I’ve made the title of this page a lie, but I’ll continue to update this page with thoughts on the literature I read.

Currently reading:
Tymoczko, Julianna. The geometry and combinatorics of Springer fibers. arXiv preprint arXiv:1606.02760 (2016).
I’m trying to learn about Springer fibers and Schubert varieties, and this paper gives a good overview of the combinatorics underlying their geometric structure. There appear to be some interesting analogies between the combinatorics defining some topological stratifications of Springer fibers and moduli spaces of curves, which might upgrade to some analogies between their S_n-equivariant cohomology? It’s quite lovely that the top-dimensional cohomology of Springer fibers recovers the irreducible representations of S_n, and I’d like to learn the method to prove this.

Lim, Bryan, and Stefan Zohren. Time-series forecasting with deep learning: a survey. Philosophical Transactions of the Royal Society A 379, no. 2194 (2021): 20200209.
I’ve been thinking lately about parametrizing time series in some way; obviously, there are the classical curve-fitting approaches, but I’d like a way to do so without having to manually select a type of curve to fit. Deep learning is an easy default, so I’m learning about the existing models. Evidently, this survey is mostly about forecasting, while I’m interested in classification (and using auxiliary data), so I’ll have to extend anything I find here.
Salinas, David, Valentin Flunkert, Jan Gasthaus, and Tim Januschowski. DeepAR: Probabilistic forecasting with autoregressive recurrent networks. International Journal of Forecasting 36, no. 3 (2020): 1181-1191.
This is pretty much what I was looking for. Is clustering the learned latent representation a good idea? How easy would it be to extract relevances of various TF/genome contacts at given time points?

Schiebinger, Geoffrey, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshua Gould et al. Optimal-transport analysis of single-cell gene expression identifies developmental trajectories in reprogramming. Cell 176, no. 4 (2019): 928-943.
I haven’t totally digested this paper yet, but optimal transport does seem to be a fairly natural approach to the task of analyzing single-cell trajectories. I’m trying to asses if it might be useful for formalizing changes in the epigenomics of various genomic loci rather than cells.

September 16th, 2022
Lee, DS., Luo, C., Zhou, J. et al. Simultaneous profiling of 3D genome structure and DNA methylation in single human cells. Nat Methods 16, 999–1006 (2019). https://doi.org/10.1038/s41592-019-0547-z
I was curious precisely how joint profiling increases information in this particular context. It seems that there are two primary advantages: single-cell methylation data allows for a relatively fine partition of cell types which can then be analyzed to determine cell-type specific conformations, and it turns out that certain DNA-binding proteins (in this paper, CTCF) are heavily influenced by DNA methylation so that methylation appears to have some causal relationship with loop formation and conformation more broadly.

Proudfoot, Nicholas. The equivariant Orlik–Solomon algebra. Journal of Algebra 305, no. 2 (2006): 1186-1196.

September 15th, 2022
Boninsegna, L., Yildirim, A., Polles, G. et al. Integrative genome modeling platform reveals essentiality of rare contact events in 3D genome organizations. Nat Methods 19, 938–949 (2022). https://doi.org/10.1038/s41592-022-01527-x
Finally got around to reading this paper, but sadly it was actually not the most interesting. I think the three big takeaways are: in general, incorporating more data as additional constraints in a maximum likelihood estimation problem improves structure predictions; highly expressed genes have low cell-to-cell variability in microenvironment; “rare” contacts are vital for correct structure prediction. I suppose this opens the question of: in single-cell data, where noise dominates, is it still possible to obtain accurate predictions? I’ll study the numerical solution to the MLE problem in more detail later.

Matherne, Jacob P., Dane Miyata, Nicholas Proudfoot, and Eric Ramos. Equivariant log concavity and representation stability. arXiv preprint arXiv:2104.00715 (2021).

September 3rd-September 14th, 2022
Break due to moving and settling in.

