Living cells and tissues are complex machinery, where biochemical signaling pathways, genetic networks, molecular motors, and cytoskeletal elements orchestrate to drive specific functions. Rapid advancements in microscopy, molecular biology, and genetics have improved the resolution and scale of our understanding of biological processes. Biophysical modeling techniques are often instrumental in obtaining physical insights, and provide predictive power to future experiments. My talk will use the developing fruit fly as an example model system, where the starting single cell divides into multiple cells, which take different identities to make tissues, and ultimately deform into shapes to generate functional organs. In this talk, I will specifically focus on a fluid dynamics model, which demonstrates how actively driven slender fibers (microtubules) hydrodynamically interact with each other and self-organize to generate large scale flows. Finally, I will discuss an algorithm founded on polymer-physics theories to track chromosomal structures within DNA from real-time videos. Taken together, this talk will highlight the vital role of fundamental engineering principles to make the most of cutting-edge experimental data across biological contexts – from molecules to tissues.

Soft amorphous solids are everywhere around us in various forms (colloids, gels, foams, livings cells etc.). In many of these solids, quite often, the constituent particles can change their size or shape. In this thesis, we explore the fluidization of two dimensional models of such soft deformable soft materials. The different questions that we explore are: (i) what are the mechanical properties of a system of particles with active size deformation ? (ii) what happens when a collection of flexible soft objects undergo compression ? (iii) how do glassy systems of rigid or flexible objects differ in their yielding behaviour. Using extensive numerical simulations, we provide physical insights into these different phenomena, from a microscopic perspective.

Discrete homotopy theory is a homotopy theory designed for studying graphs, detecting combinatorial (rather than topological) holes. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis. In this talk, based on joint work with Chris Kapulkin, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert?van Kampen theorem, generalizing a previous result of H. Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction. Currently, a central open problem in the field is to determine whether the cubical nerve functor, which associates a cubical Kan complex to a graph is a DK-equivalence of relative categories. If true, this would allow the import of important results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. We propose a new line of attack, by breaking it into more tractable problems comparing the homotopy theories of the respective $n$-types, for each integer $n ≥ 0$. We also solve this problem for the first nontrivial case, $n = 1$.