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.

Let V be a suvariety of a manifold M. We say that V has extension property, if any bounded holomorphic function on V extends to a holomorphic function on M with the same sup-norm. In the talk we shall explain connections between this problem and operator theory (von Neumann inequality, interpolation problem) as well as with the theory of invariant functions and metrics

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.

In this lecture series, I will discuss rigidity in the long time dynamics of some evolution equation. The specific equation to be paid attention to is the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a gauged 2D cubic nonlinear Schrödinger equation that has pseudoconformal invariance and solitons. However, two distinguished features of (CSS), the self-duality and non-locality, make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We will explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem, in detail.

In this lecture series, I will discuss rigidity in the long time dynamics of some evolution equation. The specific equation to be paid attention to is the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a gauged 2D cubic nonlinear Schrödinger equation that has pseudoconformal invariance and solitons. However, two distinguished features of (CSS), the self-duality and non-locality, make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We will explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem, in detail.

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.

We consider the nonlinear Schrödinger equation and the nonlinear wave equation with initial data decaying slower than L^2 functions. However, the L^p-spaces for p \neq 2 are not invariant under the linear propagation. We consider function spaces, which allow for decay like in L^p, p > 2, and which are invariant under the linear propagation. We show L^p-smoothing estimates using \ell^2-decoupling due to Bourgain-Demeter. The results on nonlinear wave equations are joint work with Jan Rozendaal (IMPAN).

In this lecture series, I will discuss rigidity in the long time dynamics of some evolution equation. The specific equation to be paid attention to is the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a gauged 2D cubic nonlinear Schrödinger equation that has pseudoconformal invariance and solitons. However, two distinguished features of (CSS), the self-duality and non-locality, make the long time dynamics of (CSS) surprisingly rigid. For instance, (i) any finite energy spatially decaying solutions to (CSS) decompose into at most one modulated soliton and a radiation. Moreover, (ii) in the high equivariance case (i.e., the equivariance index ≥ 1), any smooth finite-time blow-up solutions even have a universal blow-up speed, namely, the pseudoconformal one. We will explore this rigid dynamics using modulation analysis, combined with the self-duality and non-locality of the problem, in detail.

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.

We review constructions of Manolescu’s Floer homotopy type, which gives a homotopical refinement of monopole Floer homology. Based on it, we will introduce some homology cobordism/ knot concordance invariant. Using these invariants, we provide relative versions of 10/8 inequalities for 4-manifolds with boundary or surfaces in 4-manifolds. In particular, I’ll explain Manolescu’s relative 10/8 inequality, real 10/8 inequality, and Montague’s 10/8 inequality.

Let X be a semistable p-adic formal scheme. In this talk, we will discuss a prismatic description of semistable local systems on the generic fiber of X. A main new ingredient is a purity result. This is based on a joint work with Heng Du, Tong Liu, Koji Shimizu.

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.

We first review fundamental concepts about Seiberg-Witten theory for closed 4-manifolds. Subsequently, we will introduce a refinement of Seiberg-Witten invariant, called Bauer—Furuta invariant. Using Bauer—Furuta invariant, I will explain how to prove Furuta’s 10/8 inequality and its variant for group actions proven by Bryan and Kato.

The essential dimension of an algebraic object E over a field L is heuristically the number of parameters it takes to define it. This notion was formalized and developed by Buhler and Reichstein in the late 90s, who noticed at the time, that several classical results could be interpreted as theorems about essential dimension. Since the paper of Buhler and Reichstein, most of the progress on essential dimension has had to do with essential dimension of versal G-torsors for an algebraic group G. But recently Farb, Kisin and Wolfson showed that interesting theorems can be proved for certain (usually) non-versal torsors arising from congruence covers of Shimura varieties. I'll explain this work, some extensions of it proved by me and Fakhruddin, and a conjecture on period maps which generalizes the picture.

A loose cycle is a cyclic ordering of edges such that every two consecutive edges share exactly one vertex. A cycle is Hamilton if it spans all vertices. A codegree of a $k$-uniform hypergraph is the minimum nonnegative integer $t$ such that every subset of vertices of size $k-1$ is contained in $t$ distinct edges. We prove "robust" versions of Dirac-type theorems for Hamilton cycles and optimal matchings. Let $\mathcal{H}$ be a $k$-uniform hypergraph on $n$ vertices with $n \in (k-1)\mathbb{N}$ and codegree at least $n/(2k-2)$, and let $\mathcal{H}_p$ be a spanning subgraph of $\mathcal{H}$ such that each edge of $\mathcal{H}$ is included in $\mathcal{H}_p$ with probability $p$ independently at random. We prove that a.a.s. $\mathcal{H}_p$ contains a loose Hamilton cycle if $p = \Omega(n^{-k+1} \log n)$, which is asymptotically best possible. We also present similar results for Hamilton $\ell$-cycles for $\ell \geq 2$. Furthermore, we prove that if $\mathcal{H}$ is a $k$-uniform hypergraph on $n$ vertices with $n \notin k \mathbb{N}$ and codegree at least $\lfloor n/k \rfloor$, then a.a.s. $\mathcal{H}_p$ contains a matching of size $\lfloor n/k \rfloor$ if $p = \Omega(n^{-k+1} \log n)$. This is also asymptotically best possible. This is joint work with Michael Anastos, Debsoumya Chakraborti, Abhishek Methuku, and Vincent Pfenninger.

