For a graph $F$, the Turán number is the maximum number of edges in an $n$-vertex simple graph not containing $F$. The celebrated Erdős-Stone-Simonovits Theorem gives that \[ \text{ex}(n,F)=\bigg(1-\frac{1}{\chi(F)-1}+o(1)\bigg)\binom{n}{2},\] where $\chi(F)$ is the chromatic number of $H$. This theorem asymptotically solves the problem when $\chi(F)\geqslant 3$. In case of bipartite graphs $F$, not even the order of magnitude is known in general. In this talk, I will introduce some recent progress on Turán numbers of bipartite graphs and related generalizations and discuss several methods developed in recent years. Finally, I will introduce some interesting open problems on this topic.

The driving passion of molecular cell biologists is to understand the molecular mechanisms that control important aspects of cell physiology, but this ambition is – paradoxically – limited by the very wealth of molecular details currently known about these mechanisms. Their complexity overwhelms our intuitive notions of how molecular regulatory networks might respond under normal and stressful conditions. To make progress we need a new paradigm for connecting molecular biology to cell physiology. I will outline an approach that uses precise mathematical methods to associate the qualitative features of dynamical systems, as conveyed by ‘bifurcation diagrams’, with ‘signal–response’ curves measured by cell biologists.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Cell growth, DNA replication, mitosis and division are the fundamental processes by which life is passed on from one generation of eukaryotic cells to the next. The eukaryotic cell cycle is intrinsically a periodic process but not so much a ‘clock’ as a ‘copy machine’, making new daughter cells as warranted. Cells growing under ideal conditions divide with clock-like regularity; however, if they are challenged with DNA-damaging agents or mitotic spindle disruptors, they will not progress to the next stage of the cycle until the damage is repaired. These ‘decisions’ (to exit and re-enter the cell cycle) are essential to maintain the integrity of the genome from generation to generation. A crucial challenge for molecular cell biologists in the 1990s was to unravel the genetic and biochemical mechanisms of cell cycle control in eukaryotes. Central to this effort were biochemical studies of the clock-like regulation of ‘mitosis promoting factor’ during synchronous mitotic cycles of fertilized frog eggs and genetic studies of the switch-like regulation of ‘cyclin-dependent kinases’ in yeast cells. The complexity of these control systems demands a dynamical approach, as described in the first lecture. Using mathematical models of the control systems, I will uncover some of the secrets of cell cycle ‘clocks’ and ‘switches’.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

We determine the maximum number of copies of $K_{s,s}$ in a $C_{2s+2}$-free $n$-vertex graph for all integers $s \ge 2$ and sufficiently large $n$. Moreover, for $s\in\{2,3\}$ and any integer $n$ we obtain the maximum number of cycles of length $2s$ in an $n$-vertex $C_{2s+2}$-free bipartite graph. This is joint work with Ervin Győri (Renyi Institute), Zhen He (Tsinghua University), Zequn Lv (Tsinghua University), Casey Tompkins (Renyi Institute), Kitti Varga (Technical University of Budapest BME), and Xiutao Zhu (Nanjing University).

TBA

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Cellular, chemical, and population processes are all often represented via networks that describe the interactions between the different population types (typically called the “species”). If the counts of the species are low, then these systems are often modeled as continuous-time Markov chains on the d-dimensional integer lattice (with d being the number of species), with transition rates determined by stochastic mass-action kinetics. A natural (broad) mathematical question is: how do the qualitative properties of the dynamical system relate to the graph properties of the network? For example, it is of particular interest to know which graph properties imply that the stochastically modeled reaction network is positive recurrent, and therefore admits a stationary distribution. After a general introduction to the models of interest, I will discuss this problem, giving some of the known results. I will also discuss recent progress on the Chemical Recurrence Conjecture, which has been open for decades, which is the following: if each connected component of the network is strongly connected, then the associated stochastic model is positive recurrent.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

We’re all familiar with sleep, but how can we mathematically model it? And what determines how long and when we sleep? In this talk I’ll introduce the nonsmooth coupled oscillator systems that form the basis of current models of sleep-wake regulation and discuss their dynamical behaviour. I will describe how we are using models to unravel environmental, societal and physiological factors that determine sleep timing and outline how we are using models to inform the quantitative design of light interventions for mental health disorders and address contentious societal questions such as whether to move school start time for adolescents.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)

Several years ago, Chi Li introduced the local volume of a klt singularity in his work on K-stability. The local-global analogy between klt singularities and Fano varieties, together with recent study in K-stability lead to the conjecture that klt singularities whose local volumes are bounded away from zero are bounded up to special degeneration. In this talk, I will discuss some recent work on this conjecture through the minimal log discrepancies of Kollár components. * Zoom information will not be provided. Please send an email to Jinhyung Park if you are interested in.

