Max Planck Institute for Mathematics

Short tandem repeats (STRs) are low-entropy regions in the genome, consisting of a short (1-6 bp) unit that is consecutively repeated multiple times. They are known for high mutational instability, due to so-called stutter-mutations, in which the number of units in the run increases or descreases. In particular, STRs with repeat unit length of 1-2 bp are prone to mutate even within several cell divisions. The extremely rapid accumulation of variation makes them interesting phylogenetic markers for retrospective single-cell lineage reconstruction. Here we model their mutational dynamics at the level of individual repeat unit type and then aggregate length variations over many STR loci with the aim of obtaining a very fast ``molecular clock''. We calibrate our model based on several datasets with known lineage structure prepared from cultured cells. We find that the mutational dynamics of STRs are reasonably consistent for a given cell line, but vary among different ones. This suggests that the dynamics are not entirely explained by mutations in caretaker genes, rather, various other factors play a role -- possibly tissue origin and differentiation state. Further data and research is necessary to asses their relative effects.

There are large health inequalities between neighborhoods in many cities of the world. This paper studies individuals' sorting based on health amenities and exposes an important connection between health preferences and the housing market. I estimate a neighborhood choice model using geolocated data from a health survey in New York City and neighborhood characteristics from different data sources. I find that individuals with higher preferences for being healthy are more likely to choose neighborhoods close to health amenities such as parks, and this sorting explains a large part of observed geographical inequalities in health behaviors. Moreover, the model predicts that a higher demand for health amenities capitalizes into house prices and displaces poor individuals to neighborhoods further from these amenities. This implies that health awareness campaigns might actually worsen the inequality in access to public health amenities.

Abstract In this paper, we study the oriented swap process on the positive integers and its asymptotic properties. Our results extend a theorem by Angel, Holroyd, and Romik regarding the trajectories of particles in the finite oriented swap process. Furthermore, we study the evolution of the type of a particle at the leftmost position over time. Our approach relies on a relationship between multi-species particle systems and Hecke algebras, complemented by a detailed asymptotic analysis.

We investigate the geometry of a family of log-linear statistical models called quasi-independence models. The toric fiber product is useful for understanding the geometry of parameter inference in these models because the maximum likelihood degree is multiplicative under the TFP. We define the coordinate toric fiber product, or cTFP, and give necessary and sufficient conditions under which a quasi-independence model is a cTFP of lower-order models. We show that the vanishing ideal of every 2-way quasi-independence model with ML-degree 1 can be realized as an iterated toric fiber product of linear ideals. We also classify which Lawrence lifts of 2-way quasi-independence models are cTFPs and give a necessary condition under which a $k$-way model has ML-degree 1 using its facial submodels.

A bstract We present a formulation of the three- and four-point amplitudes on the Coulomb branch of $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 SYM as integrals over the symplectic Grassmannian. We demonstrate that their kinematic spaces are equivalent to symplectic Grassmannians SpGr( n , 2 n ). For the three-point case, we express the amplitude as an integral over the symplectic Grassmannian in a specific little group frame. In the four-point case, we show that the integral yields the amplitude up to a known kinematic factor. Building on the four-dimensional analysis, we also express the six-dimensional $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = (1, 1) SYM amplitude in terms of four-dimensional variables in a form that makes its symplectic Grassmannian structure manifest.

We show that for every genus g ≥ 0 , there exist quasipositive knots K 0 g and K 1 g such that there is a cobordism of genus g = | g 4 ( K 1 g ) − g 4 ( K 0 g ) | between K 0 g and K 1 g , but there is no ribbon cobordism of genus g in either direction and thus no complex cobordism between these two knots. This gives a negative answer to a question posed by Feller in 2016. Nous montrons que pour chaque genre g ≥ 0 , il existe des nœuds quasi-positifs K 0 g et K 1 g tels qu'il y ait un cobordisme de genre g = | g 4 ( K 1 g ) − g 4 ( K 0 g ) | entre K 0 g et K 1 g , mais qu'il n'existe aucun cobordisme ruban de genre g dans aucune des deux directions. Par conséquent, il n'y a pas de cobordisme complexe entre ces deux nœuds. Cela donne une réponse négative à une question posée par Feller en 2016.

Abstract In this paper we are interested in the isotopy classes of symplectic log Calabi–Yau divisors in a fixed symplectic rational surface. We give several equivalent definitions and prove the stability, finiteness and rigidity results. Motivated by the problem of counting toric actions, we obtain a general counting formula of symplectic log Calabi–Yau divisors in a restrictive region of c 1 -nef cone. A detailed count in the case of 2- and 3-point blow-ups of complex projective space for all symplectic forms is also given. In our framework the complexity of the combinatorics of analyzing Delzant polygons is reduced to the arrangement of homology classes. Then we study its relation with almost toric fibrations. We raise the problem of realizing all symplectic log Calabi–Yau divisors by some almost toric fibrations and verify it together with another conjecture of Symington in a special region.

Abstract Let p and q be two distinct fixed prime numbers and $$(n_i)_{i\ge 0}$$ the sequence of consecutive integers of the form $$p^a\cdot q^b$$ with $$a,b\ge 0$$ . Tijdeman gave a lower bound (1973) and an upper bound (1974) for the gap size $$n_{i+1}-n_i$$ , with each bound containing an unspecified exponent and implicit constant. We will explicitly bound these four quantities. Earlier Langevin (1976) gave weaker estimates for (only) the exponents. Given a real number $$\alpha &gt;1$$ , there exists a smallest number m such that for every $$n\ge m$$ , there exists an integer $$n_i$$ in $$[n,n\alpha )$$ . Our effective version of Tijdeman’s result immediately implies an upper bound for m , which using the Koksma–Erdős–Turan inequality we will improve on. We present a fast algorithm to determine m when $$\max \{p,q\}$$ is not too large and demonstrate it with numerical material. In an appendix we explain, given $$n_i$$ , how to efficiently determine both $$n_{i-1}$$ and $$n_{i+1}$$ , something closely related to work of Bérczes, Dujella and Hajdu.

Abstract This work concerns representations of a finite flat group scheme G defined over a noetherian commutative ring R . The focus is on lattices, namely, finitely generated G -modules that are projective as R -modules, and on the full subcategory of all G -modules projective over R generated by the lattices. The stable category of such G -modules is a rigidly-compactly generated, tensor triangulated category. The main result is that this stable category is stratified and costratified by the natural action of the cohomology ring of G . Applications include formulas for computing the support and cosupport of tensor products and the module of homomorphisms, and a classification of the thick ideals in the stable category of lattices.