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Deep convolutional neural networks (CNNs) have emerged as a new paradigm for Mammogram diagnosis. Contemporary CNN-based computer-aided-diagnosis (CAD) for breast cancer directly extract latent features from input mammogram image and ignore the importance of morphological features. In this paper, we introduce a novel deep learning framework for mammogram image processing, which computes mass segmentation and simultaneously predict diagnosis results. Specifically, our method is constructed in a dual-path architecture that solves the mapping in a dual-problem manner, with an additional consideration of important shape and boundary knowledge. One path called the Locality Preserving Learner (LPL), is devoted to hierarchically extracting and exploiting intrinsic features of the input. Whereas the other path, called the Conditional Graph Learner (CGL) focuses on generating geometrical features via modeling pixel-wise image to mask correlations. By integrating the two learners, both the semantics and structure are well preserved and the component learning paths in return complement each other, contributing an improvement to the mass segmentation and cancer classification problem at the same time. We evaluated our method on two most used public mammography datasets, DDSM and INbreast. Experimental results show that DualCoreNet achieves the best mammography segmentation and classification simultaneously, outperforming recent state-of-the-art models.