Bruce Fontaine

About me

Teaching

Cornell

• Math 2220 - Multivariable Calculus (Spring 2013)
• Math 1910 - Calculus for Engineers (Spring 2013)
• Math 2210 - Linear Algebra (Fall 2013)
• Math 4320 - Abstract Algebra (Spring 2014)
• Math 2220 - Multivariable Calculus (Spring 2014)
• Math 2310 - Linear Algebra with Applications (Fall 2014)
• Math 7580 - Perverse Sheaves (Spring 2015)

Toronto

• MAT244 - Differential Equations (Winter 2011)
• MAT244 - Differential Equations (Summer 2011)
• MAT237 - Advanced Calculus (Fall 2011/Winter 2012)

Research Interests

• Friezes & Cluster Algebras

  At MSRI Pierre-Guy Plamondon and I starting thinking about counting the number of Coxeter-Conway type friezes in cluster types other than A_n. Recently we have solved the problem for all types except E_n and F_2. As an extension of this work I have been considering what must be done to count friezes over rings of integers and as a first step have been generalizing our results to the non-zero integers.

  Papers:
  Counting friezes in type D_n. (with Pierre-Guy Plamondon) arXiv:1409.3698
  Non-negative integer friezes. In Preperation

• Geometric Representation Theory

  Webs and Spiders

  In 1997 Greg Kuperberg introduced the ideas of webs and spiders. These are directed labeled graphs that correspond to elements of the G invariants in a tensor product of irreducible G representations. Together with my supervisor Joel Kamnitzer, we created an algebraic variety naturally associated to each web. Under the geometric Satake correspondence this variety is linked to the invariant associated to the web. Using it, I established a set of webs for SL(3) whose invariant vectors span the space of invariants and show that the change of basis to the natural basis given by the geometric Satake correspondence is upper unitriangular.

  Papers:
  Generating basis webs for SL_n. Adv. in Math., 229 no. 5 (2012) 2792-2817. arXiv:1108.4616
  Buildings, spiders, and geometric Satake. (with Joel Kamnitzer and Greg Kuperberg) Compos. Math., 149 no. 11 (2013) 1871-1912. arXiv:1103.3519

  Cyclic Sieving & Satake Basis

  Papers:
  Cyclic Sieving, Rotation and Geometric Representation Theory. (with Joel Kamnitzer) Selecta Math., 20 no. 2 (2014) 609-625. arXiv:1212.1314

• Knot Theory

  My research in this area is a collaboration with Stuart Rankin and Ortho Flint. We developed an efficient computable complete invariant for prime alternating links and then used it to enumerate up to 24 crossings (A059739). Slight modification allowed the enumeration of the oriented prime alternating links up to 23 crossings. Computation of the symmetry group of a prime alternating link is a byproduct of the algorithm that produces the complete invariant. Our latest project was Knotilus, a website that provides automated drawings of links, the complete database of prime alternating links up to 23 crossings and computes a variety of interesting information. See this page for more information.

Other Interests

Other documents