Department of

# Mathematics

Seminar Calendar
for events the day of Thursday, December 6, 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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Thursday, December 6, 2018

11:00 am in 241 Altgeld Hall,Thursday, December 6, 2018

#### Divisibility of the central binomial coefficients

###### Kevin Ford (Illinois Math)

Abstract: Abstract:That $(n+1)| binom{2n}{n}$ for every positive integer $n$ is a consequence of the integrality of the Catalan numbers. Pomerance conjectured that $n| binom{2n}{n}$ for a set of positive density. In joint work with Sergei Konyagin, we prove Pomerance's conjecture and give an explicit form of the density. The proof uses a new method for accounting for the largest prime factors of integers.

12:30 pm in 464 Loomis,Thursday, December 6, 2018

#### To Be Announced

###### Jennie Tracshen (University of Massachusetts at Amherst)

3:00 pm in 345 Altgeld Hall,Thursday, December 6, 2018

#### Ising model and total positivity

###### Pavel Galashin (MIT)

Abstract: The Ising model, introduced in 1920, is one of the most well-studied models in statistical mechanics. It is known to undergo a phase transition at critical temperature, and has attracted considerable interest over the last two decades due to special properties of its scaling limit at criticality. The totally nonnegative Grassmannian is a subset of the real Grassmannian introduced by Postnikov in 2006. It arises naturally in Lusztig's theory of total positivity and canonical bases, and is closely related to cluster algebras and scattering amplitudes. I will give some background on the above objects and then explain a precise relationship between the planar Ising model and the totally nonnegative Grassmannian, obtained in our recent work with P. Pylyavskyy. We will see how various known results from total positivity provide answers to old and new questions regarding the Ising model. We will also explore the topology of the underlying spaces (joint work with S. Karp and T. Lam), and discuss how several other topics (such as electrical networks and the amplituhedron) fit into our story.

4:00 pm in 314 Altgeld Hall,Thursday, December 6, 2018

#### Department-wide meeting to discuss the Altgeld-Illini building projects.

Abstract: All members of the department of mathematics are welcome to attend a department-wide meeting to discuss the Altgeld-Illini building projects.