Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, September 1, 2020.

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Tuesday, September 1, 2020

11:00 am in online Zoom, people may self-subscribe,Tuesday, September 1, 2020

Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros

Kevin Ford (UIUC Math)

Abstract: Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.

2:00 pm in Zoom,Tuesday, September 1, 2020

Splits with Forbidden Graphs

Ryan R. Martin (Iowa State University)

Abstract: In this talk, we fix a graph $H$ and ask into how many vertices each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex set is a pairwise disjoint union of $n$ parts of size at most $k$, such that there is an edge between any two distinct parts. Let $$ f(n,H) = \min \{k \in \mathbb N : \mbox{there is an $(n,k)$-graph $G$ such that $H\not\subseteq G$}\} . $$ Barbanera and Ueckerdt observed that $f(n, H)=2$ for any graph $H$ that is not bipartite. If a graph $H$ is bipartite and has a well-defined Turan exponent, i.e., ${\rm ex}(n, H) = \Theta(n^r)$ for some $r$, we show that $\Omega (n^{2/r-1}) = f(n, H) = O (n^{2/r -1} \log ^{1/r} n)$. We extend this result to all bipartite graphs for which upper and a lower Turan exponents do not differ by much. In addition, we prove that $f(n, K_{2,t}) =\Theta(n^{1/3})$ for any fixed integer $t\geq 2$.

This is joint work with Maria Axenovich, Karlsruhe Institute of Technology.

For the Zoom ID, please contact Sean at SEnglish (at) illinois (dot) edu.