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Department of Mathematical Sciences, University of Liverpool
Mathematical Sciences Building, Liverpool L69 7ZL, United Kingdom
+44 (0)151 794 4001
Selected Topics in Mathematics
Speakers
7 March (Friday, 1pm)
Dr. Derek Kitson, Lancaster University

Rigidity theory for bar-joint frameworks

Abstract. A bar-joint framework in Euclidean space is a structure composed of rigid bars of fixed lengths which are joined together at their endpoints by rotational hinges. In 1970, G. Laman obtained a complete combinatorial characterisations for generically rigid finite bar-joint frameworks in the Euclidean plane, sparking a rapid development in combinatorial rigidity theory and its applications. In this talk I will survey some of this development and discuss recent results for 1) non-Euclidean norms, and, 2) infinite crystallographic frameworks. In the former case, replacing the Euclidean norm with a polyhedral or l^p norm alters the class of generically rigid graphs and leads to new combinatorial characterisations of infinitesimal rigidity. For the latter case, I will outline connections to operator theory and Bohr's theory of almost periodic functions. This includes joint work with Stephen Power and Bernd Schulze.
21 March (Friday, 1pm, room MATH211)
Dr. Alexei Lisitsa, University of Liverpool

A SAT Attack on the Erdos Discrepancy Conjecture

Abstract. In 1930s Paul Erdos conjectured that for any positive integer C in any infinite \pm 1 sequence (x_n) there exists a subsequence (x_d, x_{2d}, x_{3d},\dots, x_{kd}), for some positive integers k and d, such that \mid \sum_{i=1}^k x_{id} \mid >C. The conjecture has been referred to as one of the major open problems in combinatorial number theory and discrepancy theory. For the particular case of C=1 a human proof of the conjecture exists; for C=2 a bespoke computer program had generated sequences of length 1124 of discrepancy 2, but the status of the conjecture remained open even for such a small bound. We show that by encoding the problem into Boolean satisfiability and applying the state of the art SAT solver, one can obtain a discrepancy 2 sequence of length 1160 and a proof of the Erdos discrepancy conjecture for C=2, claiming that no discrepancy 2 sequence of length 1161, or more, exists. We also present our partial results for the case of C=3. A short introduction to discrepancy theory and SAT solvers will be given.
28 March (Friday, 1pm)
Prof. Dmitrii V Pasechnik, University of Oxford

Rational moment generating functions and polyhedra in R^d

Abstract. The problem of reconstructing a measure in R^d from a (truncated) multi-sequence of its moments has important applications, and is in general very hard to solve. We concentrate on a natural case of a measure mu with piecewise-polynomial density supported on a compact polyhedron P, and show that such problems can be solved exactly, due to existence of a natural integral transform of the measure, which is a rational function F_mu(u). The denominator of F_mu(u) is the product of powers of linear functions of the form 1-, with v belonging to certain finite set V(P). There are interesting applications of F_mu(u) to compact (not necessarily convex) polyhedra. Let I(P) be the indicator function of P. Then I(P) can be decomposed (up to a measure 0 subset) as a sum, with real coefficients, of I(D), where D runs through simplices with vertices in V(P). This can be viewed as a non-convex generalisation of triangulations of convex polytopes. On the other hand, Laplace transforms of such decompositions arise in the theory of hyperplane arrangements. Further refinements and applications will be discussed.
2 May (Friday, 1pm, Room MATH104)
Daniel Brady, University of Liverpool

Morse theory and its applications to the topology of SO(n)

Abstract. The special orthogonal group is a classical Lie group which finds applications in many areas of mathematics due to its geometrical interpretation as the group of direct isometries of R^n. Morse theory gives us a way to study the topology of manifolds by looking at how certain functions behave on them. In this talk we shall look at how to apply Morse theory to obtain topological information about SO(n), as well as looking at some approaches to the same problem using algebraic topology.
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