Knutson's proposal makes use of degeneration of algebraic varieties to study algebraic combinatorics, particularly combinatorial representation theory. Littelmann's path model in representation theory already has such an interpretation: Chirivi gave a degeneration of each flag manifold G/P (plus ample line bundle) to a reduced, seminormal union of toric varieties. Unfortunately, this used properties of Lusztig's canonical basis, so was specific to representation theory applications. Knutson hopes to replace these properties with Samuel-Rees-Nagata degenerations, to achieve similar reducedness results for degenerations of much more general varieties. One specific application is in computing branching rules from a group G to a symmetric subgroup K, using a K-equivariant degeneration of G/P, and Knutson, jointly with Jiang-Hua Liu, states a conjectural rule based on this.
Many interesting numbers and polynomials are associated to irreducible algebraic sets, where "irreducible" essentially means "not glued from smaller pieces in a nontrivial way". A circle x^2 + y^2 - 1 = 0 is such an example, as its equation does not factor, and one can associate the degree 2 to this equation. But if we degenerate it to x^2 + y^2 - 0 = 0, then the equation does factor (over the complex numbers), giving the union of two lines x = +/- iy. Those two equations are degree 1, and from the degenerative geometry we obtain the combinatorial result 2 = 1 + 1. Knutson uses this technique to study much more interesting irreducible sets, e.g. the space of all maximal nested chains of subspaces in a vector space; degenerating them to highly reducible unions of simple pieces replaces their geometric complexity with combinatorial complexity, and gives much more interesting formulae for their degrees (and generalizations thereof). In some of the work proposed, he specifically plans to pursue this technique to understand how quantum-mechanical systems with noncompact symmetry (e.g. under special relativity transformations, which allow boosts by speeds up to but not including light-speed) decompose when one only considers their compact symmetries (e.g. under rotation, by angles that are trapped on a circle and cannot run off to infinity). This is the same sort of problem (though the hoped-for results would be wholly complementary) as studied recently by the ATLAS team in the study of the representations of E_8.
The primary goal of this research was to understand why some known degenerations of geometric sets were better behaved than expected. Consider a circle in the plane, which we stretch out horizontally to an ellipse of width W. As W goes to infinity, the ellipse limits to a pair of lines (becoming disconnected, which in more serious examples is useful for divide-and-conquer strategies). The bad situation is W to zero, where the sides of the ellipse fall atop one another. Algebraically, the equation x^2 + W*(y^2 - 1) = 0 limits to x^2 = 0, which is not as numerically stable as x = 0. The PI developed a framework to exclude such possibilities, based on (temporary) refuge from real/complex numbers to integers modulo a prime. This is based on the well-known "Freshman's Dream", that (a + b)^p = a^p + b^p mod p, which gives one a powerful tool for controlling equations with rogue exponents, such as x^2 = 0. It turned out to be useful not to consider a single set (like the circle), but an entire stratification of a vector space R^n, with strata of every dimension, analogous to the stratification of a polyhedron by its faces. In the most common setup, the stratification degenerates to the one defined by all 2^n coordinate subspaces, replacing geometric complexity by combinatorial complexity (which in particular makes it more studiable by computer). As a small example, consider the three stratifications of the plane in figure 1; the left is a cubic curve (whose real points are disconnected, but whose complex solutions are connected), the middle and right being successively more degenerate versions. Each stratum breaks into pieces; stated in reverse, we glue degenerate strata together as we go from right to left. This gluing is recorded in figure 2. In a separate project the PI studied a specific stratified space, that of all k-dimensional subspaces of n-space. (In the k=1, n=2 example these are lines in the plane through the origin, which one indexes by their slope -- a number or infinity.) To stratify, we group k-planes according to how they intersect a fixed "flag" of subspaces, meaning a chain 0 < V_1 < V_2 < ... < V_n where V_m is m-dimensional. Where this intersection jumps up (by at most one dimension) we write 1, otherwise 0, resulting in a word of length n with k 1s and n-k 0s. This particular stratified space has been studied since the 19th century and has been a driving example for topology since that field's inception. For example, take k=2, n=4, which is easier to visualize in projective geometry where it means lines L (not required to go through the origin) in 3-dimensional space. A flag is now a point V_1 inside a line V_2 inside a plane V_3 inside space V_4. If L = V_2, the associated word is 1100. If contains the point V_1, but is otherwise generic, the word is 1001. There has been much work on imposing multiple such linear conditions, since solving a nonlinear problem generally requires solving an associated linear approximation. For an example look at lines L touching two other lines, V_2 and W_2. (The associated words are both 0101.) If we consider the degenerate case that V_2 and W_2 themselves touch at a point b, then L must have one of two forms: either L goes through b (word = 1001), or L lies in the plane containing V_2 and W_2 (word = 0110). This replacement of "L does this AND this" by "L does this OR this" is used to define a multiplication, written (S_0101)^2 = S_1001 + S_0110 in this example. There are many computational rules for this multiplication and its generalizations. One due to the PI and Terence Tao in prior work uses tilings called "puzzles", where the puzzle pieces are triangles or rhombi labeled with 0s and 1s. In figure 3 one can see the above computation done with puzzles (red = 0, blue = 1). In the present work, the PI found the correct refinement of the standard stratification within which to use degenerative techniques, extending work of Vakil, and puzzles fall out almost automatically. The broader impacts of the puzzle work, and its direct connection to the linear geometry, have been at many levels. In 2012 the PI lectured on this at a week-long summer school to about 250 participants (graduate students to professors). It has also been well received in introductory linear algebra classes, as the stratification above is exactly that coming out of Gaussian elimination of matrices (the words give the "pivot columns"); students were especially interested to see 21st century mathematics in their classroom. Finally, if one takes the puzzles themselves as a starting point, there are many combinatorial theorems one can explain about them, as the PI has done in elementary and even pre-school classes. (For this it is very helpful to have physical, laser-cut puzzle pieces, as seen in figure 3.)
