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Session 4 of 4 Please be aware of the change of venue.

In this talk, we will discuss the problem of determining the maximum number of edges in an n-vertex k-critical graph. A graph is considered k-critical if its chromatic number is k, but any proper subgraph has a chromatic number less than k. The problem remains open for any integer k ≥ 4. Our presentation will showcase an improvement on previous results achieved by employing a combination of extremal graph theory and structural analysis. We will introduce a key lemma, which may be of independent interest, as it sheds light on the partial structure of dense k-critical graphs. This is joint work with Cong Luo and Jie Ma.

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Session 3 of 4

Fast and accurate predictions for complex physical dynamics are a big challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in the real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. In this study, we propose HyperDeepONet, which uses the expressive power of the hypernetwork to enable learning of a complex operator with smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a special case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet was successfully applied to various operator learning problems using low computational resources compared to other benchmarks.

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Session 2 of 4

Small regulatory RNA molecules such as microRNA modulate gene expression through inhibiting the translation of messenger RNA (mRNA). Such post-transcriptional regulation has been recently hypothesized to reduce the stochastic variability of gene expression around average levels. Here we quantify noise in stochastic gene expression models with and without such regulation. Our results suggest that silencing mRNA post-transcriptionally will always increase rather than decrease gene expression noise when the silencing of mRNA also increases its degradation as is expected for microRNA interactions with mRNA. In that regime we also find that silencing mRNA generally reduces the fidelity of signal transmission from deterministically varying upstream factors to protein levels. These findings suggest that microRNA binding to mRNA does not generically confer precision to protein expression

Many questions in everyday life as well as in research are causal in nature: How would the climate change if we lower train prices or will my headache go away if I take an aspirin? Inherently, such questions need to specify the causal variables relevant to the question and their interactions. However, existing algorithms for learning causal graphs from data are often not scaling well both with the number of variables or the number of observations. This talk will provide a brief introduction to causal structure learning, recent efforts in using continuous optimization to learn causal graphs at scale and systematic approaches for causal experimental design at scale.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

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Session 1 of 4

The standard approach to problem-solving in physics consists of identifying state variables of the system, setting differential equations governing the state evolution, and solving the obtained. The behavior of the system for different values of parameters can be examined only as a fourth step. On the contrary, the modern approach to studying dynamical systems relies on Morphological/Topological analysis which alleviates the necessity for the explicit solution of differential equations. The stability analysis of the parabolic swing will demonstrate the merit of such an approach. It will be shown how to construct a qualitatively correct model of system dynamics that is surprisingly quantitatively correct as well. The sudden (catastrophic) change in the swing’s stability, caused by a slight change in the critical value of system parameters, will be linked to the drastic topological change of the corresponding phase-space portraits. It will be shown that for a system’s parameters close to critical ones, the system’s behavior is identical to a specific simple universal prototype given by catastrophe theory. A short survey of the simplest elementary catastrophes will be given that represents the basis for applying catastrophe theory in other fields of science.

The seasonal outbreaks of influenza infection cause globally respiratory illness, or even death in all age groups. Given early-warning signals preceding the influenza outbreak, timely intervention such as vaccination and isolation management effectively decrease the morbidity. However, it is usually a difficult task to achieve the real-time prediction of influenza outbreak due to its complexity intertwining both biological systems and social systems. By exploring rich dynamical and high-dimensional information, our dynamic network marker/biomarker (DNM/DNB) method opens a new way to identify the tipping point prior to the catastrophic transition into an influenza pandemics. In order to detect the early-warning signals before the influenza outbreak by applying DNM method, the historical information of clinic hospitalization caused by influenza infection between years 2009 and 2016 were extracted and assembled from public records of Tokyo and Hokkaido, Japan. The early-warning signal, with an average of 4-week window lead prior to each seasonal outbreak of influenza, was provided by DNM-based on the hospitalization records, providing an opportunity to apply proactive strategies to prevent or delay the onset of influenza outbreak. Moreover, the study on the dynamical changes of hospitalization in local district networks unveils the influenza transmission dynamics or landscape in network level.

