Abstract: We shall discuss one of the most famous problems in Combinatorics; the folklore conjecture that for each prime number p, there is a unique projective plane of order p. The focus will be on the coding theoretic approach to this problem.
Abstract: Low-dimensional topology is full of algorithmic problems whose best known running times are exponential, or doubly-exponential, or worse - and in some cases even undecidable. Nevertheless, there is a need to have software that can actually solve these problems in practice, despite the theoretical limitations. For this we use “cheap tricks” - techniques that might not always work, but when they do will give fast, correct and certified results.
Combinatorial techniques play an important and powerful role in these tricks. We will talk through some key ideas, including well-structured triangulations, graph structure theory and properties of the dual skeleton, and how these are used in practice to find new results in knot theory, 3-manifold and 4-manifold topology.
Abstract: Stackedness and tightness are attractive properties of triangulated manifolds, which Basudeb has a lot of contributions to them. These properties are defined from combinatorial and topological point of view, but they actually have a nice connection to commutative algebra theory. In this talk, I will talk how Basudeb's works on stacked and tight triangulations connect to commutative algebra theory.
Abstract: In this talk, we review the concept of triangulation and quasitoric manifolds. Then, we discuss some explicit triangulation of certain 4-dimensional quasitoric manifolds and 8-dimensional complex projective space. The results are joint work with Basudeb Datta.
∗ The slides of Wolfgang Kühnel will be presented by Biplab Basak and Nitin Singh.
Abstract: The crosscap number of a knot is the non-orientable counterpart of its genus. It is defined as the minimum of one minus the Euler characteristic of S, taken over all non-orientable surfaces S bounding the knot. Computing the crosscap number of a knot is tricky, since normal surface theory - the usual tool to prove computability of problems in 3-manifold topology, does not deliver the answer "out-of-the-box".
In this talk, I will review the strengths and weaknesses of normal surface theory, focusing on why we need to work to obtain an algorithm to compute the crosscap number. I will then explain the theorem stating that an algorithm due to Burton and Ozlen can be used to give us the answer.
This is joint with work with Jaco, Rubinstein, and Tillmann.