Department of

Mathematics


Seminar Calendar
for Algebra Seminar events the year of Tuesday, October 26, 2021.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
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Monday, August 23, 2021

5:00 pm in Zoom,Monday, August 23, 2021

Cluster Expansion Formulas from Snake Graphs

Elizabeth Kelley (University of Illinois )

Abstract: One of the most celebrated properties of cluster algebras is the Laurent Phenomenon, which states that every cluster variable can be written as a Laurent polynomial with integral coefficients in terms of any choice of cluster. Although the Laurent Phenomenon itself was proved in the original paper of Fomin and Zelevinsky, in 2002, the accompanying positivity property, which states that the expansion coefficients are in fact non-negative, was not proven in full generality until 2018. In 2011, Musiker, Schiffler, and Williams offered the first proof of positivity for the subclass of cluster algebras from surfaces via the construction of snake graphs, which can be used to give explicit combinatorial cluster expansion formulas. In this talk, we will use concrete examples to define cluster algebras from surfaces (both unpunctured and punctured), explain how to construct snake graphs, and then state the cluster expansion formula(s) of Musiker, Schiffler, and Williams. We will largely treat material found in "Positivity for cluster algebras from surfaces", https://arxiv.org/abs/0906.0748.

Wednesday, September 1, 2021

4:00 pm in Zoom seminar,Wednesday, September 1, 2021

Generalized Snake Graphs from Triangulated Orbifolds

Elizabeth Kelley (University of Illinois)

Abstract: Cluster algebras, as originally defined by Fomin and Zelevinsky, are characterized by binomial exchange relations. A natural generalization of cluster algebras, due to Chekhov and Shapiro, allows the exchange relations to have arbitrarily many terms. For generalized cluster algebras that can be modeled by unpunctured triangulated orbifolds, we generalize the snake graph construction of Musiker, Schiffler, and Williams and obtain explicit combinatorial formulas for the Laurent expansion of any arc or closed curve. For ordinary arcs, this gives a combinatorial proof of positivity for the associated generalized cluster algebra. This talk is based on joint work with Esther Banaian.

Wednesday, September 8, 2021

4:00 pm in Zoom seminar,Wednesday, September 8, 2021

Generalized Snake Graphs from Triangulated Orbifolds

Elizabeth Kelley (University of Illinois)

Abstract: We will continue the talk from last week. Cluster algebras, as originally defined by Fomin and Zelevinsky, are characterized by binomial exchange relations. A natural generalization of cluster algebras, due to Chekhov and Shapiro, allows the exchange relations to have arbitrarily many terms. For generalized cluster algebras that can be modeled by unpunctured triangulated orbifolds, we generalize the snake graph construction of Musiker, Schiffler, and Williams and obtain explicit combinatorial formulas for the Laurent expansion of any arc or closed curve. For ordinary arcs, this gives a combinatorial proof of positivity for the associated generalized cluster algebra. This talk is based on joint work with Esther Banaian.