Equivalence relations are among the most basic concepts in mathematics. Nevertheless, it turns out that there are some pretty complicated equivalence relations out there. In this talk we will discuss different ways of classifying them - whatever that means. We will look at some (more or less) concrete examples. Finally, we will construct some fairly natural equivalence relations for which we can nevertheless prove a strong nonclassifiability result.

In this talk we will show a scattering - type result for Schrodinger maps with small Besov norm. The proof extends an earlier result of Bejenaru, Ionescu, and Kenig.

One of the fundamental problems in combinatorics involves deciding whether some given linear equation has solutions with all its variables lying in some restricted set, and if so, estimating how many such solutions there are. In this talk, we will describe some of the old and new results in this area, as well as discuss a number of unsolved problems.

The generalized Nash equilibrium problem (GNEP for short)is an extension of Nash equilibrium problem where the feasible set of each player may depend on the rivals strategies. It has many applications in areas such as economics, engineering, transportation and management sciences. In this talk, we present an exact penalty function method to reduce the GNEP into a Nash equilibrium problem. Here the penalty function is smooth, which is different from the most existing function. We also report numerical results so as to illustrate the efficiency of the proposed method.

Independent sets in hypergraphs can be encoded as 0-1 configurations on the set of vertices such that each hyperedge is adjacent to at least one 0. This model has been studied in the CS community for its large gap between efficient MCMC algorithms (previously $d <(k-1)/2$) and the conjectured onset of computational hardness ($d > O(2^{k/2})$ ), where $d$ is the largest degree of vertices and $k$ is the minimum size of hyperedges. In this talk we use a percolation approach to show that the Glauber dynamics is rapid mixing for $d < O(2^{k/2}$), closing the gap up to a multiplicative constant.

The three-dimensional spatiotemporal organization of genetic material inside the cell nucleus remains an open question in cellular biology. During the time between two cell divisions, the functional form of DNA in cells, known as chromatin, fills the cell nucleus in its uncondensed polymeric form, which allows the transcriptional machinery to access DNA. Recent in vivo imaging experiments have cast light on the existence of coherent chromatin motions inside the nucleus, in the form of large-scale correlated displacements on the scale of microns and lasting for seconds. To elucidate the mechanisms for such motions, we have developed a coarse-grained active polymer model where chromatin is represented as a confined flexible chain acted upon by active molecular motors, which perform work and thus exert dipolar forces on the system. Numerical simulations of this model that account for steric and hydrodynamic interactions as well as internal chain mechanics demonstrate the emergence of coherent motions in systems involving extensile dipoles, which are accompanied by large-scale chain reconfigurations and local nematic ordering. Comparisons with experiments show good qualitative agreement and support the hypothesis that long-ranged hydrodynamic couplings between chromatin-associated active motors are responsible for the observed coherent dynamics. The connection between our model and mechanisms proposed for self-organization of other biological systems including bacterial suspensions will also be discussed.

We consider the eigenvalue problem for octonionic $3\times3$ Hermitian
matrices (the exceptional Jordan algebra, also known as the Albert
algebra). For real eigenvalues, most of the properties expected by
analogy with the complex case still hold, provided they are
reinterpreted to take into account of the lack of commutativity and
associativity. There are nevertheless some interesting surprises along
the way, among them the existence of nonreal eigenvalues, and the fact
that the components of primitive idempotents (elements of $OP^2$, the
Cayley--Moufang plane) always associate.

Applications of these results will be briefly discussed, both to the
study of exceptional Lie groups (the Albert algebra is the minimal
representation of $e_6$) and to physics ($OP^2$ can be interpreted as
the solution space of the Dirac equation in 10 spacetime dimensions).

Geometric Recursion is a very general machinery for constructing mapping class group invariants objects associated to two dimensional surfaces. After presenting the general abstract definition we shall see how a number of constructions in low dimensional geometry and topology fits into this setting. These will include the Mirzakhani-McShane identies, mapping class group invariant closed forms on Teichmuller space (including the Weil-Petterson symplectic form) and the Goldman symplectic form on moduli spaces of flat connections for general compact simple Lie groups. If time permits we shall also discuss the process which establishes that any application of Topological Recursion can be lifted to a Geometric Recursion setting involving continuous functions on Teichmuller space, where the connection back to Topological Recursion is obtained by integration over the moduli space of curve. The work presented is joint with G. Borot and N. Orantin.