Georgia Institute of Technology College of Sciences

Cell cycle duration changes dramatically during development, starting out fast to generate cells quickly and slowing down over time as the organism matures. The cell cycle can also act as a transcriptional filter to control the expression of long gene transcripts which are partially transcribed in short cycles. Using mathematical simulations of cell proliferation, we identify an emergent property, that this filter can act as a tuning knob to control gene transcript expression, cell diversity and the number and proportion of different cell types in a tissue. Our predictions are supported by comparison to single-cell RNA-seq data captured over embryonic development. Additionally, evolutionary genome analysis shows that fast developing organisms have a narrow genomic distribution of gene lengths while slower developers have an expanded number of long genes. Our results support the idea that cell cycle dynamics may be important across multicellular animals for controlling gene transcript expression and cell fate.

Recording link: https://bluejeans.com/s/QhCWmELH6AC

Let A be a subset of the integers of size n. In 1983, Erdos and Szemeredi conjectured that either A+A or A*A must have size nearly n^2. We discuss ideas towards this conjecture, such as an older connection to incidence geometry as well as somewhat newer breakthroughs in additive combinatorics. We further highlight applications of the sum-product phenomenon. 

Link: https://bluejeans.com/398474745/0225

In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t \ge 1$. Hadwiger's Conjecture is a vast generalization of the Four Color Theorem and one of the most important open problems in graph theory. Only the cases when $t$ is at most 6 are known. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t (\log t)^{0.5})$ and hence is $O(t (\log t)^{0.5})$-colorable.  In a recent breakthrough, Norin, Song, and I proved that every graph with no $K_t$ minor is $O(t (\log t)^c)$-colorable for every $c > 0.25$,  Subsequently I showed that every graph with no $K_t$ minor is $O(t (\log \log t)^6)$-colorable.  Delcourt and I improved upon this further by showing that every graph with no $K_t$ minor is $O(t \log \log t)$-colorable. Our main technical result yields this as well as a number of other interesting corollaries.  A natural weakening of Hadwiger's Conjecture is the so-called Linear Hadwiger's Conjecture that every graph with no $K_t$ minor is $O(t)$-colorable.  We prove that Linear Hadwiger's Conjecture reduces to small graphs. In 2005, Kühn and Osthus proved that Hadwiger's Conjecture for the class of $K_{s,s}$-free graphs for any fixed positive integer $s \ge 2$. Along this line, we show that Linear Hadwiger's Conjecture holds for the class of $K_r$-free graphs for every fixed $r$.