Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, October 1, 2015.

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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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Thursday, October 1, 2015

11:00 am in 241 Altgeld Hall,Thursday, October 1, 2015

#### On some problems of Cameron and Erdős

###### József Balogh (UIUC Math)

Abstract: Cameron and Erdős proposed several of enumeration problems in additive combinatorics, for example what is the number of sum-free sets in [n], or what is the number of sets in [n] which do not contain k-term arithmetic progression. I plan to survey the recent progress on these type of questions, focusing on the applications of a recent powerful tool, the hyper graph container lemma of Balogh-Morris-Samotij, and of Saxton-Thomason. It is joint work with H. Liu, M. Sharifzadeh, and A. Treglown.

1:00 pm in Altgeld Hall 243,Thursday, October 1, 2015

#### Probability groups

###### Anush Tserunyan (UIUC Math)

Abstract: I will introduce a class of groups equipped with an invariant probability measure that respects the group structure in an appropriate sense; call such groups probability groups. This class contains all compact groups and is closed under taking ultraproducts with the induced Loeb measure. I will discuss the use of probability groups as a potential alternative to Furstenberg's correspondence principle. As an example, I will define a notion of mixing for probability groups and mention a double recurrence result for mixing probability groups that generalizes a theorem of Bergelson-Tao proved for ultra quasirandom groups, nevertheless having a considerably shorter proof.

1:00 pm in 347 Altgeld Hall,Thursday, October 1, 2015

#### Anomalous unrooted gene trees

###### James Degnan   [email]

Abstract: The coalescent and multispecies coalescent model rooted genealogies backward through time. Often, the direction of time is unknown in trees estimated from molecular sequences due to reversible mutation models, absence of an appropriate outgroup, and the absence of the molecular clock. In this talk, probabilities of unrooted gene tree topologies under the multispecies coalescent are considered. The main result is that for any species tree topology with seven or more taxa, there exist branch lengths such that there are unrooted gene tree topologies that are more likely than the one that has the same unrooted topology as the species tree. Species trees with such anomalous unrooted gene trees (AUGTs) are characterized for trees with five and six taxa, and patterns of branch lengths leading to AUGTs and rooted anomalous gene trees are explored. The results could be useful for understanding gene tree discordance and designing simulations studies for inferring challenging species trees.

2:00 pm in 347 Altgeld Hall,Thursday, October 1, 2015

#### Transfers

###### Cary Malkiewich   [email] (UIUC Math)

Abstract: A transfer map is a map that goes the "wrong way." So if E -> B is a covering space, a transfer is some kind of map from B back to E. The intuition is that you take each point of B to the sum of its preimages in E. Of course, this requires that there are only finitely many preimages, but more importantly, we have to pass to some context where that sum makes sense. The easy way out is to use homology, but a much more interesting approach uses spectra. I'll give a couple of different definitions and show that they give the same map up to homotopy. Then I'll review duality of spectra, show how to define Becker and Gottlieb's generalization of this transfer map, and talk about some open problems.

4:00 pm in 245 Altgeld Hall,Thursday, October 1, 2015

#### Fall Department Faculty Meeting

Abstract: The Fall Department Faculty Meeting will be held at 4 p.m. in 245 Altgeld Hall, followed by a reception.