In many scenarios, especially biomedical applications, correct delineation ofcomplex fine-scaled structures such as neurons, tissues and vessels is critical fordownstream analysis. Despite the strong prediction power of deep learning methods, theydo not provide a satisfactory representation of these structures, thus creating significantbarriers in scalable annotation and downstream analysis. In this report, we tackle suchchallenges by proposing novel representations of these topological structures in a deeplearning framework. We leverage the mathematical tool from topological data analysis,i.e., persistent homology and discrete morse theory, to develop principled methods forbetter segmentation and uncertainty estimation, which will become powerful tools forscalable annotation.We focus on a few specific problems. First, we propose novel topological lossesfor fully supervised segmentation. Although deep-learning-based segmentation methodshave achieved satisfactory segmentation performance in terms of per-pixel accuracy, mostof them are still prone to structural errors, e.g., broken connections and missing connectedcomponents. We propose topological losses to teach neural network to segment withcorrect topology. The continuous-valued loss functions enforce a segmentation to have thebetter topology by penalizing topologically critical pixels/locations. Second, we focus onthe iterative setting, where uncertainty measurement of a neural network segmenter iscrucial for scalable annotation. Existing methods only learn pixel-wise featurerepresentations. We move from pixel space to structure space using the classical discreteMorse theory. We decompose an input image into the structural elements such as branchesand patches, and then learn a probabilistic model over such structural space. Our methodeffectively identifies hypothetical structures that a model is uncertain about and asks adomain expert to confirm. This will significantly improve the annotation speed. Finally,we also extend the study to topological constraints involving multiple semantic labels, andapply the proposed methods to other applications such as the detection of backdoor attacks.