We consider the problem of graph matching, or learning vertex correspondence, between two correlated stochastic block models (SBMs). The graph matching problem arises in various fields, including computer vision, natural language processing and bioinformatics, and in particular, matching graphs with inherent community structure has significance related to de-anonymization of correlated social networks. Compared to the correlated Erdos-Renyi (ER) model, where various efficient algorithms have been developed, among which a few algorithms have been proven to achieve the exact matching with constant edge correlation, no low-order polynomial algorithm has been known to achieve exact matching for the correlated SBMs with constant correlation. In this work, we propose an efficient algorithm for matching graphs with community structure, based on the comparison between partition trees rooted from each vertex, by extending the idea of Mao et al. (2021) to graphs with communities. The partition tree divides the large neighborhoods of each vertex into disjoint subsets using their edge statistics to different communities. Our algorithm is the first low-order polynomial-time algorithm achieving exact matching between two correlated SBMs with high probability in dense graphs.

We prove that every finite colouring of the plane contains a monochromatic pair of points at an odd integral distance from each other.

: For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasi-isometry type: all saddle connection graphs form a single quasi-isometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.

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심사위원장: 김동환, 심사위원: 이창옥, 홍영준, 곽도영, 오덕순(충남대학교)

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심사위원장: 김동환, 심사위원: 이창옥, 홍영준, 곽도영, 오덕순(충남대학교)

The talk will focus on a part of the frontier of our current understanding of nonlinear stability of traveling waves of partial differential equations, especially on how spectral stability implies nonlinear stability and which kind of dynamics may be expected. We shall highlight main expected difficulties related to the stability of discontinuous waves of hyperbolic systems, and show a few significant steps obtained by the speaker with respectively Vincent Duchêne (Rennes), Gregory Faye (Toulouse) and Louis Garénaux (Karslruhe).

The Gauss-Bonnet theorem implies that the two dimensional torus does not have nonnegative Gauss curvature unless it is flat, and that the two dimensional sphere does not a metric which has Gaussian curvature bounded below by one and metric bounded below by the standard round metric. Gromov proposed a series of conjectures on generalizing the Gauss-Bonnet theorem in his four lectures. I will report my work with Gaoming Wang (now Tsinghua) on Gromov dihedral rigidity conjecture in hyperbolic 3-space and scalar curvature comparison of rotationally symmetric convex bodies with some simple singularities.

In this lecture, we aim to delve deep into the emerging landscape of 'Foundation Models'. Distinct from traditional deep learning models, Foundation Models have ushered in a new paradigm, characterized by their vast scale, versatility, and transformative potential. We will uncover the key differences between these models and their predecessors, delving into the intricate mechanisms through which they are trained and the profound impact they are manifesting across various sectors. Furthermore, the talk will shed light on the invaluable role of mathematics in understanding, optimizing, and innovating upon these models. We will explore the symbiotic relationship between Foundation Models and mathematical principles, elucidating how the latter not only underpins their functioning but also paves the way for future advancements.

Graph product structure theory describes complex graphs in terms of products of simpler graphs. In this talk, I will introduce this subject and talk about some of my recent results in this area. The focus of my talk will be on a new tool in graph product structure theory called `blocking partitions.’ I’ll show how this tool can be used to prove stronger product structure theorems for powers of planar graphs as well as k-planar graphs, resolving open problems of Dujmović, Morin and Wood, and Ossona de Mendez.

In this talk, we study the non-cutoff Boltzmann collision kernel for the inverse power law potentials $U_s(r)=1/r^{s-1}$ for $s>2$ in dimension $d=3$. We will study the formal derivation of the non-cutoff collision kernel. Then we will prove the limit of the non-cutoff kernel to the hard-sphere kernel and check the angular singularity would vanish. We will also see precise asymptotic formulas of the singular layer near $\theta\simeq 0$ in the limit $s\to \infty$. Consequently, we will also see that solutions to the homogeneous Boltzmann equation converge to the respective solutions weakly in $L^1$ globally in time as $s\to \infty$.

