CU Boulder, Fall 2019

R1. Notes on the Octonions by Salamon and Walpuski

R2. Some Remarks on ${\rm G}_2$ Structures by R. L. Bryant

9/4       Speaker: Andrew Stocker   (Summary: Let $V$ be a $n$-dimensional Hilbert space with a cross product. One can define an associated $3$-form $\phi(u,v,w):= \langle u\times v, w\rangle$. For any vector $u\ne 0\in V$, the restriction of $\iota(u)\phi$ to $\{u\}^\perp$ is symplectic, thus $n$ is odd, say $n = 2k+1$; furthermore, when $n\ge 3$, the $n$-form $(\iota(u)\phi)^{k-1}\wedge\phi$ corresponds to the same orientation for all $u\ne 0$, hence $n$ is of the form $4m+3$. An injectivity result in Clifford algebra implies that $n\le 7$. Consequently, cross product could only exist in dimensions $0,1,3$ or $7$. Explicit examples in these dimensions demonstrate the converse. This result is related to the real dimensions of the division algebras and the dimensions of the $n$-spheres that are parallelizable. )

9/12     Speaker: Andrew Stocker   (Summary: On a $7$-dimensional real vector space $V$, a special role is played by a nondegenerate $3$-form $\phi$. In fact, the $7$-form $\frac{1}{6}\iota(u)\phi\wedge\iota(u)\phi\wedge\phi$ is never zero for $u\ne 0\in V$. This allows us to choose a volume form $\nu$ on $V$ such that the symmetric bilinear form defined by $g(u,v) = \frac{1}{6\nu}\iota(u)\phi\wedge\iota(v)\phi\wedge\phi$ is positive-definite. By scaling $\nu$, we can always arrange that $\nu = \nu_g$, the oriented volume form associated to $g$. This uniquely determines an inner product $g_\phi$ on $V$ in the sense that the bilinear map $(u,v)\mapsto u\times v$ determined by $\phi(u,v,w) = g_\phi(u\times v, w)$ is a cross product. A final note is that ${\rm GL}(V)$ acts transitively on the space of nondegenerate $3$-forms on $V$.)

9/19    Speaker: Evan Wickenden   (Summary: Let $V$ be a $7$-dimensional real vector space with an associative calibration $\phi\in \Lambda^3V^*$ with its uniquely induced inner product. The induced cross product can be used to define the associator bracket on $V$. A $3$-dimensional vector subspace $\Lambda\subset V$ is closed under the cross product if and only if the associator bracket vanishes when restricted to $\Lambda$. Such a $\Lambda$ is called an associative subspace. One can also define the coassociator bracket on $V$. A $4$-dimensional vector subspace $H\subset V$ pulls back $\phi$ to be zero if and only if the coassociator bracket restricts to $H$ to vanish. Such an $H$ is called a coassociative subspace. The orthogonal complement of an associative subspace must be coassociative, and vice versa. The Hodge dual of $\phi$, say, $\psi\in \Lambda^4V^*$, is known as the coassociative calibration induced by $\phi$. Two associative calibrations $\phi,\hat\phi$ with the same induced coassociative calibration can at most differ by a sign. )

9/26    Speaker: Evan Wickenden   (Summary: In this talk, Evan presented the proof of the theorem: Two associative calibrations $\phi,\hat\phi$ with the same induced coassociative calibration can at most differ by a sign. Furthermore, he presented a proof of the following theorem: Let $V$ be a vector space, $\dim_\mathbb{R}V = 7$. Given $\psi\in \Lambda^4V^*$, $\pm \psi$ is the induced coassociative calibration for some nondegenerate $\phi\in \Lambda^3V^*$ if and only if, for any linearly independent $\alpha,\beta\in V^*$, $\exists \gamma\in V^*$ such that $\alpha\wedge\beta\wedge\gamma\wedge\psi\ne 0$.)

10/3    Speaker: Taylor Klotz   (Summary: A normed algebra is a finite-dimensional Hilbert space $W$ with a `product' and a unit $1\in W$ that satisfy $|uv| = |u||v|$ and $1u=u1 = u$ for any $u,v\in W$. A $n$-dimensional normed algebra is intimately related to an $(n-1)$-dimensional Hilbert space with a compatible cross product. This observation leads to the classification of normed algebras into being the reals, complexes, quaternions and octonions. Furthermore, on a $8$-dimensional normed algebra $W$, one can define 'triple cross product' and 'four-fold cross product'. Their relation with other operations and quantities, in particular, the associative and coassociative calibrations on $1^\perp$ are discussed.)

10/10  (No meeting on this day due to lockdown of the building)

10/17  Speaker: Taylor Klotz   (Summary: Analogous to the usual cross product, one can define a 'triple cross product' (t.c.p) on a Hilbert space $W$ in a axiomatic way. By specifying a unit vector $e\in W$, a t.c.p. on $W$ induces a usual cross product on $e^\perp$ compatible with the induced metric. This tells us that the dimensions that support a t.c.p. are $0,1,2,4,8$. In fact, from any t.c.p. on an $8$-dimensional $W$ and any unit vector $e\in W$, one can explicitly introduce a product on $W$, making it a normed algebra with $e$ being the unit.)

10/24  (Talk rescheduled.)

10/31  Speaker: Andrew Stocker   (Summary: On an $8$-dimensional vector space $W$, a nondegenerate $4$-form $\Phi$ is called a Cayley calibration if there exists an inner product $\langle,\rangle$ on $W$ such that the multilinear map $(u,v,w)\mapsto P(u,v,w)$ defined by $\langle x,P(u,v,w)\rangle := \Phi(x,u,v,w)$ is a triple cross product. There is a somewhat technical criterion for $\Phi$ being Cayley. Positivity of a Cayley form can be defined. A theorem says that ${\rm GL}(W)$ acts transitively on the spaces of positive Cayley forms.)

11/6    Speaker: Andrew Stocker   (Summary: Let $V$ be a $7$-dimensional Hilbert space with a cross product. Let $\phi$ be the corresponding associative calibration. The subgroup of ${\rm GL}(V)$ formed by the stabilizers of $\phi$ is called ${\rm G}_2$. This talk presents the following aspects of ${\rm G}_2$: that ${\rm G}_2$ is a $14$-dimensional Lie group; that it is simple, connected and simply connected; and that it acts transitively on $\mathbb{S}^6$ and on the Stiefel manifold of orthonormal pairs in $V$.)

11/14  Speaker: Evan Wickenden   

11/21  Speaker: Evan Wickenden

12/5    Speaker: Taylor Klotz


$*$ Summary by Y. Hu.