Graph pebbling is a combinatorial game played on an undirected graph with an initial configuration of pebbles. A pebbling move consists of removing two pebbles from one vertex and placing one pebbling on an adjacent vertex. The pebbling number of a graph is the smallest number of pebbles necessary such that, given any initial configuration of pebbles, at least one pebble can be moved to a specified target vertex. In this talk, we will give a survey of several streams of research in pebbling, including describing a theoretical and computational framework that uses mixed-integer linear programming to obtain bounds for the pebbling numbers of graphs. We will also discuss improvements to this framework through the use of newly proved weight functions that strengthen the weight function technique of Hurlbert. Finally, we will discuss some open extremal problems in pebbling, specifically related to Class 0 graphs and describe how structural graph theoretic techniques such as discharging can be used to obtain results. Collaborators on these projects include Dan Cranson, Dominic Flocco, Luke Postle, Jonad Pulaj, Chenxiao Xue, Marshall Yang, Daniel Zhou.

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.

Typically, the mathematical description of reaction networks involves a system of parameter-dependent ordinary differential equations. Generally, one is interested in the qualitative and quantitative behavior of solutions in various parameter regions. In applications, identifying the reaction parameters is a fundamental task. Reduction of dimension is desirable from a practical perspective, and even necessary when different timescales are present. For biochemical reaction networks, a classical reduction technique assumes quasi-steady state (QSS) of certain species. From a general mathematical perspective, singular perturbation theory – involving a small parameter – is often invoked. The talk is mathematically oriented. The following points will be discussed: Singular perturbation reduction in general coordinates. (“How does one compute reductions?”) Critical parameters for singular perturbations. (“How does one find small parameters?”) Quasi-steady state and singular perturbations. (“What is applicable, what is correct?”)

Even delta-matroids generalize matroids, as they are defined by a certain basis exchange axiom weaker than that of matroids. One natural example of even delta-matroids comes from a skew-symmetric matrix over a given field $K$, and we say such an even delta-matroid is representable over the field $K$. Interestingly, a matroid is representable over $K$ in the usual manner if and only if it is representable over $K$ in the sense of even delta-matroids. The representability of matroids got much interest and was extensively studied such as excluded minors and representability over more than one field. Recently, Baker and Bowler (2019) integrated the notions of $K$-representable matroids, oriented matroids, and valuated matroids using tracts that are commutative ring-like structures in which the sum of two elements may output no element or two or more elements. We generalize Baker-Bowler's theory of matroids with coefficients in tracts to orthogonal matroids that are equivalent to even delta-matroids. We define orthogonal matroids with coefficients in tracts in terms of Wick functions, orthogonal signatures, circuit sets, and orthogonal vector sets, and establish basic properties on functoriality, duality, and minors. Our cryptomorphic definitions of orthogonal matroids over tracts provide proofs of several representation theorems for orthogonal matroids. In particular, we give a new proof that an orthogonal matroid is regular (i.e., representable over all fields) if and only if it is representable over $\mathbb{F}_2$ and $\mathbb{F}_3$, which was originally shown by Geelen (1996), and we prove that an orthogonal matroid is representable over the sixth-root-of-unity partial field if and only if it is representable over $\mathbb{F}_3$ and $\mathbb{F}_4$. This is joint work with Tong Jin.

In many stochastic service systems, decision-makers find themselves making a sequence of decisions, with the number of decisions being unpredictable. To enhance these decisions, it is crucial to uncover the causal impact these decisions have through careful analysis of observational data from the system. However, these decisions are not made independently, as they are shaped by previous decisions and outcomes. This phenomenon is called sequential bias and violates a key assumption in causal inference that one person’s decision does not interfere with the potential outcomes of another. To address this issue, we establish a connection between sequential bias and the subfield of causal inference known as dynamic treatment regimes. We expand these frameworks to account for the random number of decisions by modeling the decision-making process as a marked point process. Consequently, we can define and identify causal effects to quantify sequential bias. Moreover, we propose estimators and explore their properties, including double robustness and semiparametric efficiency. In a case study of 27,831 encounters with a large academic emergency department, we use our approach to demonstrate that the decision to route a patient to an area for low acuity patients has a significant impact on the care of future patients.

The square of a graph $G$, denoted $G^2$, has the same vertex set as $G$ and has an edge between two vertices if the distance between them in $G$ is at most $2$. Wegner's conjecture (1977) states that for a planar graph $G$, the chromatic number $\chi(G^2)$ of $G^2$ is at most 7 if $\Delta(G) = 3$, at most $\Delta(G)+5$ if $4 \leq \Delta(G) \leq 7$, and at most $\lfloor \frac{3 \Delta(G)}{2} \rfloor$ if $\Delta(G) \geq 8$. Wegner's conjecture is still wide open. The only case for which we know tight bound is when $\Delta(G) = 3$. Thomassen (2018) showed that $\chi(G^2) \leq 7$ if $G$ is a planar graph with $\Delta(G) = 3$, which implies that Wegner's conjecture is true for $\Delta(G) = 3$. A natural question is whether $\chi_{\ell}(G^2) \leq 7$ or not if $G$ is a subcubic planar graph, where $\chi_{\ell}(G^2)$ is the list chromatic number of $G^2$. Cranston and Kim (2008) showed that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 7. We prove that $\chi_{\ell}(G^2) \leq 7$ if $G$ is a subcubic planar graph of girth at least 6. This is joint work with Xiaopan Lian (Nankai University).