Almost all biological systems possess the ability to gather environmental information and modulate their behaviors to adaptively respond to changing environments. While animals excel at sensing odors, even simple bacteria can detect faint chemicals using stochastic receptors. They then navigate towards or away from the chemical source by processing this sensed information through intracellular reaction systems. In the first half of our talk, we demonstrate that the E. coli chemotactic system is optimally structured for sensing noisy signals and controlling taxis. We utilize filtering theory and optimal control theory to theoretically derive this optimal structure and compare it to the quantitatively verified biochemical model of chemotaxis. In the latter half, we discuss the limitations of traditional information theory, filtering theory, and optimal control theory in analyzing biological systems. Notably, all biological systems, especially simpler ones, have constrained computational resources like memory size and energy, which influence optimal behaviors. Conventional theories don’t directly address these resource constraints, likely because they emerged during a period when computational resources were continually expanding. To address this gap, we introduce the “memory-limited partially observable optimal control,” a new theoretical framework developed by our group, and explore its relevance to biological problems.

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

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심사위원장: 변재형, 심사위원 : 강문진, 김용정, 배명진, 권오상(충북대학교)

An edge-coloured graph is said to be rainbow if it uses no colour more than once. Extremal problems involving rainbow objects have been a focus of much research over the last decade as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on n vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of $(\log n)^{2+o(1)}$ for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the $o(1)$ term in Tomon's bound. We show that the answer to the question is equal to $(\log n)^{1+o(1)}$. A key tool we use is the theory of robust sublinear expanders. In addition, we observe a connection between this problem and several questions in additive number theory, allowing us to extend existing results on these questions for abelian groups to the case of non-abelian groups. Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.

With the success of deep learning technologies in many scientific and engineering applications, neural network approximation methods have emerged as an active research area in numerical partial differential equations. However, the new approximation methods still need further validations on their accuracy, stability, and efficiency so as to be used as alternatives to classical approximation methods. In this talk, we first introduce the neural network approximation methods for partial differential equations, where a neural network function is introduced to approximate the PDE (Partial Differential Equation) solution and its parameters are then optimized to minimize the cost function derived from the differential equation. We then present the approximation error and the optimization error behaviors in the neural network approximate solution. To reduce the approximation error, a neural network function with a larger number of parameters is often employed but when optimizing such a larger number of parameters the optimization error usually pollutes the solution accuracy. In addition to that, the gradient-based parameter optimization usually requires computation of the cost function gradient over a tremendous number of epochs and it thus makes the cost for a neural network solution very expensive. To deal with such problems in the neural network approximation, a partitioned neural network function can be formed to approximate the PDE solution, where localized neural network functions are used to form the global neural network solution. The parameters in each local neural network function are then optimized to minimize the corresponding cost function. To enhance the parameter training efficiency further, iterative algorithms for the partitioned neural network function can be developed. We finally discuss the possibilities in this new approach as a way of enhancing the neural network solution accuracy, stability, and efficiency by utilizing classical domain decomposition algorithms and their convergence theory. Some interesting numerical results are presented to show the performance of the partitioned neural network approximation and the iteration algorithms.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

Maximal functions of various forms have played crucial roles in harmonic analysis. Various outstanding open problems are related to Lp boundedness (estimate) of the associated maximal functions. In this talk, we discuss Lp boundedness of maximal functions given by averages over curves.

In physics, Bohr’s correspondence principle asserts that the theory of quantum mechanics can be reduced to that of classical mechanics in the limit of large quantum numbers. This rather vague statement can be formulated explicitly in various ways. In this talk, focusing on an analytic point of view, we discuss the correspondence between basic inequalities and that between measures. Then, as an application, we present the convergence from quantum to kinetic white dwarfs.

The k-color induced size-Ramsey number of a graph H is the smallest number of edges a (host) graph G can have such that for any k-coloring of its edges, there exists a monochromatic copy of H which is an induced subgraph of G. In 1995, in their seminal paper, Haxell, Kohayakawa and Łuczak showed that for cycles these numbers are linear for any constant number of colours, i.e., for some C=C(k), there is a graph with at most Cn edges whose any k-edge-coloring contains a monochromatic induced cycle of length n. The value of C comes from the use of the sparse regularity lemma and has a tower-type dependence on k. In this work, we obtain nearly optimal bounds for the required value of C. Joint work with Nemanja Draganić and Benny Sudakov.

