Department of

# Mathematics

Seminar Calendar
for events the day of Tuesday, April 10, 2018.

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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, April 10, 2018

12:00 pm in 243 Altgeld Hall,Tuesday, April 10, 2018

#### Central Limit Theorem for odometers and B-free integers

###### Francesco Cellarosi (Queens University Math)

Abstract: Odometers (or von Neumann–Kakutani adding machines) are classical examples of dynamical systems of low complexity, much alike irrational rotations of the circle. We consider generalized adding machines. In spite of their rigid behaviour (zero entropy, not weakly mixing), we are able to prove a Central Limit Theorem for the ergodic sums corresponding to certain (randomly chosen) observables, generalizing the work of M.B. Levin and E. Merzbach.
Time permitting, I will describe the connections of odometers to the dynamical systems naturally arising when studying the statistical properties of B-free integers and explain why it would be interesting to obtain a central limit theorem for these systems. Joint work with Maria Avdeeva.

1:00 pm in 345 Altgeld Hall,Tuesday, April 10, 2018

#### "On dp-minimal ordered structures" by P. Simon (part 4)

###### Elliot Kaplan (Illinois Math)

Abstract: We continue through "On dp-minimal ordered structures" by P. Simon [arXiv].

3:00 pm in 243 Altgeld Hall,Tuesday, April 10, 2018

#### Equal sums of higher powers of binary quadratic forms, II

###### Bruce Reznick (UIUC)

Abstract: A continuation of last week, but newcomers are welcome and will be brought up to speed. I promise to completely satisfy any curiosity you might have about the representation of binary sextic forms as a sum of two cubes of binary forms and as a sum of three cubes of binary forms. A theorem about universal representations as a sum of three cubes resolves the first non-trivial case of a conjecture of Boris Shapiro.

3:00 pm in 241 Altgeld Hall,Tuesday, April 10, 2018

#### Directed hypergraphs

###### Gyorgy Turan (UIC Math)

Abstract: Directed graphs can be generalized to directed hypergraphs in different ways. The version where hyperedges can have several vertices in their tail, but only a single head, comes up in many contexts, such as reasoning with implications and closure operators. Paths are defined using a forward chaining process. We discuss some extremal, algorithmic and probabilistic aspects of directed hypergraphs and mention several open problems.

4:00 pm in 1 Illini Hall,Tuesday, April 10, 2018

#### GIT quotients of flag varieties

###### Joshua Wen (UIUC)

Abstract: As we’ve seen, GIT quotients depend on a choice of line equivariant line bundle, and varying this choice can lead to drastic or subtle changes between quotients. After introducing a framework for ‘variation of GIT’ by Dolgachev and Hu, I want to consider a case of the flag variety and its action either by a torus or semisimple group. Here, one already knows many equivariant line bundles, and studying dimensions of invariant sections leads to results of representation-theoretic significance.