Background Microbial community simulations using genome scale metabolic networks (GSMs) are relevant for many application areas, such as the analysis of the human microbiome. Such simulations rely on assumptions about the culturing environment, affecting if the culture may reach a metabolically stationary state with constant microbial concentrations. They also require assumptions on decision making by the microbes: metabolic strategies can be in the interest of individual community members or of the whole community. However, the impact of such common assumptions on community simulation results has not been investigated systematically. Results Here, we investigate four combinations of assumptions, elucidate how they are applied in literature, provide novel mathematical formulations for their simulation, and show how the resulting predictions differ qualitatively. Our results stress that different assumption combinations give qualitatively different predictions on microbial coexistence by differential substrate utilization. This fundamental mechanism is critically under explored in the steady state GSM literature with its strong focus on coexistence states due to crossfeeding (division of labor). Furthermore, investigating a realistic synthetic community, where the two involved strains exhibit no growth in isolation, but grow as a community, we predict multiple modes of cooperation, even without an explicit cooperation mechanism. Conclusions Steady state GSM modelling of microbial communities relies both on assumed decision making principles and environmental assumptions. In principle, dynamic flux balance analysis addresses both. In practice, our methods that address the steady state directly may be preferable, especially if the community is expected to display multiple steady states.

Mathematical models of biological systems, including neurons, often feature components that evolve on very different timescales. Mathematical analysis of these multi-timescale systems can be greatly simplified by partitioning them into subsystems that evolve on different time scales. The subsystems are then analyzed semi-independently, using a technique called fast-slow analysis. I will briefly describe the fast-slow analysis technique and its application to neuronal bursting oscillations and basic coupled neuron modeling. After this, I will discuss fancier forms of dynamics such as canard oscillations, mixed-mode oscillations, and three-timescale dynamics. Although these examples all involve neural systems, the methods can and have been applied to other biological, chemical, and physical systems.

Pluripotential theory, namely positive closed and positive ddc-closed currents, is a fundamental tool in the theory of iteration of holomorphic maps and the theory of foliations. We will start with a crash course on positive closed and positive ddc-closed currents focusing on some recent progress of the pluripotential theory. We then discuss applications in complex dynamics. We will explain how the pluripotential theory allows to obtain equidistribution results, the unique ergodicity or other fine statistical properties. (2 of 2)

In 1986, Robertson and Seymour proved a generalization of the seminal result of Erdős and Pósa on the duality of packing and covering cycles: A graph has the Erdős-Pósa property for minor if and only if it is planar. In particular, for every non-planar graph $H$ they gave examples showing that the Erdős-Pósa property does not hold for $H$. Recently, Liu confirmed a conjecture of Thomas and showed that every graph has the half-integral Erdős-Pósa property for minors. In this talk, we start the delineation of the half-integrality of the Erdős-Pósa property for minors. We conjecture that for every graph $H$ there exists a finite family $\mathfrak{F}_H$ of parametric graphs such that $H$ has the Erdős-Pósa property in a minor-closed graph class $\mathcal{G}$ if and only if $\mathcal{G}$ excludes a minor of each of the parametric graphs in $\mathfrak{F}_H$. We prove this conjecture for the class $\mathcal{H}$ of Kuratowski-connected shallow-vortex minors by showing that, for every non-planar $H\in\mathcal{H}$ the family $\mathfrak{F}_H$ can be chosen to be precisely the two families of Robertson-Seymour counterexamples to the Erdős-Pósa property of $H$.

Pluripotential theory, namely positive closed and positive ddc-closed currents, is a fundamental tool in the theory of iteration of holomorphic maps and the theory of foliations. We will start with a crash course on positive closed and positive ddc-closed currents focusing on some recent progress of the pluripotential theory. We then discuss applications in complex dynamics. We will explain how the pluripotential theory allows to obtain equidistribution results, the unique ergodicity or other fine statistical properties. (1 of 2)

The Michaelis–Menten (MM) rate law has been the dominant paradigm of modeling biochemical rate processes for over a century with applications in biochemistry, biophysics, cell biology, and chemical engineering. The MM rate law and its remedied form stand on the assumption that the concentration of the complex of interacting molecules, at each moment, approaches an equilibrium much faster than the molecular concentrations change. Yet, this assumption is not always justified. Here, we relax this quasi-steady state requirement and propose the generalized MM rate law for the interactions of molecules with active concentration changes over time. Our approach for time-varying molecular concentrations, termed the effective time-delay scheme (ETS), is based on rigorously estimated time-delay effects in molecular complex formation. With particularly marked improvements in protein– protein and protein–DNA interaction modeling, the ETS provides an analytical framework to interpret and predict rich transient or rhythmic dynamics (such as autogenously-regulated cellular adaptation and circadian protein turnover), which goes beyond the quasi-steady state assumption.

