La recherche

We study the Maximum Zero-Sum Partition problem (or MZSP), defined as follows: given a multiset S={a1,a2,…,an} of integers ai∈Z⁎ (where Z⁎ denotes the set of non-zero integers) such that ∑i=1nai=0, find a maximum cardinality partition {S1,S2,…,Sk} of S such that, for every 1≤i≤k, ∑aj∈Siaj=0. Solving MZSP is useful in genomics for computing evolutionary distances between pairs of species. Our contributions are a series of algorithmic results concerning MZSP, in terms of complexity, (in)approximability, with a particular focus on the fixed-parameter tractability of MZSP with respect to either (i) the size k of the solution, (ii) the number of negative (resp. positive) values in S and (iii) the largest integer in S.

We consider Riesz energy problems with radial external fields. We study the question of whether or not the equilibrium is the uniform distribution on a sphere. We develop general necessary as well as general sufficient conditions on the external field that apply to powers of the Euclidean norm as well as certain Lennard--Jones type fields. Additionally, in the former case, we completely characterize the values of the power for which dimension reduction occurs in the sense that the support of the equilibrium measure becomes a sphere. We also briefly discuss the relation between these problems and certain constrained optimization problems. Our approach involves the Frostman characterization, the Funk--Hecke formula, and the calculus of hypergeometric functions.

In this thesis, we study the problem of extending a Morse function on a closed manifold to a Lefschetz fibration on the cotangent bundle of said manifold, as well as certain Floer-theoretic questions related to this fibration. In the first chapter, we present the extension theorem due to E. Giroux, we extend it to the real-analytic setting and prove a theorem in the local model on the result of parallel transport along the unit semi-circle. In chapters 2 and 3, we re-examine a conjecture of P. Seidel in this context, which establishes a relation between the flow category of the Morse function and the directed Donaldson-Fukaya category of the Lefschetz fibration, particularly in dimension 2, resp. 3. In the last chapter, we give a negative answer to a natural question of E. Giroux, on a relation between the Heegaard-Floer homology of a 3-manifold and the Lefschetz fibration which extends a Morse function which induces a Heegaard diagram.