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)