University of Waterloo

Tue 10 Sep 2019, 4:00pm
Discrete Math Seminar
ESB 4127

SAT Solving with Computer Algebra: A Powerful Combinatorial Search Method

ESB 4127
Tue 10 Sep 2019, 4:00pm5:00pm
Abstract
Solvers for the Boolean satisfiability problem have been increasingly used to solve hard problems from many fields and now routinely solve problems with millions of variables. Combinatorial problems are a natural target, as SAT solvers contain excellent combinatorial search algorithms. Despite this, SAT solvers can fail on small problems, for example when properties of the problem cannot be concisely expressed in Boolean logic. We describe a new combinatorial search method that allows properties to be specified using a computer algebra system (CAS), thereby combining the expressiveness of a CAS with the search power of SAT solvers.
In this talk we describe how our SAT+CAS system MathCheck has verified, partially verified, or found new counterexamples to conjectures from design theory, graph theory, and number theory. In particular, we have classified Williamson matrices up to order 70, quaternary Golay sequence pairs up to length 28, best matrices up to order 7, verified the Ruskey–Savage and Norine conjectures up to larger bounds than had previously been verified, found the smallest counterexample of the Williamson conjecture, and found three new counterexamples to a conjecture on good matrices. Currently we are using the system to verify the nonexistence of projective planes of order 10.
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University of Washington

Tue 17 Sep 2019, 4:00pm
Discrete Math Seminar
ESB 4127

Resolving Stanley’s conjecture on kfold acyclic complexes

ESB 4127
Tue 17 Sep 2019, 4:00pm5:00pm
Abstract
In 1993, Stanley showed that if a simplicial complex is acyclic over some field, then its face poset can be decomposed into disjoint rank 1 boolean intervals whose minimal faces together form a subcomplex. Stanley further conjectured that complexes with a higher notion of acyclicity could be decomposed in a similar way using boolean intervals of higher rank. We provide an explicit counterexample to this conjecture. We also prove a special case of the conjecture, and show that a weaker decomposition into boolean trees always exists. This is joint work with Joseph Doolittle.
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Mathematics, UBC

Tue 24 Sep 2019, 4:00pm
Discrete Math Seminar
ESB 4127

Normal lattice supercharacter theories and Hopf structures

ESB 4127
Tue 24 Sep 2019, 4:00pm5:00pm
Abstract
The concept of Hopf algebras originated from the theory of algebraic groups and algebraic topology in the mid 20th century. Hopf structures have numerous applications in many other mathematical branches, and now it is a familiar concept in representation theory as the class functions and superclass functions of some tower of groups have Hopf structures. In these Hopf structures, representation theoretic functors give the product and coproduct. In this talk, we give a brief introduction to normal lattice supercharacter theories, and then we construct a Hopf structure by using these supercharacter theories.
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