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.

Undergraduate research fellowship opportunities

Meredith Monaghan, Director of Academic Fellowships, writes:

I am happy to announce the latest competition for two sources of funding designed to support undergraduate research at Brandeis University. Applications for both the Schiff Undergraduate Fellows Program and the Undergraduate Research Program are due in March; specific details for each are below. For your reference, I have also attached to this email the info sheets/applications for each.

Schiff Fellows work closely with a Faculty Mentor on a year-long research or pedagogical project; Fellows earn $2000 and their Faculty Mentors receive $500. Current and past Schiff Fellows describe this as an excellent opportunity to pursue independent research in collaboration with a caring and knowledgeable expert in their field. In past years, faculty members have been particularly helpful in identifying excellent candidates for the Schiff Fellowship, and have often approached a student directly with an idea for a project. Applications for academic year 2011-2012 are available in Academic Services (Usdan 130) or by emailing Meredith Monaghan. The submission deadline is 5pm on Monday, March 7, 2011.

This cycle of the Undergraduate Research Program competition is for summer 2011 grants. This award is open to students in all disciplines, and funds can be used to pay for research materials, travel to conferences, and other research-related expenses. Students need a recommendation from a faculty mentor, but the role of the faculty member is less hands-on for the URP than for the Schiff Fellowship Program. Applications are available in Academic Services (Usdan 130) or by emailing Meredith Monaghan. The submission deadline is 5pm on Wednesday, March 16, 2011.

For information about other fellowship opportunities, see the Academic Services website.

Last year’s winners, the 2010-2011 Schiff Fellows, are:

  • BENJAMIN G. COOPER ’11, Chemistry & Biology (with Prof. Christine Thomas) — “Catalyst Design for Environmentally-Friendly Production of Fuels”
  • USMAN HAMEEDI ’12, Biology & HSSP (with Prof. Bruce Foxman) — “Temperature Sensitive Ferrocene Complexes”
  • JUNE ALLISON HE ’11, Psychology (with Prof. Nicolas Rohleder) — “Investigating the Link Between Subjective Conceptions of Stress and Health and Age-Related Declines in Cognitive Functioning”
  • MAYA KOENIG ’11, IIM Medical Anthropology (with Prof. Sarah Lamb) — “Bringing Medical Anthropology to Brandeis / Using CAM to Conceptualize Health”
  • ALEXANDRA KRISS ’11, HSSP (with Prof. Sara Shostak) — “College-Aged Women & Contraceptives: What Does Advertising Have To Do With It?”
  • ALEXANDRU PAPIU ’12, Mathematics (with Prof. Bong Lian) — “Structural Properties of a Certain Kind of Semigroup”
  • Géraldine Rothschild ’12, Economics & French (with Prof. Edward Kaplan) — “Jewish Identities in France During 1945”
  • MARTHA SOLOMON ’11, Biology (with Prof. Lawrence Wangh) — “Barrett’s Adenocarcinoma and its Effects on Mitochondrial DNA”
  • ILANA SPECTOR ’11, Economics & Philosophy (with Prof. Marion Smiley) — “The Meaning of Life: Revealing Individual Perspectives Behind Broader Philosophical Notions”
  • JOSEPH POLEX WOLF ’11, Neuroscience & HSSP (with Prof. Angela Gutchess) — “Cognition at the Cross-Roads: Bicultural Cognitive Processing in Turkish Individuals”

Chiral Equivariant Cohomology

Prof. Bong Lian from Math writes:

In the 1950’s, French mathematicians Henri Cartan and Armand Borel defined a new topological invariant that was capable of distinguishing symmetries of certain geometric spaces known as G-manifolds. Cartan and Borel called their invariant the Equivariant Cohomology of a G-manifold. It was new in that it was able to capture essential aspects of geometric operations, called Lie group actions (after Sophus Lie), on manifolds that ordinary cohomology theory was unable to detect. Hence it provides a new conceptual framework for studying symmetries of spaces on the one hand, and offers a powerful tool for computing ordinary cohomology of these spaces, on the other.

In the late 80’s, physicists invented String Theory in their attempt to construct a grand unified field theory. They found that certain solutions to String Theory are essentially governed by an algebraic structure called a Chiral Algebra. This turns out to be a new structure that generalizes many fundamental algebraic constructs in mathematics, including commutative algebras and Lie algebras. A question was then raised as to whether there exists a natural theory that integrates both the Cartan-Borel invariant and Vertex Algebras. This hypothetical theory, which I learned about as a graduate student at Yale University, was dubbed the stringy analogue of the Equivariant Cohomology theory.

In 2004, Andrew Linshaw, a Brandeis PhD student (now Research Fellow, U. Darmstadt), and I constructed such a theory, which we coined the Chiral Equivariant Cohomology (CEC) of a G-manifold. In our latest paper, joint with another Brandeis PhD student, Bailin Song (now Assoc. Prof., Univ. of Science and Technology of China), we showed that not only does the CEC subsumes the Cartan-Borel theory, it goes well beyond that. For example, we have found an infinite family of Lie group actions on spheres that the Cartan-Borel theory is too weak to distinguish, but have non-isomorphic CEC. This proves that the CEC theory is strictly stronger as a topological invariant than the Cartan-Borel invariant. The paper appears in the December 2010 issue of the American Journal of Mathematics (Volume 132, Number 6).

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