Multi-omics technologies, and in particular those with single-cell and spatial resolution, provide unique opportunities to study the deregulation of intra- and inter-cellular signaling processes in disease. I will present recent methods and applications from our group toward this aim, focusing on computational approaches that combine data with biological knowledge within statistical and machine learning methods. This combination allows us to increase both the statistical power of our analyses and the mechanistic interpretability of the results. These approaches allow us to identify key processes, that can be in turn studied in detailed with dynamic mechanistic models. I will then present how cell-specific logic models, trained with measurements upon perturbations, can provides new biomarkers and treatment opportunities. Finally, I will show how, using novel microfluidics-based technologies, this approach can also be applied directly to biopsies, allowing to build mechanistic models for individual cancer patients, and use these models to prose new therapies.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

We give a summary on the work of the last months related to Frankl's Union-Closed conjecture and its offsprings. The initial conjecture is stated as a theorem in extremal set theory; when a family F is union-closed (the union of sets of F is itself a set of $\mathcal F$), then $\mathcal F$ contains an (abundant) element that belongs to at least half of the sets. Meanwhile, there are many related versions of the conjecture due to Frankl. For example, graph theorists may prefer the equivalent statement that every bipartite graph has a vertex that belongs to no more than half of the maximal independent sets. While even proving the ε-Union-Closed Sets Conjecture was out of reach, Poonen and Cui & Hu conjectured already stronger forms. At the end of last year, progress was made on many of these conjectures. Gilmer proved the ε-Union-Closed Sets Conjecture using an elegant entropy-based method which was sharpened by many others. Despite a sharp approximate form of the union-closed conjecture as stated by Chase and Lovett, a further improvement was possible. In a different direction, Kabela, Polak and Teska constructed union-closed family sets with large sets and few abundant elements. This talk will keep the audience up-to-date and give them insight in the main ideas. People who would like more details, can join the Ecopro-reading session on the 7th of March (10 o'clock, B332) as well. Here we go deeper in the core of the proofs and discuss possible directions for the full resolution.

The morphological method – based on the topology and singularity theory and originally developed for the analysis of the scattering experiments – was extended to be applicable for the analysis of biological data. The usefulness of the topological viewpoint was demonstrated by quantification of the changes of collagen fiber straightness in the human colon mucosa (healthy mucosa, colorectal cancer, and uninvolved mucosa far from cancer). This has been done by modeling the distribution of collagen segment angles by the polymorphic beta-distribution. Its shapes were classified according to the number and type of critical points. We found that biologically relevant shapes could be classified as shapes without any preferable orientation (i.e. shapes without local extrema), transitional forms (i.e. forms with one broad local maximum), and highly oriented forms (i.e. forms with two minima at both ends and one very narrow maximum between them). Thus, changes in the fiber organization were linked to the metamorphoses of the beta-distribution forms. The obtained classification was used to define a new, shape-aware/based, measure of the collagen straightness, which revealed a slight, and moderate increase of the straightness in mucosa samples taken 20 cm and 10 cm away from the tumor. The largest increase of collagen straightness was found in samples of cancer tissue. Samples of the healthy individuals have a uniform distribution of beta-distribution forms. We found that this distribution has the maximal information entropy. At 20 cm and 10 cm away from cancer, the transition forms redistribute into unoriented and highly oriented forms. Closer to cancer the number of unoriented forms decreases rapidly leaving only highly oriented forms present in the samples of the cancer tissue, whose distribution has minimal information entropy. The polarization of the distribution was followed by a significant increase in the number of quasi-symmetrical forms in samples 20 cm away from cancer which decreases closer to cancer. This work shows that the evolution of the distribution of the beta-distribution forms – an abstract construction of the mind – follows the familiar laws of statistical mechanics. Additionally, the polarization of the beta-distribution forms together with the described change in the number of quasi-symmetrical forms, clearly visible in the parametric space of the beta-distribution and very difficult to notice in the observable space, can be a useful indicator of the early stages in the development of colorectal cancer.

