Department of

# Mathematics

Seminar Calendar
for events the day of Wednesday, October 2, 2013.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
September 2013          October 2013          November 2013
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2  3  4  5  6  7          1  2  3  4  5                   1  2
8  9 10 11 12 13 14    6  7  8  9 10 11 12    3  4  5  6  7  8  9
15 16 17 18 19 20 21   13 14 15 16 17 18 19   10 11 12 13 14 15 16
22 23 24 25 26 27 28   20 21 22 23 24 25 26   17 18 19 20 21 22 23
29 30                  27 28 29 30 31         24 25 26 27 28 29 30

Wednesday, October 2, 2013

3:00 pm in 143 Altgeld Hall,Wednesday, October 2, 2013

#### The Boson-Fermion Correspondence and Plücker Relations

Abstract: I will give an overview of the Boson-Fermion correspondence which gives an isomorphism between the representation of the Heisenberg Lie algebra on $\mathbb{C}[x_{1},x_{2,}\cdots]$ and the representation of the Heisenberg Lie algebra on an infinite dimensional wedge space. I will briefly discuss how this isomorphism can be extended to isomorphisms between representations of larger Lie algebras. I will then discuss how to use the Boson-Fermion correspondence to understand the orbit of 1 under the action of the Lie group $GL_{\infty}$, and will mention the fact that elements in the orbit are solutions to an infinite set of differential equations. This parallels our previous discussion of representations of the Lie algebra $sl_{n}$ on the finite exterior algebra, $\Lambda(V)=\oplus_{k=0}^{n}\Lambda^{k}V$, which we used to obtain the Plücker relations. Our discussion of the Boson-Fermion correspondence and its uses will follow the treatment given in Kac and Raina's Highest Weight Representations of Infinite Dimensional Lie Algebras.

3:00 pm in 347 Altgeld Hall,Wednesday, October 2, 2013

#### Constructions for Quasicategories

###### Mychael Sanchez (UIUC Math)

Abstract: I'll talk about functors and limits in the setting of quasicategories. I'll also give examples of quasicategories and talk about homotopy coherent diagrams in simplicially enriched categories and their relation to quasicategories.

4:00 pm in 245 Altgeld Hall,Wednesday, October 2, 2013