Department of

Mathematics


Seminar Calendar
for events the day of Tuesday, April 13, 2021.

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Tuesday, April 13, 2021

11:00 am in Zoom,Tuesday, April 13, 2021

Bombieri-Vinogradov type theorems for primes with a missing digit

Kunjakanan Nath (U. Montreal math)

Abstract: One of the fundamental questions in number theory is to find primes in any subset of the natural numbers. In general, it's a difficult question and leads to open problems like the twin prime conjecture, Landau's problem and many more. Recently, Maynard considered the set of natural numbers with a missing digit and showed that it contains infinitely many primes whenever the base b ≥ 10. In fact, he has established the right order of the upper and the lower bounds when the base b = 10 and an asymptotic formula whenever b is large (say 2 10⁶). In this talk, we will consider the distribution of primes with a missing digit in arithmetic progressions for base b large enough. In particular, we will show an analog of the Bombieri-Vinogradov type theorems for primes with a missing digit for large base b. The proof relies on the circle method, which in turn is based on the Fourier structure of the digital set and the Fourier transform of primes over arithmetic progressions on an average. Finally, we will give its application to count the primes of the form p = 1 + m + n with a missing digit for a large base.

12:30 pm in Zoom,Tuesday, April 13, 2021

Poverty and Pandemic: Tackling current issues using actuarial methodologies

Dr. Sooie-Hoe Loke, assistant professor (Central Washington University)

Abstract: Traditionally, actuarial science involves quantifying risks in the insurance and finance industry. Its scope has broadened tremendously since its introduction as a formal mathematical discipline in the 17th century. In this talk, we discuss several applications, particularly to some current issues in our society today, such as poverty and pandemic. Questions welcomed at the end. Join the meeting at: https://illinois.zoom.us/j/87086696430?pwd=aHFaeGV0Z29PUGdpVWlDMTlzM3BoZz09.

2:00 pm in Zoom,Tuesday, April 13, 2021

Hexagon-Free Planar Graphs

Ervin Gyori (Renyi Institute for Mathematics)

Abstract: In this talk, we concentrate on determining the maximum number of edges in a hexagon-free planar graphs and we would like to show the nature of these problems (difficulties, necessary results and conjectures). Let $\mathrm{ex}_\mathcal{P}(n, H)$ denote the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. Dowden obtained exact results for $\mathrm{ex}_\mathcal{P}(n, C_4)$ and $\mathrm{ex}_\mathcal{P}(n, C_5)$, but the case of longer cycles remained open. Later on, Y. Lan, et al. proved that $\mathrm{ex}_\mathcal{P}(n, C_6)\leq \frac{18(n−2)}7$. In this talk I plan to sketch the proof of the tight result $\mathrm{ex}_\mathcal{P}(n, C_6)\leq \frac{5}2n-7$, for all $n\geq 18$, and show infinitely many examples when it is tight. Based on them, we raise a conjecture on $\mathrm{ex}_\mathcal{P}(n, C_k)$, for $k\geq 7$.

Joint work with Debarun Ghosh (CEU), Ryan R. Martin (Iowa State Univ.), Addisu Paulos (CEU), Chuanqi Xiao(CEU)

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