A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}3\] for all sufficiently large $t$.  Our proof involves a combination of results on the chromatic number of triangle-sparse graphs. Joint work with Jacob Fox and Jonathan Tidor.

Given a set of lines in $\mathbb R^d$, a joint is a point contained in d linearly independent lines. Guth and Katz showed that N lines can determine at most $O(N^{3/2})$ joints in $\mathbb R^3$ via the polynomial method. Yu and I proved a tight bound on this problem, which also solves a conjecture proposed by Bollobás and Eccles on the partial shadow problem. It is surprising to us that the only known proof of this purely extremal graph theoretic problem uses incidence geometry and the polynomial method.

TBD

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

The Nagata Conjecture governs the minimal degree required for a plane algebraic curve to pass through a collection of $r$ general points in the projective plane $P^2$ with prescribed multiplicities. The "SHGH" Conjecture governs the dimension of the linear space of these polynomials. We formulate transcendental versions of these conjectures in term of pluripotential theory and we're making some progress.

For $d\ge 2$ and an odd prime power $q$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field $\mathbb{F}_q$. The distance between two points $(x_1,\ldots,x_d)$ and $(y_1,\ldots,y_d)$ is defined to be $\sum_{i=1}^d (x_i-y_i)^2$. An influential result of Iosevich and Rudnev is: if $E \subset \mathbb{F}_q^d$ is sufficiently large and $t \in \mathbb{F}_q$, then there are a pair of points $x,y \in E$ such that the distance between $x$ and $y$ is $t$. Subsequent works considered embedding graphs of distances, rather than a single distance. I will discuss joint work with Debsoumya Chakraborti, in which we show that every sufficiently large subset of $\mathbb{F}_q^d$ contains every nearly spanning tree of distances with bounded degree in each distance. The main novelty in this result is that the distance graphs we find are nearly as large as the set $S$ itself, but even for smaller distance trees our work leads to quantitative improvements to previously known bounds. A key ingredient in our proof is a new colorful generalization of a classical result of Haxell on finding nearly spanning bounded-degree trees in expander graphs. This is joint work with Debsoumya Chakraborti.

Dirac's theorem determines the sharp minimum degree threshold for graphs to contain perfect matchings and Hamiltonian cycles. There have been various attempts to generalize this theorem to hypergraphs with larger uniformity by considering hypergraph matchings and Hamiltonian cycles. We consider another natural generalization of the perfect matchings, Steiner triple systems. As a Steiner triple system can be viewed as a partition of pairs of vertices, it is a natural high-dimensional analogue of a perfect matching in graphs. We prove that for sufficiently large integer $n$ with $n \equiv 1 \text{ or } 3 \pmod{6},$ any $n$-vertex $3$-uniform hypergraph $H$ with minimum codegree at least $\left(\frac{3 + \sqrt{57}}{12} + o(1) \right)n = (0.879... + o(1))n$ contains a Steiner triple system. In fact, we prove a stronger statement by considering transversal Steiner triple systems in a collection of hypergraphs. We conjecture that the number $\frac{3 + \sqrt{57}}{12}$ can be replaced with $\frac{3}{4}$ which would provide an asymptotically tight high-dimensional generalization of Dirac's theorem.

I will report on some recent results on modelling the heart, the external circulation, and their application to problems of clinical relevance. I will show that a proper integration between PDE-based and machine-learning algorithms can improve the computational efficiency and enhance the generality of our iHEART simulator.

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

Given a hypergraph $H=(V,E)$, we say that $H$ is (weakly) $m$-colorable if there is a coloring $c:V\to [m]$ such that every hyperedge of $H$ is not monochromatic. The (weak) chromatic number of $H$, denoted by $\chi(H)$, is the smallest $m$ such that $H$ is $m$-colorable. A vertex subset $T \subseteq V$ is called a transversal of $H$ if for every hyperedge $e$ of $H$ we have $T\cap e \ne \emptyset$. The transversal number of $H$, denoted by $\tau(H)$, is the smallest size of a transversal in $H$. The transversal ratio of $H$ is the quantity $\tau(H)/|V|$ which is between 0 and 1. Since a lower bound on the transversal ratio of $H$ gives a lower bound on $\chi(H)$, these two quantities are closely related to each other. Upon my previous presentation, which is based on the joint work with Joseph Briggs and Michael Gene Dobbins (https://www.youtube.com/watch?v=WLY-8smtlGQ), we update what is discovered in the meantime about transversals and colororings of geometric hypergraphs. In particular, we focus on chromatic numbers of $k$-uniform hypergraphs which are embeddable in $\mathbb{R}^d$ by varying $k$, $d$, and the notion of embeddability and present lower bound constructions. This result can also be regarded as an improvement upon the research program initiated by Heise, Panagiotou, Pikhurko, and Taraz, and the program by Lutz and Möller. We also present how this result is related to the previous results and open problems regarding transversal ratios. This presentation is based on the joint work with Eran Nevo.

TBD

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

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction. Here we present a systematic, computationally efficient likelihood-based workflow that addresses all three steps in a unified way. Recently developed methods for constructing profile-wise prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters, and then combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. We apply our methods to a range of synthetic data and real-world ecological data describing re-growth of coral reefs on the Great Barrier Reef after some external disturbance, such as a tropical cyclone or coral bleaching event.

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

Determining the density required to ensure that a host graph G contains some target graph as a subgraph or minor is a natural and well-studied question in extremal combinatorics. The celebrated 50-year-old Erdős-Sós conjecture states that for every k, if G has average degree exceeding k-2 then it contains every tree T with k vertices as a subgraph. This is tight as the clique with k-1 vertices contains no tree with k vertices as a subgraph. We present some variants of this conjecture. We first consider replacing bounds on the average degree by bounds on the minimum and maximum degrees. We then consider replacing subgraph by minor in the statement.

The Hypothalamic-Pituitary-Adrenal (HPA) axis is the key regulatory pathway responsible for maintaining homeostasis under conditions of real or perceived stress. Endocrine responses to stressors are mediated by adrenocorticotrophic hormone (ACTH) and corticosteroid (CORT) hormones. In healthy, non-stressed conditions, ACTH and CORT exhibit highly correlated ultradian pulsatility with an amplitude modulated by circadian processes. Disruption of these hormonal rhythms can occur as a result of stressors or in the very early stages of disease. Despite the fact that misaligned endocrine rhythms are associated with increased morbidity, a quantitative understanding of their mechanistic origin and pathogenicity is missing. Mathematically, the HPA axis can be understood as a dynamical system that is optimised to respond and adapt to perturbations. Normally, the body copes well with minor disruptions, but finds it difficult to withstand severe, repeated or long-lasting perturbations. Whilst a healthy HPA axis maintains a certain degree of robustness to stressors, its fragility in diseased states is largely unknown, and this understanding constitutes a critical step toward the development of digital tools to support clinical decision-making. This talk will explore how these challenges are being addressed by combining high-resolution biosampling techniques with mathematical and computational analysis methods. This interdisciplinary approach is helping us quantify the inter-individual variability of daily hormone profiles and develop novel “dynamic biomarkers” that serve as a normative reference and to signal endocrine dysfunction. By shifting from a qualitative to a quantitative description of the HPA axis, these insights bring us a step closer to personalised clinical interventions for which timing is key.

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