Explore the computational intersections between algebra, combinatorics, and discrete geometry through this comprehensive workshop series from the Institute for Pure & Applied Mathematics (IPAM). Delve into the transformative connections that have emerged over the past fifty years between these mathematical fields, examining how computational approaches have revolutionized theoretical mathematics research. Investigate pivotal mathematical objects including monomial ideals, affine semigroup rings, Stanley-Reisner rings, Ehrhart rings, toric rings, Cox rings, and Chow rings from the algebraic perspective, alongside graphs, matroids, simplicial complexes, polytopes, polyhedral complexes, convex bodies, posets, lattices, and hyperplane arrangements from combinatorial and discrete geometry viewpoints. Discover how Gröbner bases and other computational tools have evolved to facilitate deeper understanding of these interconnected areas. Engage with cutting-edge research presentations covering Schubert coefficients, game theory equilibria, nonlinear algebra probability, toric ideal normalizations, polytope mutations, cosmological polytopes, graph coloring theory, matroid polytopes, Presburger modules, semi-Eulerian complexes, hyperplane arrangement symmetries, type cone Minkowski bases, valuation polytopes, symmetric edge polytopes, interval complexes, complexified arrangements, tropical ideals, symbolic powers, and vertex order shellings. Learn from leading mathematicians as they demonstrate unexpected connections and collaborative opportunities across these three fundamental mathematical domains, expanding your computational and theoretical toolkit for advanced mathematical research.

Syllabus

Colleen Robichaux - Deciding positivity of Schubert coefficients - IPAM at UCLA
Irem Portakal - Combinatorics of equilibria in game theory - IPAM at UCLA
Sonja Petrovic - Probability and Randomness in Nonlinear Algebra - IPAM at UCLA
Lauren Cranton Heller - Cellular resolutions for normalizations of toric ideals - IPAM at UCLA
Laura Escobar - Families of degenerations from mutations of polytopes - IPAM at UCLA
Martina Juhnke - Algebra, Geometry and Combinatorics of Cosmological polytopes - IPAM at UCLA
Akiyoshi Tsuchiya - Interactions between commutative algebra & graph coloring theory - IPAM at UCLA
Luis Ferroni - The polytope of all matroids - IPAM at UCLA
Jonathan Montaño - Presburger modules: quasipolynomial and tameness - IPAM at UCLA
José Alejandro Samper Casas - The cd-index of a semi-Eulerian complex - IPAM at UCLA
Sarah Brauner - Symmetries of rings from hyperplane arrangements - IPAM at UCLA
Federico Castillo - Minkowski bases of type cones - IPAM at UCLA
Anastasia Chavez - The valuation polytope on height two posets - IPAM at UCLA
Benjamin Braun - Symmetric Edge Polytopes: Clustering, Degree Sequences, and Graphs with Few Edges
Anton Dochtermann - Interval complexes, linear resolutions, & spaces of digraph maps - IPAM at UCLA
Galen Dorpalen-Barry - Yoshinaga Criteron & Topology of Complexified Complement of Real Arrangement
Felipe Rincon - Tropical Ideals - IPAM at UCLA
Michael DiPasquale - Generalized Hamming weights & symbolic powers of Stanley-Reisner ideal matroids
Bennet Goeckner - Vertex order shellings - IPAM at UCLA