Cooperation means that one individual pays a cost for another to receive a benefit. Cooperation can be at variance with natural selection. Why should you help competitors? Yet cooperation is abundant in nature and is important component of evolutionary innovation. Cooperation can be seen as the master architect of evolution and as the third fundamental principle of evolution beside mutation and selection. I will present five mechanisms for the evolution of cooperation: direct reciprocity, indirect reciprocity, spatial selection, group selection and kin selection. Global cooperation and the cooperation with future generations is necessary to ensure the survival of our species. Further reading: Nowak MA (2006). Evolutionary Dynamics. Harvard University Press Nowak MA & Highfield R (2011) SuperCooperators. Simon & Schuster. Hauser OP, Rand DG, Peysakhovich A & Nowak MA (2014). Cooperating with the future. Nature 511: 220-223 Hilbe C, Šimsa Š, Chatterjee K & Nowak MA (2018). Evolution of cooperation in stochastic games. Nature 559: 246-249 Hauser OP, Hilbe C, Chatterjee K & Nowak MA (2019). Social dilemmas among unequals. Nature 572: 524-527

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

A key goal of synthetic biology is to establish functional biochemical modules with network-independent properties. Antithetic integral feedback (AIF) is a recently developed control module in which two control species perfectly annihilate each other’s biological activity. The AIF module confers robust perfect adaptation to the steady-state average level of a controlled intracellular component when subjected to sustained perturbations. Recent work has suggested that such robustness comes at the unavoidable price of increased stochastic fluctuations around average levels. We present theoretical results that support and quantify this trade-off for the commonly analyzed AIF variant in the idealized limit with perfect annihilation. However, we also show that this trade-off is a singular limit of the control module: Even minute deviations from perfect adaptation allow systems to achieve effective noise suppression as long as cells can pay the corresponding energetic cost. We further show that a variant of the AIF control module can achieve significant noise suppression even in the idealized limit with perfect adaptation. This atypical configuration may thus be preferable in synthetic biology applications.

Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal of the planar disjoint paths problem is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. This problem has been studied extensively due to its numerous applications such as VLSI layout and circuit routing. However, this problem is NP-complete even for grid graphs. This motivates the study of this problem from the viewpoint of parameterized algorithms. In this talk, I will present a $2^{O(k^2)}n$-time algorithm for the planar disjoint paths problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020]. This is joint work with Kyungjin Cho and Seunghyeok Oh.

1. The “temporal information code” of insulin action: a bottom-up approach One of the essential elements of signaling networks is to encode information from a wide variety of inputs into a limited set of molecules. We have proposed a “temporal information code” that regulates a variety of physiological functions by encoding input information in temporal patterns of molecular activity, and based on this concept, we are analyzing biological homeostasis by insulin signaling. Taking blood insulin as an example, we will explain how the temporal information of blood insulin is selectively decoded by downstream networks. 2. Transomics of insulin action: a top-down approach In order to obtain a complete picture of insulin action, we performed transomics measurements integrating metabolomics and transcriptomics, and found that metabolism is regulated by allosteric regulation in the liver of normal mice and by compensatory gene expression in the liver of obese mice. (Top-down approach). I will talk about approach the principle of homeostasis of living organisms by temporal patterns, using the analysis of systems biology of insulin action using two different approaches.

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ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Cellular dynamics and emerging biological function are governed by patterns of gene expression arising from networks of interacting genes. Inferring these interactions from data is a notoriously difficult inverse problem that is central to systems biology. The majority of existing network inference methods work at the population level and construct a static representations of gene regulatory networks; they do not naturally allow for inference of differential regulation across a heterogeneous cell population. Building upon recent dynamical inference methods that model single cell dynamics using Markov processes, we propose locaTE, an information-theoretic approach which employs a localised transfer entropy to infer cell-specific, causal gene regulatory networks. LocaTE uses high-resolution estimates of dynamics and geometry of the cellular gene expression manifold to inform inference of regulatory interactions. We find that this approach is generally superior to using static inference methods, often by a significant margin. We demonstrate that factor analysis can give detailed insights into the inferred cell-specific GRNs. In application to two experimental datasets, we recover key transcription factors and regulatory interactions that drive mouse primitive endoderm formation and pancreatic development. For both simulated and experimental data, locaTE provides a powerful, efficient and scalable network inference method that allows us to distil cell-specific networks from single cell data.

Let $X$ be a 2-dimensional simplicial complex. Denote by $\text{ex}_{\hom}(n,X)$ the maximum number of 2-simplices in an $n$-vertex simplicial complex that has no sub-simplicial complex homeomorphic to $X$. The asymptotics of $\text{ex}_{\hom}(n,X)$ have recently been determined for all surfaces $X$. I will discuss these results, as well as ongoing work bounding $\text{ex}_{\hom}(n,X)$ for arbitrary 2-dimensional simplicial complexes $X$.

