Daily Papers

We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described by Kanemitsu. We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and mathbb L-equivalence. For the case of K3 surfaces, we provide alternative interpretations for the Fourier-Mukai duality in the family of K3 surfaces of degree 12 of Mukai. In all these constructions the derived equivalence lifts to an equivalence of matrix factorizations categories.

We consider generalized homogeneous roofs, i.e. quotients of simply connected, semisimple Lie groups by a parabolic subgroup, which admit two projective bundle structures. Given a general hyperplane section on such a variety, we consider the zero loci of its pushforwards along the projective bundle structures and we discuss their properties at the level of Hodge structures. In the case of the flag variety F(1,2,n) with its projections to P^{n-1} and G(2, n), we construct a derived embedding of the relevant zero loci by methods based on the study of B-brane categories in the context of a gauged linear sigma model.

Fixed a polarised variety X, we can ask if it admits Ulrich bundles and, in case, what is their minimal possible rank. In this thesis, after recalling general properties of Ulrich sheaves, we show that any finite covering of P^n that embeds as a divisor in a weighted projective space with weights (1^{n+1},m) admits Ulrich sheaves, by using matrix factorisations. Among these varieties, we focus on double coverings of with nge3. Through Hartshorne--Serre correspondence, which we review along the way, we prove that the general such X admits a rank 2 Ulrich sheaf if and only if n=3 and m=2,3,4, and characterise the zero loci of their sections. Moreover, we construct generically smooth components of the expected dimension of their moduli spaces, analyse the action of the natural involution on them and the restriction of those bundles to low degree hypersurfaces. For m=2,3, we verify the existence of slope-stable Ulrich bundles of all the possible ranks.

Gene regulatory network inference (GRNI) is a challenging problem, particularly owing to the presence of zeros in single-cell RNA sequencing data: some are biological zeros representing no gene expression, while some others are technical zeros arising from the sequencing procedure (aka dropouts), which may bias GRNI by distorting the joint distribution of the measured gene expressions. Existing approaches typically handle dropout error via imputation, which may introduce spurious relations as the true joint distribution is generally unidentifiable. To tackle this issue, we introduce a causal graphical model to characterize the dropout mechanism, namely, Causal Dropout Model. We provide a simple yet effective theoretical result: interestingly, the conditional independence (CI) relations in the data with dropouts, after deleting the samples with zero values (regardless if technical or not) for the conditioned variables, are asymptotically identical to the CI relations in the original data without dropouts. This particular test-wise deletion procedure, in which we perform CI tests on the samples without zeros for the conditioned variables, can be seamlessly integrated with existing structure learning approaches including constraint-based and greedy score-based methods, thus giving rise to a principled framework for GRNI in the presence of dropouts. We further show that the causal dropout model can be validated from data, and many existing statistical models to handle dropouts fit into our model as specific parametric instances. Empirical evaluation on synthetic, curated, and real-world experimental transcriptomic data comprehensively demonstrate the efficacy of our method.

The advent of single-cell Assay for Transposase-Accessible Chromatin using sequencing (scATAC-seq) offers an innovative perspective for deciphering regulatory mechanisms by assembling a vast repository of single-cell chromatin accessibility data. While foundation models have achieved significant success in single-cell transcriptomics, there is currently no foundation model for scATAC-seq that supports zero-shot high-quality cell identification and comprehensive multi-omics analysis simultaneously. Key challenges lie in the high dimensionality and sparsity of scATAC-seq data, as well as the lack of a standardized schema for representing open chromatin regions (OCRs). Here, we present ChromFound, a foundation model tailored for scATAC-seq. ChromFound utilizes a hybrid architecture and genome-aware tokenization to effectively capture genome-wide long contexts and regulatory signals from dynamic chromatin landscapes. Pretrained on 1.97 million cells from 30 tissues and 6 disease conditions, ChromFound demonstrates broad applicability across 6 diverse tasks. Notably, it achieves robust zero-shot performance in generating universal cell representations and exhibits excellent transferability in cell type annotation and cross-omics prediction. By uncovering enhancer-gene links undetected by existing computational methods, ChromFound offers a promising framework for understanding disease risk variants in the noncoding genome.