Research
Preprints
Solving the area-length systems in discrete gravity using homotopy continuation
(S. K. Asante, B.)
Using tools from numerical algebraic geometry, we study the area-length systems. In particular, we show that given the ten triangular areas of a single 4-simplex, there could be up to 64 compatible sets of edge lengths. Moreover, we show that these 64 solutions do not, in general, admit formulae in terms of the areas by analyzing the Galois group, or monodromy group, of the problem. We show that by introducing additional symmetry constraints, it is possible to obtain such formulae for the edge lengths. We take the first steps toward applying our results within discrete quantum gravity, specifically for effective spin foam models.
Lawrence Lifts, Matroids, and Maximum Likelihood Degrees
(B., A. Maraj)
Quatroids and Rational Plane Cubics
(B., F. Gesmundo, A. Steiner)
Accepted/Published
Monodromy Coordinates
Max-Convolution through numerics and tropical geometry
(B., J. Hauenstein and C. Hills)
Numerical Algorithms
The algebraic matroid of the Heron variety
(S. K. Asante, B., M. Hatzel)
Code
Polyhedral Geometry in OSCAR
(B., M. Joswig)
To appear as a chapter in the OSCAR book
Enumerating chambers of hyperplane arrangements with symmetry
(B., H. Eble, L. Kuhne)
Discrete & Computational Geometry
Sparse trace tests
(T. Brysiewicz, M. Burr)
Mathematics of Computation (Accepted)
Likelihood degenerations
(D. Agostini, T. Brysiewicz, C. Fevola, L. Kuhne, B. Sturmfels, S. Telen)
Advances in Mathematics 451 (2022) 108863
Tangent Quadrics in Real 3-Space
(B., C. Fevola, B. Sturmfels)
Le Matematiche 76 (2021) 355-367
Nodes on quintic spectrahedra
(B., K. Kozhasov, M. Kummer)
Le Matematiche 76 (2021) 415-430
Solving decomposable sparse systems
(B., J. Rodriguez, F. Sottile, and T. Yahl)
Brysiewicz, Taylor & Rodriguez, Jose & Sottile, Frank & Yahl, Thomas. (2021). Solving decomposable sparse systems. Numerical Algorithms. 10.1007/s11075-020-01045-x.We present a recursive algorithm to solve decomposable sparse systems.
The degree of the Stiefel manifold
(B.)
Mathematics in Computer Science (2020)We present an implementation in Macaulay2 of numerical software which computes the Newton polytope of a hypersurface given as the image of a map.We also describe how one can use this as a tropical membership test.
Supplementary Materials
(B.)
J. Symb. Comput. (2019) https://doi.org/10.1016/j.jsc.2019.11.002We count the number of polynomial parametrized osculants to a general analytic curve in the plane: as combinatorial necklaces.
Supplementary Materials
(Brandt, Bruce, B., Krone, Robeva)
Combinatorial Algebraic Geometry, 207-224, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci., 2017.We compute the degree of SO(n) as an algebraic variety in C^(n^2). We give a combinatorial description of this number as a non-intersecting lattice path count and we relate this degree with the complexity of a low-rank relaxation of semi-definite programming.
Supplementary Materials
(Balay-Wilson, B.)
Rose-Hulman Undergraduate Mathematics Journal. Volume 15, No. 1, Spring 2014.We give an explicit geometric condition for when, for a point on a cubic, there exists another cubic intersecting it exactly at that point (with multiplicity 9).
Supplementary Materials
Dissertation
Newton polytopes and numerical algebraic geometry
Supplementary Materials