Two Brandeis Professors Receive 2017 Simons Fellowships, part II

Spectral Flow

Spectral Flow (full caption below)

Two Brandeis professors have been awarded highly prestigious and competitive Simons Fellowships for 2017. Daniel Ruberman received a 2017 Simons Fellowship in Mathematics. Matthew Headrick was awarded a 2017 Simons Fellowship in Theoretical Physics. This is the second of two articles where each recipient describes their award-winning research.

Daniel Ruberman’s research asks “What is the large-scale structure of our world?” Einstein’s unification of physical space and time tells us that the universe is fundamentally 4-dimensional. Paradoxically, the large-scale structure, or topology, of 4-dimensional spaces, is much less understood than the topology in other dimensions. Surfaces (2-dimensional spaces) are completely classified, and the study of 3-dimensional spaces is largely dominated by geometry. In contrast, problems about spaces of dimension greater than 4 are translated, using the technique called surgery theory, into the abstract questions of algebra.

Ruberman will work on several projects studying the large-scale topology of 4-dimensional spaces. His work combines geometric techniques with the study of partial differential equations arising in physics. One major project, with Nikolai Saveliev (Miami) is to test a prediction of the high-dimensional surgery theory, that there should be `exotic’ manifolds that resemble a product of a circle and a 3-dimensional sphere. The proposed method, which would show that this prediction is incorrect, is to compare numerical invariants derived from the solutions to the Yang-Mills and Seiberg-Witten equations, by embedding both in a more complicated master equation. The study of the Seiberg-Witten invariants is complicated by their instability with respect to varying geometric parameters in the theory. A key step in their analysis is the introduction of the notion of end-periodic spectral flow, which compensates for that instability, as illustrated below.

Other projects for the year will apply techniques from 4-dimensional topology to classical problems of combinatorics and geometry about configurations of lines in projective space. In recent years, combinatorial methods have been used to decide if a specified incidence relation between certain objects (“lines”) and other objects (“points”) can be realized by actual points and lines in a projective plane. For the real and complex fields, one can weaken the condition to look for topologically embedded lines (circles in the real case, spheres in the complex case) that meet according to a specified incidence relation. Ruberman’s work with Laura Starkston (Stanford) gives new topological restrictions on the realization of configurations of spheres in the complex projective plane.

Caption: Solutions to the Seiberg-Witten equations of quantum field theory provide topological information about 4-dimensional spaces. However, the set of solutions, or moduli space, can undergo a phase transition as a parameter T is varied, making those solutions hard to count. This figure illustrates a key calculation: the phase transition is equal to the end-periodic spectral flow, a new concept introduced in work of Mrowka-Ruberman-Saveliev. In the figure, the spectral set, illustrated by the red curves, evolves with the parameter T. Every time the spectral set crosses the cylinder, the moduli space changes, gaining or losing points according to the direction of the crossing.

Two Brandeis Professors Receive 2017 Simons Fellowships

Bit threads in a holographic spacetime

Bit threads in a holographic spacetime

Two Brandeis professors have been awarded highly prestigious and competitive Simons Fellowships for 2017. Daniel Ruberman received a 2017 Simons Fellowship in Mathematics. Matthew Headrick was awarded a 2017 Simons Fellowship in Theoretical Physics. This is the first of two articles where each recipient’s award-winning research is described.

Matthew Headrick’s research studies the phenomenon of entanglement in certain quantum systems and its connection to the geometry of spacetime in general relativity. This very active area of research is the culmination of three developments in theoretical physics over the past 20 years.

First, in 1997, string theorists discovered that certain quantum systems involving a large number of very strongly interacting constituents — whose analysis would normally be intractable — are secretly equivalent to general relativity — a classical theory describing gravity in terms of curved spacetime — in a space with an extra dimension. For example, if the quantum system has two dimensions of space, then the general relativity has three; the phenomenon is thus naturally dubbed “holography”.

This equivalence between two very different-looking theories is incredibly powerful, and has led to much progress in understanding both strongly-interacting quantum systems and general relativity. However, it is still not fully understood how or precisely under what conditions such an equivalence holds.

[Read more…]

Simons Foundation funds Brandeis Math, Physics collaborations

In 2014, the Simons Foundation, one of the world’s largest and most prominent basic science philanthropies, launched an unprecedented program to fund multi-year, international research collaborations in mathematics and theoretical physics. These are $10M grants over four years, renewable, that aim to drive progress on fundamental scientific questions of major importance in mathematics, theoretical physics, and theoretical computer science. There were 82 proposals in this first round. In September 2015, two were funded. Both involve Brandeis.

Matthew Headrick (Physics) is deputy director of the Simons Collaboration It from Qubit, which involves 16 faculty members at 15 institutions in six countries. This project is trying from multiple angles to bring together physics and quantum information theory, and show how some fundamental physical phenomena (spacetime, black holes etc.) emerge from the very nature of quantum information. Fundamental physics and quantum information theory remain distinct disciplines and communities, separated by significant barriers to communication and collaboration. “It from Qubit” is a large-scale effort by some of the leading researchers in both communities to foster communication, education and collaboration between them, thereby advancing both fields and ultimately solving some of the deepest problems in physics. The overarching scientific questions motivating the Collaboration include:

  • Does spacetime emerge from entanglement?
  • Do black holes have interiors?
  • Does the universe exist outside our horizon?
  • What is the information-theoretic structure of quantum field theories?
  • Can quantum computers simulate all physical phenomena?
  • How does quantum information flow in time?