We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlström proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlström, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain. We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomassé, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendez, Simon, Thomassé, Toruńczyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the "middle" box in the grid minor can be proclaimed irrelevant — not contained in the sought solution — and thus reduced.

In the context of science, the well-known adage “a picture is worth a thousand words” might well be “a model is worth a thousand datasets.” In this manuscript we introduce the SciML software ecosystem as a tool for mixing the information of physical laws and scientific models with data-driven machine learning approaches. We describe a mathematical object, which we denote universal differential equations (UDEs), as the unifying framework connecting the ecosystem. We show how a wide variety of applications, from automatically discovering biological mechanisms to solving high-dimensional Hamilton-Jacobi-Bellman equations, can be phrased and efficiently handled through the UDE formalism and its tooling. We demonstrate the generality of the software tooling to handle stochasticity, delays, and implicit constraints. This funnels the wide variety of SciML applications into a core set of training mechanisms which are highly optimized, stabilized for stiff equations, and compatible with distributed parallelism and GPU accelerators.

Hadwiger's famous coloring conjecture states that every t-chromatic graph contains a $K_t$-minor. Holroyd [Bull. London Math. Soc. 29, (1997), pp. 139-144] conjectured the following strengthening of Hadwiger's conjecture: If G is a t-chromatic graph and S⊆V(G) takes all colors in every t-coloring of G, then G contains a $K_t$-minor rooted at S. We prove this conjecture in the first open case of t=4. Notably, our result also directly implies a stronger version of Hadwiger's conjecture for 5-chromatic graphs as follows: Every 5-chromatic graph contains a $K_5$-minor with a singleton branch-set. In fact, in a 5-vertex-critical graph we may specify the singleton branch-set to be any vertex of the graph. Joint work with Anders Martinsson (ETH).

Mapping individual differences in behavior is fundamental to personalized neuroscience, but quantifying complex behavior in real world settings remains a challenge. While mobility patterns captured by smartphones have increasingly been linked to a range of psychiatric symptoms, existing research has not specifically examined whether individuals have person-specific mobility patterns. We collected over 3000 days of mobility data from a sample of 41 adolescents and young adults (age 17–30 years, 28 female) with affective instability. We extracted summary mobility metrics from GPS and accelerometer data and used their covariance structures to identify individuals and calculated the individual identification accuracy—i.e., their “footprint distinctiveness”. We found that statistical patterns of smartphone-based mobility features represented unique “footprints” that allow individual identification (p < 0.001). Critically, mobility footprints exhibited varying levels of person-specific distinctiveness (4–99%), which was associated with age and sex. Furthermore, reduced individual footprint distinctiveness was associated with instability in affect (p < 0.05) and circadian patterns (p < 0.05) as measured by environmental momentary assessment. Finally, brain functional connectivity, especially those in the somatomotor network, was linked to individual differences in mobility patterns (p < 0.05). Together, these results suggest that real-world mobility patterns may provide individual-specific signatures relevant for studies of development, sleep, and psychopathology.

The syzygies of a curve are the algebraic relation amongst the equation defining it. They are an algebraic concept but they have surprising applications to geometry. For example, the Green-Lazarsfeld secant conjecture predicts that the syzygies of a curve of sufficiently high degree are controlled by its special secants. We prove this conjecture for all curves of Clifford index at least two and not bielliptic and for all line bundles of a certain degree. Our proof is based on a classic result of Martens and Mumford on Brill-Noether varieties and on a simple vanishing criterion that comes from the interpretation of syzygies through symmetric products of curves.

Ordinary differential equation (ODE) models are widely used to describe chemical or biological processes. This article considers the estimation and assessment of such models on the basis of time-course data. Due to experimental limitations, time-course data are often noisy and some components of the system may not be observed. Furthermore, the computational demands of numerical integration have hindered the widespread adoption of time-course analysis using ODEs. To address these challenges, we explore the efficacy of the recently developed MAGI (MAnifold-constrained Gaussian process Inference) method for ODE inference. First, via a range of examples we show that MAGI is capable of inferring the parameters and system trajectories, including unobserved components, with appropriate uncertainty quantification. Second, we illustrate how MAGI can be used to assess and select different ODE models with time-course data based on MAGI’s efficient computation of model predictions. Overall, we believe MAGI is a useful method for the analysis of time-course data in the context of ODE models, which bypasses the need for any numerical integration.