"어떻게 하면 더 좋은 제품을 더 빠르게 개발할 수 있을까?"라는 문제는 모든 제조업이 안고 있는 숙제입니다. 최근 DX를 통해 많은 데이터들이 디지털화되고, AI의 급격한 발전을 통해 제품개발프로세스를 혁신하려는 시도가 일어나고 있습니다. 과거의 시뮬레이션 기반 설계에서 AI 기반 설계로의 패러다임 전환을 통해 제품개발 기간을 단축함과 동시에 제품의 품질을 향상시킬 수 있습니다. 본 세미나는 딥러닝을 통해 제품 설계안을 생성/탐색/예측/최적화/추천할 수 있는 생성형 AI 기반의 설계 프로세스(Deep Generative Design)를 소개하고, 모빌리티를 비롯한 제조 산업에 적용된 다양한 사례들을 소개합니다.

In this talk, we discuss the Neural Tangent Kernel. The NTK is closely related to the dynamics of the neural network during training via the Gradient Flow(or Gradient Descent). But, since the NTK is random at initialization and varies during training, it is quite delicate to understand the dynamics of the neural network. In relation to this issue, we introduce an interesting result: in the infinite-width limit, the NTK converge to a deterministic kernel at initialization and remains constant during training. We provide a brief proof of the result for the simplest case.

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9월 14일, 10월 4일, 5일 세 번에 걸친 발표.

In this talk, I will explain the setting of online convex optimization and the definition of regret and constraint violation. I then will introduce various algorithms and their theoretical guarantees under various assumptions. The connection with some topics in machine learning such as stochastic gradient descent, multi-armed bandit, and reinforcement learning will also be briefly discussed.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

In this talk, we discuss the Neural Tangent Kernel. The NTK is closely related to the dynamics of the neural network during training via the Gradient Flow(or Gradient Descent). But, since the NTK is random at initialization and varies during training, it is quite delicate to understand the dynamics of the neural network. In relation to this issue, we introduce an interesting result: in the infinite-width limit, the NTK converge to a deterministic kernel at initialization and remains constant during training. We provide a brief proof of the result for the simplest case.

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9월 14일, 10월 4일, 5일 세 번에 걸친 발표로, 본 시간에는 주로 9월 14일 내용의 리뷰를 주로 다룸.

We prove that the twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the annulus, is stable to perturbations. In fact, it is possible to prove the stability of the whole of the lifted dynamics to non-autonomous perturbations, though single particle paths are generically unstable. These all-time stability facts are used to establish a number of results related to the long-time behavior of fluid flows. (Joint work with T. Drivas and T. Elgindi)

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.

최근의 생성모델에 관하여 스탠포드대학의 Ermon교수팀에서 NeurIPS2019, ICLR2021에 발표한 아래의 2편의 논문을 집중 리뷰하면서 SDE를 이용한 Generative Modeling의 연구동향과 발전 방향을 심층토의하게 됩니다.

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2주 연속 강의 (2번째) *공지: 22일 (금) 강연이 취소되고, 25일 (월) 강연으로 변경됨.

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.

In this talk, we consider the blow-up dynamics of co-rotational solutions for energy-critical wave maps with the 2-sphere target. We briefly introduce the (2+1)-dimensional wave maps problem and its co-rotational symmetry, which reduces the full wave map to the (1+1)-dimensional semilinear wave equation. Under such symmetry, we see that this problem has a unique explicit stationary solution, so-called "harmonic map". Then we point out some of the works of analyzing the long-term dynamics of the flow near the harmonic map. Among them, we focus on the smooth blow-up result that corresponds to the stable regime. In particular, the case where the homotopy index is one has a distinctive nature from the other cases, which allows us to exhibit the smooth blow-up with the quantized blow-up rates corresponding to the excited regime.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

We will discuss certain main problems concerning group actions on 1-dimensional manifolds (the circle and the interval) and perspectives for future research.

