Abstract:
We present the results of a numerical search for periodic orbits of three equal masses moving in a plane under the influence of Newtonian gravity, with zero angular momentum. A topological method is used to classify periodic three-body orbits into families, which fall into four classes, with all three previously known families belonging to one class. The classes are defined by the orbits geometric and algebraic symmetries. In each class we present a few orbits initial conditions, 15 in all; 13 of these correspond to distinct orbits.

Abstract:
The low-energy spectrum of three particles interacting via nearly resonant two-body interactions in the Efimov regime is set by the so-called three-body parameter. We show that the three-body parameter is essentially determined by the zero-energy two-body correlation. As a result, we identify two classes of two-body interactions for which the three-body parameter has a universal value in units of their effective range. One class involves the universality of the three-body parameter recently found in ultracold atom systems. The other is relevant to short-range interactions that can be found in nuclear physics and solid-state physics.

Abstract:
The configuration space of the planar three-body problem when collisions are excluded has a rich topology which supports a large set of free homotopy classes. Most classes survive modding out by rotations. Those that survive are called the reduced free homotopy classes and have a simple description when projected onto the shape sphere. They are coded by syzygy sequences. We prove that every reduced free homotopy class, and thus every reduced syzygy sequence, is realized by a reduced periodic solution to the Newtonian planar three-body problem. The realizing solutions have nonzero angular momentum, repeatedly come very close to triple collision, and have lots of "stutters"--repeated syzygies of the same type. The heart of the proof is contained in the work by one of us on symbolic dynamics arising out of the central configurations after the triple collision is blown up using McGehee's method.

Abstract:
Ab-initio predictions of nuclei with masses up to A~100 or more is becoming possible thanks to novel advances in computations and in the formalism of many-body physics. Some of the most fundamental issues include how to deal with many-nucleon interactions, how to calculate degenerate--open shell--systems, and pursuing ab-initio approaches to reaction theory. Self-consistent Green's function (SCGF) theory is a natural approach to address these challenges. Its formalism has recently been extended to three- and many-body interactions and reformulated within the Gorkov framework to reach semi-magic open shell isotopes. These exciting developments, together with the predictive power of chiral nuclear Hamiltonians, are opening the path to understanding large portions of the nuclear chart, especially within the $sd$ and $pf$ shells. The present talk reviews the most recent advances in ab-initio nuclear structure and many-body theory that have been possible through the SCGF approach.

Abstract:
This expository note describes McGehee blow-up \cite{McGehee} in its role as one of the main tools in my recent proof with Rick Moeckel \cite{RM2} that every free homotopy class for the planar three-body problem can be realized by a periodic solution. The main novelty is my use of energy-balance to motivate the transformation of McGehee. Another novelty is an explicit description of the blown-up reduced phase space for the planar N-body problem, $N \ge 3$ as a complex vector bundle over the half-line times complex projective $N-2$-space. The half line coordinate is the size of the labelled planar N-gon whose vertices are the instantaneous positions of the N bodies and the projective space coordinatizes the shape of this N-gon body.

Abstract:
Closed classes of three-valued logic generated by symmetric funtions that equal $1$ in almost all tuples from $\{1,2\}^n$ and equal $0$ on the rest tuples are considered. Criteria for bases existence for these classes is obtained.

Abstract:
The three-body problem in one-dimension with a repulsive inverse square potential between every pair was solved by Calogero. Here, the known results of supersymmetric quantum mechanics are used to propose a number of new three-body potentials which can be solved algebraically. Analytic expressions for the eigenspectrum and the eigenfunctions are given with and without confinement.

Abstract:
The method we have applied in "A. Bernini, L. Ferrari, R. Pinzani, Enumerating permutations avoiding three Babson-Steingrimsson patterns, Ann. Comb. 9 (2005), 137--162" to count pattern avoiding permutations is adapted to words. As an application, we enumerate several classes of words simultaneously avoiding two generalized patterns of length 3.

Abstract:
Currently, the fifteen new periodic solutions of Newtonian three-body problem with equal mass were reported by \v{S}uvakov and Dmitra\v{s}inovi\'{c} (PRL, 2013) [1]. However, using a reliable numerical approach (namely the Clean Numerical Simulation, CNS) that is based on the arbitrary-order Taylor series method and data in arbitrary-digit precision, it is found that at least seven of them greatly depart from the periodic orbits after a long enough interval of time. Therefore, the reported initial conditions of at least seven of the fifteen orbits reported by \v{S}uvakov and Dmitra\v{s}inovi\'{c} [1] are not accurate enough to predict periodic orbits. Besides, it is found that these seven orbits are unstable.

Abstract:
We consider the Newtonian planar three--body problem with positive masses $m_1$, $m_2$, $m_3$. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases $ \sum m_i m_j/(\sum m_k)^2= 1/3$, $2^3/3^3$, $2/3^2$ where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in our previous papers and based of the Morales-Ramis-Ziglin approach.