Bong Lian (Mathematics) is a member of the Simons Collaboration on Homological Mirror Symmetry, which involves nine investigators from eight different institutions in three countries. Mirror Symmetry, first discovered by theoretical physicists in late ‘80s, is a relationship between two very different-looking physical models of Nature, a remarkable equivalence or “duality” between different versions of a particular species of multidimensional space or shape (Calabi-Yau manifolds) that seemed to give rise to the same physics. People have been trying to give a precise and general mathematical description of this mirroring ever since, and in the process have generated a long list of very surprising and far-reaching mathematical predictions and conjectures. The so-called “Homological Mirror Symmetry Conjecture” (HMS) may be thought of as a culmination of these efforts, and Lian was a member of the group (including S.-T. Yau) that gave a proof of a precursor to HMS in a series of papers in the late ‘90s.

Lian and his Simons collaborators are determined to prove HMS in full generality and explore its applications. One consequence of HMS says that if one starts from a “complex manifold” (a type of even-dimensioned space that geometers have been studying since Riemann described the first examples in 1845), then all its internal geometric structures can in fact be described using a certain partner space, called a “symplectic manifold”. The latter type of space was a mathematical edifice invented to understand classical physics in the mid-1900s. This connection goes both ways: any internal geometric structure of the symplectic partner also has an equally compelling description using the original complex partner. No one had even remotely expected such a connection, especially given that the discoveries of the two types of spaces — complex and symplectic — were separated by more than 100 years and were invented for very different reasons. If proven true, HMS will give us ways to answer questions about the internal geometric structure of a complex manifold by studying its symplectic partner, and vice versa.

Proving HMS will also help resolve many very difficult problems in enumerative geometry that for more than a century were thought to be intractable. Enumerative geometry is an ancient (and until recently moribund) branch of geometry in which people count the number of geometric objects of a particular type that can be contained inside a space. Mirror symmetry and HMS have turned enumerative geometry into a new way to characterize and relate shapes and spaces.

Why do so many powers of 2 start with “1”?

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384 …

If you liked math in middle school, odds are that maybe you memorized the powers of two. But did you ever think about the fact that so many of them start with the digit “1”? Is there a reason for it? How would you go about stating the problem in more formal mathematical terms?

Dmitry Kleinbock from the Brandeis Math department explains in this Numberphile video:

Be sure to watch the extra content (below) for a slightly more technical, but still completely approachable, additional explanation, where the problem reduces to the so-called equidistribution property of irrational rotations of the unit circle.

Are there other numbers more likely to start with the digit “1”?  It’s pretty easy to convince yourself that there are.

SciFest V is in the books

The Brandeis University Division of Science held its annual undergraduate research poster session SciFest V on July 30, 2015. Despite the 90 degree heat (and the steam leak) outside, the student presenters in the Shapiro Science atrium admirably kept their cool and showed off the results of their summer’s (or last year’s) worth of independent research. We had a great audience of grad students, postdocs, faculty, proud parents, members of the Brandeis senior administration, visiting neuroscientists at Brandeis helping evaluate our Computational Neuroscience training program, and physicists at Brandeis attending the IGERT Summer Institute.

IMG_1295

If you’re a student who didn’t get to present, or you’re a community member who just wanted a chance to talk about science with our energized undergrads, we’re planning another session for Fall Fest 2015. Stay tuned for details.

For a few more impressions of the event, see the story at Brandeis NOW. More pictures and abstract books are available at the SciFest site.

SciFest V by numbers

 

IGERT Summer Institute – July 27 to August 7, 2015

IGERTBrandeis is hosting a two-week summer institute for graduate students in the mathematical sciences from July 27-August 7.  This will combine the annual summer institute of Brandeis’ Geometry and Dynamics IGERT program, with a sequel to the US-India Advanced Studies Institute on thermalization, held two years ago in Bangalore.

Topics:

  • Large deviation theory
  • Statistics of extreme events
  • The large N expansion in statistical and quantum physics
  • Statistical fluid dynamics
  • Quantum information and quantum gravity
  • Thermalization in Quantum Systems

Lecturers:

Sumit Das (U. Kentucky)
Chandan Dasgupta (IISC, Bangalore)
Rajesh Gopakumar (HRI, Allahabad and ICTS)
Alex Maloney (McGill University)
Satya Majumdar (LPTMS, Paris)
Sanjib Sabhapandit (Raman Research Institute, Bangalore)
Peter Weichman (BAE systems)

Organizers:

Albion Lawrence
Bulbul Chakraborty

Registration:

There will be no registration fee, but the venue will have limited capacity, so interested students should register by sending an email to Catherine Broderick (cbroderi@brandeis.edu) by July 4. Please list your affiliation, your year in graduate school, any publications, and the name of your PhD advisor.

Additional information can be found at www.brandeis.edu/igert/.

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