The Structural Theorem of the Graph Minors series of Robertson and Seymour asserts that, for every $t\in\mathbb{N},$ there exists some constant $c_{t}$ such that every $K_{t}$-minor-free graph admits a tree decomposition whose torsos can be transformed, by the removal of at most $c_{t}$ vertices, to graphs that can be seen as the union of some graph that is embeddable to some surface of Euler genus at most $c_{t}$ and "at most $c_{t}$ vortices of depth $c_{t}$". Our main combinatorial result is a "vortex-free" refinement of the above structural theorem as follows: we identify a (parameterized) graph $H_{t}$, called shallow vortex grid, and we prove that if in the above structural theorem we replace $K_{t}$ by $H_{t},$ then the resulting decomposition becomes "vortex-free". Up to now, the most general classes of graphs admitting such a result were either bounded Euler genus graphs or the so called single-crossing minor-free graphs. Our result is tight in the sense that, whenever we minor-exclude a graph that is not a minor of some $H_{t},$ the appearance of vortices is unavoidable. Using the above decomposition theorem, we design an algorithm that, given an $H_{t}$-minor-free graph $G$, computes the generating function of all perfect matchings of $G$ in polynomial time. This algorithm yields, on $H_{t}$-minor-free graphs, polynomial algorithms for computational problems such as the {dimer problem, the exact matching problem}, and the computation of the permanent. Our results, combined with known complexity results, imply a complete characterization of minor-closed graphs classes where the number of perfect matchings is polynomially computable: They are exactly those graph classes that do not contain every $H_{t}$ as a minor. This provides a sharp complexity dichotomy for the problem of counting perfect matchings in minor-closed classes. This is joint work with Dimitrios M. Thilikos.

Katona's intersection theorem states that every intersecting family $\mathcal F\subseteq[n]^{(k)}$ satisfies $\vert\partial\mathcal F\vert\geq\vert\mathcal F\vert$, where $\partial\mathcal F=\{F\setminus x:x\in F\in\mathcal F\}$ is the shadow of $\mathcal F$. Frankl conjectured that for $n>2k$ and every intersecting family $\mathcal F\subseteq [n]^{(k)}$, there is some $i\in[n]$ such that $\vert \partial \mathcal F(i)\vert\geq \vert\mathcal F(i)\vert$, where $\mathcal F(i)=\{F\setminus i:i\in F\in\mathcal F\}$ is the link of $\mathcal F$ at $i$. Here, we prove this conjecture in a very strong form for $n> \binom{k+1}{2}$. In particular, our result implies that for any $j\in[k]$, there is a $j$-set $\{a_1,\dots,a_j\}\in[n]^{(j)}$ such that \[ \vert \partial \mathcal F(a_1,\dots,a_j)\vert\geq \vert\mathcal F(a_1,\dots,a_j)\vert.\]A similar statement is also obtained for cross-intersecting families.

The activation of Ras depends upon the translocation of its guanine nucleotide exchange factor, Sos, to the plasma membrane. Moreover, artificially inducing Sos to translocate to the plasma membrane is sufficient to bring about Ras activation and activation of Ras’s targets. There are many other examples of signaling proteins that must translocate to the membrane in order to relay a signal. One attractive idea is that translocation promotes signaling by bringing a protein closer to its target. However, proteins that are anchored to the membrane diffuse more slowly than cytosolic proteins do, and it is not clear whether the concentration effect or the diffusion effect would be expected to dominate. Here we have used a reconstituted, controllable system to measure the association rate for the same binding reaction in 3D vs. 2D to see whether association is promoted, and, if so, how.

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This talk will be presented online. ZOOM ID: 997 8258 4700 (Biomedical Mathematics Online Colloquium), (pw: 1234)