**Thesis and internships**

This page has a partial list of proposed internships and thesis:

**Internship and PhD projects proposed by: A. Kashani-Poor (LPTENS)**

String theory in its most common incarnations lives in 10 dimensions. To reduce this number down to the 4 dimensions that we observe (a process called compactification), the geometry of the 6 supplementary dimensions must be specified. 4d physics depends sensitively on the choice of this geometry, giving rise to a beautiful interplay between physics and mathematics.

The research projects I propose are centered around compactification studies, ranging from flux compactifications with a distinctly differential geometric flavor to questions in topological string theory with an emphasis on algebraic geometry methods.

**Internship and PhD topics proposed by : S. Rychkov (LPTENS)**

See here

**Stage and PhD projects in Theoretical Biophysics proposed by : A. Walczak (LPTENS)**

These theoretical biophysics projects are aimed understanding the recognition of pathogens by the receptors of the B- and T-cells of the immune systems using methods of nonequilibrium statistical mechanics. B- and T-cells have receptors on their surface which bind pathogens (or parts of pathogen proteins) and based on this interaction, either "recognize" them as proteins that are natural to the organism ("self") or foreign. The foreign proteins trigger an immune response, aimed at eliminating the pathogen from the organism. If a receptor recognizes a pathogen it proliferates, if it doesn’t recognize anything for a long time - it dies. The repertoire is a set of all the receptors of a given type (B- or T-cells) of a given organism. The projects aim to understand the composition of the repertoire, the distribution of the diversity of the receptors that make up the repertoire, how the repertoire is shaped and maintained and how degenerate are the receptors.

Some of these projects are more theoretical and some are closer to data. All project will involve using methods of nonequilibrium statistical mechanics: probabilistic modeling, stochastic equations, nonlinear dynamics. The data (mainly sequence data) comes from two main experimental collaborators. The data analysis projects will involve learning how to work with this kind of data.

**PhD topics proposed by : K. Wiese (LPTENS)**

Topic 1 : Unzipping of RNA

One of the most exciting experimental developments at the frontier between physics and biology is our possiblity to attach the ends of an RNA or DNA molecule to beads, trap the beads in an optical tweezer, pull on the beads, and measure the ensuing forces as a function of bead-position (see figure). This force in the pico-Newton range contains a wealth of information on the biological system. It is also an exciting testing ground for theoretical concepts. In this internship, we propose to study the effects of thermal fluctuations onto the sawtooth like unzipping curves. Are these thermal effects negligible, or do they even change the large-scale physics?

Topic 2 : Extreme-Value Statistics for Fractional Brownian Motion

Brownian motion is a stochastic process which is Gaussian, scale invariant, translationally invariant, and Markovian, i.e. has independent increments. Apart from simple expectations, one is particularly interested in extremal properties: Does the process exceed a curtain threshold, and if yes, when? How long is the process prositive? This is important to correctly model the necessary capacity for dams, estimate the time one has to heat, etc. While these questions can often be answered for Brownian motion, which is Markovian (a property essential in the solution), many processes in nature are not Markovian, i.e. have memory. Fractional Brownian motion (see figure) is such a generalization, indexed by the Hurst exponent H, and reducing for H = 1/2 to Brownian motion. For H < 1/2, the process is anticorrelated and rougher (red curves in the figure), while for H > 1/2 it is positively correlated. We have recently been able to construct a systematic perturbative expansion around H = 1/2. The stage proposes to study the limit of Hurst-index H = 1, and to try to build a systematic perturbation expansion in that limit.