2022 highlighted by business growth, technical milestones, leadership expansion and more.
  • "The momentum in which we entered the New Year, with a 140% in bookings increase and a nearly doubling of our headcount, has positioned ColdQuanta for a strong year," said Scott Faris, ColdQuanta CEO.
  • "ColdQuanta continues to demonstrate its industry leadership by attracting esteemed quantum experts, attaining notable product milestones, and garnering third party endorsements.
  • The name ColdQuanta and the ColdQuanta logo are both registered trademarks of ColdQuanta, Inc.
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SEC Charges CohnReznick LLP and Three Partners with Improper Professional Conduct

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Wednesday, June 8, 2022
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Washington, D.C.--(Newsfile Corp. - June 8, 2022) - The Securities and Exchange Commission today charged audit firm CohnReznick LLP with improper professional conduct on engagements for two clients in 2017.

Key Points: 
  • Washington, D.C.--(Newsfile Corp. - June 8, 2022) - The Securities and Exchange Commission today charged audit firm CohnReznick LLP with improper professional conduct on engagements for two clients in 2017.
  • The Commission also charged firm partners Stephen M. Wyss, Stephen H. Jackson, and Robert G. Hilbert with improper professional conduct for violating numerous professional standards in their third quarter 2017 interim review and 2017 annual audit of Sequentials financial statements.
  • Despite CohnReznicks national office partners and the firms own valuation specialists expressing concerns with Sequentials conclusion, the firm failed to obtain sufficient evidence or conduct additional procedures.
  • The order finds that CohnReznicks deficient system of quality control led to failures to adhere to professional auditing standards.

ColdQuanta Acquires Super.tech and Announces the Commercial Availability of Hilbert, the World's First Cold Atom Quantum Computer

Retrieved on: 
Tuesday, May 10, 2022
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BOULDER, Colo., and CHICAGO, May 10, 2022 /PRNewswire/ -- ColdQuanta, the global quantum ecosystem leader, today announced its acquisition of Chicago-based Super.tech, a world leader in quantum software application and platform development, and the beta launch of Hilbert, the world's first gate-based cold atom quantum computer. Super.tech's full team of quantum information scientists and software developers, including CEO Pranav Gokhale and Chief Scientist Fred Chong, will join ColdQuanta and lead the company's new Chicago office. The acquisition of Super.tech transforms ColdQuanta into a multi-platform, quantum hardware and software company.

Key Points: 
  • The combination of Super.tech's software applications and algorithms and ColdQuanta's ground-breaking quantum computer, Hilbert, will enable customers to accelerate time to value."
  • ColdQuanta's Hilbert is the world's first commercial cold atom quantum computer.
  • "The commercial release of Hilbert marks an important and exciting milestone for ColdQuanta and for the cold atom quantum computing modality," said Paul Lipman, President of Quantum Information Platforms at ColdQuanta.
  • Super.tech's software makes quantum computing solutions accessible without the need for quantum expertise, accelerating the commercial adoption of this burgeoning technology.

Collaboration Between Leading Quantum Organizations Achieves Industry First for Neutral Atom Quantum Computing

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Wednesday, April 20, 2022
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BOULDER, Colo., April 20, 2022 /PRNewswire/ -- ColdQuanta, Riverlane and the University of Wisconsin–Madison, today announced they have successfully run a quantum algorithm on a cold atom qubit array system, codenamed "AQuA," an industry first that brings quantum computing one step closer to real world applications. The milestone was conducted at the University of Wisconsin–Madison in the research group led by Prof. Mark Saffman. A paper on the research was published today in Nature, the world's leading multidisciplinary science journal.

Key Points: 
  • Cold atom hyperfine qubits provide inherent scalability due to their identical characteristics, long coherence times, and ability to be trapped in dense multi-dimensional arrays.
  • The team is the first in the world to demonstrate quantum algorithms on a programmable gate model cold atom quantum computer.
  • "This is the first quantum computer using cold atoms with multiple qubits and running quantum algorithms, so it's a significant step forward for the cold atom approach."
  • This starts with breaking through the biggest barrier to quantum computing today: quantum error correction.