September 2nd, 2022
Beagan, Jonathan A., et al. Local genome topology can exhibit an incompletely rewired 3D-folding state during somatic cell reprogramming. Cell stem cell 18.5 (2016): 611-624.
I think I’m starting to arrive at the realization that genome conformation is actually fairly well understood (from a certain perspective). Granted this paper isn’t the “highest novelty”, but I was able to predict most of their results (OSKM genes experience more enhancer contacts, genes exhibiting this pattern are associated with plutipotency, etc) based on the trends I’ve been seeing in previous papers. Maybe the interesting questions that remain are things like incomplete rewiring and the actual principles of folding, so I should shift my focus slightly.

Madeline Brandt. Martin Ulirsch. Divisorial Motivic Zeta Functions for Marked Stable Curves. Michigan Math. J. 71 (2) 271 – 282, May 2022. https://doi.org/10.1307/mmj/20195792
Didn’t finish this one today. I’m realizing that one math paper a day might be a little overly ambitious.

September 1st, 2022
Takei, Y., Yun, J., Zheng, S. et al. Integrated spatial genomics reveals global architecture of single nuclei. Nature 590, 344–350 (2021). https://doi.org/10.1038/s41586-020-03126-2
I’ve never really worked with imaging technologies before, so I can’t say I understand this paper perfectly. Nonetheless, it’s given me a sincere appreciation for the power of imaging; its power to resolve “fixed loci” (as seen in Figure 2) is quite remarkable. The takeaway that nuclear “anchors” are the real invariants of genome structure is enlightening. I’m really interested in how the questions asked differ between sequencing and imaging modalities: sequencing-based papers are mostly curious about loci-loci contacts, while this imaging-based paper has the flexibility to ask about interactions with other nuclear structures (something to think about with joint profiling sequencing assays?) This is a paper to return to again and again.

Clader, Emily, Dante Luber, and Kyla Quillin. Boundary complexes of moduli spaces of curves in higher genus. Proceedings of the American Mathematical Society 150.05 (2022): 1837-1848.
This paper involves some fairly intricate combinatorics relating to the structure of various dual graphs of boundary strata in M_{g,n}-bar. The upshot is that the “nice” behavior of pairwise non-empty intersections implying non-empty total intersection of divisors in M_{0,n}-bar holds only in very small genus. I think a similar argument would extend this result to Hassett heavy-light spaces..?

August 31st, 2022
Stadhouders, R., Vidal, E., Serra, F. et al. Transcription factors orchestrate dynamic interplay between genome topology and gene regulation during cell reprogramming. Nat Genet 50, 238–249 (2018). https://doi.org/10.1038/s41588-017-0030-7
This paper makes the observation that changes in gene expression should be (and often are) consistently preceded by changes in genome conformation, both at the compartment and TAD level (e.g. A/B compartment switching, formation of SOX2-SE subdomains). The overall suggestion is that genome conformation is therefore an important consideration in the dynamics of reprogramming. It also mentions a tendency for the PSC genome to be highly plastic in shape. Why is this?

Ramadas, Rohini, and Silversmith, Rob. Two-dimensional cycle classes on \overline{\mathcal{M}_{0,n}}. Mathematische Zeitschrift (2022).
Still digesting this one. The argument is mostly combinatorial on stable marked graphs, so the translation back to curves requires some thought. The filtration on cohomology is really interesting! Definitely will return to better understand this.

August 30th, 2022
Winick-Ng, W., Kukalev, A., Harabula, I. et al. Cell-type specialization is encoded by specific chromatin topologies. Nature 599, 684–691 (2021). https://doi.org/10.1038/s41586-021-04081-2
This paper discusses gene melting (for long, multiple isoform neuron genes) and contact frequency differences in neurons relative to mES. I think this supports the idea of using structure to predict function of genomic loci, although there would have to be some sort of relative perspective. They also consider regions of differential contact frequency, which they find to be enriched for certain TF motifs. There’s a new method for probing genome conformation (immunoGAM).

Aharoni, Ron, Eli Berger, and Roy Meshulam. Eigenvalues and homology of flag complexes and vector representations of graphs. Geometric & Functional Analysis GAFA 15.3 (2005): 555-566.
This paper establishes a vanishing condition for the homology of a flag complex based on its Laplacian. It seems like it could guarantee some triviality of a shifted flag complex’s homology.
Update: this paper by Meshulam establishes a simpler (but less powerful) vanishing condition based directly on the connectivity of the graph. It might be more useful.