Take a prime power $q$, an integer $n\geq 2$, and a coordinate subspace $S\subseteq GF(q)^n$ over the Galois field $GF(q)$. One can associate with $S$ an $n$-partite $n$-uniform clutter $\mathcal{C}$, where every part has size $q$ and there is a bijection between the vectors in $S$ and the members of $\mathcal{C}$. In this paper, we determine when the clutter $\mathcal{C}$ is ideal, a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether $q$ is $2,4$, a higher power of $2$, or otherwise. Each characterization uses crucially that idealness is a minor-closed property: first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $\mathcal{C}$ depends solely on the underlying matroid of $S$. Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and $\tau=2$ Conjectures for this class of clutters. This is joint work with Ahmad Abdi (London School of Economics).

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph. The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $\omega(G)$ has chromatic number at most $f(\omega(G))$. Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker, Charbit, and Naserasr's $\overrightarrow{\chi}$-boundedness conjecture states that for every oriented forest $F$, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(\omega(D))$, where $\omega(D)$ is the size of a maximum clique in the graph underlying $D$. In this talk, we perform the first step towards proving Aboulker, Charbit, and Naserasr's $\overrightarrow{\chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.

Ecosystems are complex systems of various physical, biological, and chemical processes. Since ecosystem dynamics are composed of a mixture of different levels of stochasticity and nonlinearity, handling these data is a challenge for existing methods of time series–based causal inferences. Here, we show that, by harnessing contemporary machine learning approaches, the concept of Granger causality can be effectively extended to the analysis of complex ecosystem time series and bridge the gap between dynamical and statistical approaches. The central idea is to use an ensemble of fast and highly predictive artificial neural networks to select a minimal set of variables that maximizes the prediction of a given variable. It enables decomposition of the relationship among variables through quantifying the contribution of an individual variable to the overall predictive performance. We show how our approach, EcohNet, can improve interaction network inference for a mesocosm experiment and simulated ecosystems. The application of the method to a long-term lake monitoring dataset yielded interpretable results on the drivers causing cyanobacteria blooms, which is a serious threat to ecological integrity and ecosystem services. Since performance of EcohNet is enhanced by its predictive capabilities, it also provides an optimized forecasting of overall components in ecosystems. EcohNet could be used to analyze complex and hybrid multivariate time series in many scientific areas not limited to ecosystems.

How can one arrange a collection of convex sets in d-dimensional Euclidean space? This guiding question is fundamental in discrete geometry, and can be made concrete in a variety of ways, for example the study of hyperplane arrangements, embeddability of simplicial complexes, Helly-type theorems, and more. This talk will focus on the classical topic of d-representable complexes and its more recent generalization to convex codes. We will show how these frameworks differ, share some novel Helly-type results, and offer several tantalizing open questions.

The well-known Internal Model Principle (IMP) is a cornerstone of modern control theory. It stipulates the necessary conditions for asymptotic robustness of disturbance-prone dynamical systems by asserting that such a system must embed a subsystem in a feedback loop, and this subsystem must be able to reduplicate the dynamic disturbance using only the regulated variable as the input. The insights provided by IMP can help in both designing suitable controllers and also in analysing the regulatory mechanisms in complex systems. So far the application of IMP in biology has been case-specific and ad hoc, primarily due to the lack of generic versions of the IMP for biomolecular reaction networks that model biological processes. In this short article we highlight the need for an IMP in biology and discuss a recently developed version of it for biomolecular networks that exhibit maximal Robust Perfect Adaptation (maxRPA) by being robust to the maximum number of disturbance sources.

Ramsey’s Theorem guarantees for every graph H that any 2-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of H. As probabilistic constructions often provide good bounds on quantities in extremal combinatorics, we say that a graph H is common if the random 2-edge-coloring asymptotically minimizes the number of monochromatic copies of H. This notion goes back to the work of Erdős in the 1960s, who conjectured that every complete graph is common. The conjecture was disproved by Thomason in the 1980s, however, a classification of common graphs remains one of the most intriguing problems in extremal combinatorics. Sidorenko’s Conjecture (if true) would imply that every bipartite graph is common, and in fact, no bipartite common graph unsettled for Sidorenko’s Conjecture is known. Until Hatami et al. showed that a 5-wheel is common about a decade ago, all graphs known to be common had chromatic number at most three. The existence of a common graph with chromatic number five or more has remained open for three decades. We will present a construction of (connected) common graphs with arbitrarily large chromatic number. At the end of the talk, we will also briefly discuss the extension of the notion to more colors and particularly its relation to Sidorenko’s Conjecture. The main result presented in the talk is based on joint work with Jan Volec and Fan Wei.