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KAI-X Distinguished lecture

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.

최근의 생성모델에 관하여 스탠포드대학의 Ermon교수팀에서 NeurIPS2019, ICLR2021에 발표한 아래의 2편의 논문을 집중 리뷰하면서 SDE를 이용한 Generative Modeling의 연구동향과 발전 방향을 심층토의 하게 됩니다.

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2주 연속 강의(1번째)

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.

In this talk, we discuss the Neural Tangent Kernel. The NTK is closely related to the dynamics of the neural network during training via the Gradient Flow(or Gradient Descent). But, since the NTK is random at initialization and varies during training, it is quite delicate to understand the dynamics of the neural network. In relation to this issue, we introduce an interesting result: in the infinite-width limit, the NTK converge to a deterministic kernel at initialization and remains constant during training. We provide a brief proof of the result for the simplest case.

(information) "Introduction to Oriented Matroids" Series Thursdays 14:30-15:45

Questions of parameter estimation – that is, finding the parameter values that allow a model to best fit some data – and parameter identifiability – that is, the uniqueness of such parameter values – are often considered in settings where experiments can be repeated to gain more certainty about the data. In this talk, however, I will consider parameter estimation and parameter identifiability in situations where data can only be collected from a single experiment or trajectory. Our motivation comes from medical settings, where data comes from a patient; such limitations in data also arise in finance, ecology, and climate, for example. In this setting, we can try to find the best parameters to fit our limited data. In this talk, I will introduce a novel, alternative goal, which we refer to as a qualitative inverse problem. The aim here is to analyze what information we can gain about a system from the available data even if we cannot estimate its parameter values precisely. I will discuss results that allow us to determine whether a given model has the ability to fit the data, whether its parameters are identifiable, the signs of model parameters, and/or the local dynamics around system fixed points, as well as how much measurement error can be tolerated without changing the conclusions of our analysis. I will consider various classes of model systems and will illustrate our latest results with the classic Lotka-Volterra system.

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).

In the past decades, there has been considerable progress in the theory of random walks on groups acting on hyperbolic spaces. Despite the abundance of such groups, this theory is inherently not preserved under quasi-isometry. In this talk, I will present our study of random walks on groups that satisfy a certain QI-invariant property that does not refer to an action on hyperbolic spaces. Joint work with Kunal Chawla, Kasra Rafi, and Vivian He.

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.

In this talk, we discuss some concepts that are used to study (hyperbolic) holomorphic dynamics on K3 surfaces. These concepts include Green currents, their laminations, and Green measures, which emerge as the natural choice for measuring maximal entropy. These tools effectively establish Kummer rigidity – that is, when the Green measure is absolutely continuous to the volume measure, the surface is Kummer, and the dynamics is linear. We provide an overview of the techniques employed to establish this principle and provide a glimpse into their extension within the hyperkähler context – one of the higher-dimensional analogues of K3 surfaces.

This talk aims to explore the application of calculus of variations in materials sciences. We will discuss the physics behind solid-solid phase transitions and elastic energy. Then the Allen-Cahn model in studying interfaces and their energy will be introduced. Finally, we will examine the Ohta-Kawasaki model and its role in understanding self-assembly in block copolymers. Recent research advancements in the Ohta-Kawasaki problem will also be presented.

Hénon maps were introduced by Michel Hénon as a simplified model of the Poincaré section of the Lorenz model. They are among the most studied discrete-time dynamical systems that exhibit chaotic behavior. Complex Hénon maps in any dimension have been extensively studied over the last three decades, in parallel with the development of pluripotential theory. We will present the dynamical properties of these maps such as the behavior of point orbits, variety orbits, equidistribution of periodic points and fine ergodic properties of the systems. This talk is based on the work of Bedford, Fornaess, Lyubich, Sibony, Smillie, and on recent work of the speaker in collaboration with Bianchi and Sibony.

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.