For hyperbolic systems of conservation laws in one space dimension endowed with a single convex entropy, it is an open question if it is possible to construct solutions via convex integration. Such solutions, if they exist, would be highly non-unique and exhibit little regularity. In particular, they would not have the strong traces necessary for the nonperturbative $L^2$ stability theory of Vasseur. Whether convex integration is possible is a question about large data, and the global geometric structure of genuine nonlinearity for the underlying PDE. In this talk, I will discuss recent work which shows the impossibility, for a large class of 2x2 systems, of doing convex integration via the use of $T_4$ configurations. Our work applies to every well-known 2x2 hyperbolic system of conservation laws which verifies the Liu entropy condition. This talk is based on joint work with László Székelyhidi.

This three-day lecture series aims to explore some topics in mathematical image processing before the era of neural networks, highlighting the techniques and applications that were prevalent at that time. From the classical filter-based models to PDE-based or minimization-based models, a variety of example-driven explanations and underlying mathematical theories are provided. By attending the lecture series, participants will gain a comprehensive understanding of image processing techniques used before the advent of neural networks, exploring the challenges, innovations and applications of classical algorithms. This knowledge will provide a foundation for further exploration in the field of image processing and its evolution into the AI-driven era.

We prove an arithmetic path integral formula for the inverse p-adic absolute values of the p-adic L-functions of elliptic curves over the rational numbers with good ordinary reduction at odd prime p. This is joint work with Jeehoon Park

This three-day lecture series aims to explore some topics in mathematical image processing before the era of neural networks, highlighting the techniques and applications that were prevalent at that time. From the classical filter-based models to PDE-based or minimization-based models, a variety of example-driven explanations and underlying mathematical theories are provided. By attending the lecture series, participants will gain a comprehensive understanding of image processing techniques used before the advent of neural networks, exploring the challenges, innovations and applications of classical algorithms. This knowledge will provide a foundation for further exploration in the field of image processing and its evolution into the AI-driven era.

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3-day lecture series (2 of 3)

This three-day lecture series aims to explore some topics in mathematical image processing before the era of neural networks, highlighting the techniques and applications that were prevalent at that time. From the classical filter-based models to PDE-based or minimization-based models, a variety of example-driven explanations and underlying mathematical theories are provided. By attending the lecture series, participants will gain a comprehensive understanding of image processing techniques used before the advent of neural networks, exploring the challenges, innovations and applications of classical algorithms. This knowledge will provide a foundation for further exploration in the field of image processing and its evolution into the AI-driven era.

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3-day lecture series (1 of 3)

In a rainbow variant of the Turán problem, we consider $k$ graphs on the same set of vertices and want to determine the smallest possible number of edges in each graph, which guarantees the existence of a copy of a given graph $F$ containing at most one edge from each graph. In other words, we treat each of the $k$ graphs as a graph in one of the $k$ colors and consider how many edges in each color force a rainbow copy of a given graph $F$. In the talk, we will describe known results on the topic, as well as present recent developments, obtained jointly with Sebastian Babiński and Magdalen Prorok, solving the rainbow Turán problem for a path on 4 vertices and a directed triangle with any number of colors.

The well-known 1-2-3 Conjecture by Karoński, Łuczak and Thomason states that the edges of any connected graph with at least three vertices can be assigned weights 1, 2 or 3 so that for each edge $uv$ the sums of the weights at $u$ and at $v$ are distinct. The list version of the 1-2-3 Conjecture by Bartnicki, Grytczuk and Niwczyk states that the same holds if each edge $e$ has the choice of weights not necessarily from $\{1,2,3\}$, but from any set $\{x(e),y(e),z(e)\}$ of three real numbers. The goal of this talk is to survey developments on the 1-2-3 Conjecture, especially on the list version of the 1-2-3 Conjecture.

I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang.

A theoretical dynamical system is a pair (X,T) where X is a compact metric space and T is a self homeomorphism of X. The topological entropy of a theoretical dynamical system (X,T), first introduced in 1965 by Adler, Konheim and McAndrew, is a nonnegative real number that measures the complexity of the system. Systems with positive entropy are random in certain sense, and systems with zero entropy are said to be deterministic. To distinguish between deterministic systems, Huang and Ye (2009) introduced the concept of maximal pattern entropy of a theoretical dynamical system. At the heart of their argument is a Sauer-Shelah-type lemma. We will discuss this lemma and its surprising connection to a recent breakthrough in communication complexity. Joint work with Guorong Gao, Jie Ma, and Mingyuan Rong.