Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, September 3, 2019.

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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, September 3, 2019

12:50 pm in 347 Altgeld Hall,Tuesday, September 3, 2019

#### Multi-frequency class averaging for three-dimensional cryo-electron microscopy

###### Zhizhen Jane Zhao (Illinois ECE)

Abstract: We introduce a novel intrinsic classification algorithm--multi-frequency class averaging (MFCA)--for clustering noisy projection images obtained from three-dimensional cryo-electron microscopy (cryo-EM) by the similarity among their viewing directions. This new algorithm leverages multiple irreducible representations of the unitary group to introduce additional redundancy into the representation of the transport data, extending and outperforming the previous class averaging algorithm that uses only a single representation. We will discuss the formal algebraic model and representation theoretic patterns of the proposed MFCA algorithm. We conceptually establish the consistency and stability of MFCA by inspecting the spectral properties of a generalized localized parallel transport operator on the two-dimensional unit sphere through the lens of Wigner D-matrices. We will also show how this algorithm can be applied to directly denoise the real data.

1:00 pm in 345 Altgeld Hall,Tuesday, September 3, 2019

#### Ergodic theorems and more fun with countable Borel equivalence relations

###### Jenna Zomback (UIUC Math)

Abstract: I will discuss my current and future work, which lies in the study of countable Borel equivalence relations (CBERs) and its applications to ergodic theory and measured group theory. In the first section of the talk, I will discuss a tiling result for amenable groups along Tempelman Følner sequences and explain how this result implies the corresponding pointwise ergodic theorem (this is joint work with Jonathan Boretsky). In the second section, I will introduce the notion of cost of an equivalence relation, and state a few important results in this field. In each of the two sections, I will state some proposed avenues for future work. This talk is part of a preliminary examination.

2:00 pm in 243 Altgeld Hall,Tuesday, September 3, 2019

#### Large Monochromatic Components in Sparse Random Hypergraphs

###### Sean English (Illinois Math)

Abstract: It is known, due to Gyárfás and Füredi, that for any $r$-coloring of the edges of $K_n$, there is a monochromatic component of order $(1/(r-1)+o(1))n$. Recently, Bal and DeBiasio, and independently Dudek and Prałat showed that the Erdős-Rényi random graph $\mathcal{G}(n,p)$ behaves very similarly with respect to the size of the largest monochromatic component. More precisely, it was shown that a.a.s. for any $r$-coloring of the edges of $\mathcal{G}(n,p)$ and arbitrarily small constant $\alpha>0$, there is a monochromatic component of order $(1/(r-1)-\alpha)n$, provided that the average degree goes to infinity with $n$. As before, this result is best possible.

In this talk we present a generalization of this result to hypergraphs. Specifically we show that in the $k$-uniform random hypergraph, $\mathcal{H}^{(k)}(n,p)$ a.a.s. for any $k$-coloring of the edges, there is a monochromatic component of order $(1-\alpha)n$. Furthermore, for any $k+1$ coloring, there is a monochromatic component of order $(1-\alpha)\frac{k}{k+1}n$. These results hold as long as the average degree goes to infinity.

It is also known Gyárfás, Sárközy and Szemerédi that the Ramsey number for loose cycles on $n$ vertices in $k$-uniform hypergraphs is asymptotically $\frac{2k-1}{2k-2}n$, which implies that in any $2$-coloring of $K^{(k)}_n$, for large $n$, we can find a loose cycle on about $\frac{2k-2}{2k-1}n$ vertices. We will present a generalization of this which shows that even if the host graph is $\mathcal{H}^{(k)}_{n,p}$, this result still holds a.a.s. provided that the average degree goes to infinity.

This project is joint work with Patrick Bennett, Louis Debiasio and Andrzej Dudek.