Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, October 6, 2020.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 6, 2020

11:00 am in Zoom,Tuesday, October 6, 2020

#### Higher moments of primes in progressions

###### Daniel Fiorilli (CNRS and Université Paris-Saclay)

Abstract: Since the work of Barban, Davenport and Halberstam, the variance of primes in arithmetic progressions has been widely studied and continues to be an active topic of research. However, much less is known about higher moments. Hooley established a bound on the third moment, which was later sharpened by Vaughan for a variant involving a major arcs approximation. Little is known for moments of order four or higher, other than a conjecture of Hooley. In this talk I will discuss recent joint work with Régis de la Bretèche on weighted moments of moments of primes in progressions. In particular we will show how to deduce sharp unconditional omega results on all weighted even moments in certain ranges.

2:00 pm in Zoom,Tuesday, October 6, 2020

#### Subgraphs of large connectivity and chromatic number

###### Bhargav Narayanan (Rutgers University)

Abstract: Resolving a problem raised by Norin, we show that for each $k\in \mathbb{N}$, there exists an $f(k)\leq 7k$ such that every graph $G$ with chromatic number at least $f(k)+1$ contains a subgraph $H$ with both connectivity and chromatic number at least $k$. This result is best-possible up to multiplicative constants, and sharpens earlier results of Alon-Kleitman-Thomassen-Saks-Seymour from 1987 showing that $f(k)=O(k^3)$, and of Chudnovsky-Penev-Scott-Trotignon from 2013 showing that $f(k)=O(k^2)$.

For Zoom information please email Sean at SEnglish (at) illinois (dot) edu

2:00 pm in Zoom Meeting (email daesungk@illinois.edu for info),Tuesday, October 6, 2020

#### Degenerate diffusions and optimal convex bodies

###### Yair Shenfeld (MIT)

Abstract: It has been known since antiquity that the ball is the unique shape which minimizes surface area among all shapes of equal volume. Remarkably, if in addition to the volume, the average width of the shape is fixed, the optimal solutions become non-unique and non-smooth. This is just a very special instance of optimization problems arising in convex geometry whose optimal solutions are bizarre and still conjectural. In this talk, I will explain how the study of these optimal shapes is intertwined with the spectral structure of a certain diffusion operator on the sphere, and how we (with Ramon van Handel) solved many of these problems. No prior knowledge of convex geometry is assumed.