In order to study a certain 3-dimensional analogue of the Mandelbrot set, I needed a notion of holomorphy for the corresponding 3-dimensional real algebra. While this motivation (to generalize the basic elements of complex dynamics) is analytic, the focus of this talk will be the connection between holomorphic functions over a real algebra and first-order properties of the algebra. It turns out that, under the assumption of a single (left or right) divisible element, the holomorphic functions over an arbitrary real algebra form a vector space. Properties such as the Product Rule and Chain Rule are equivalent to purely algebraic properties of the underlying real algebra. When our real algebra has a multiplicative identity element, the Product Rule is precisely equivalent to being both commutative and associative. I will present proofs of some of these statements, which are quite succinct once the proper groundwork has been done.
Holomorphy over real algebras
Tue, Oct. 13 1:30pm (Math 3…
Homotopy
Theo Johnson-Freyd (Perimeter Institute for Theoretical Physics and Dalhousie University)
X
Abstract: Kitaev has argued that the spaces of n-dimensional invertible phases of matter, as n varies, fit together into the structure of an Omega-spectrum. It follows that each n-dimensional phase determines a (possibly trivial) character on the nth stable homotopy group of spheres. I will present an enforced-gaplessness theorem (conditional on some reasonable semisimplicity assumptions about the category of topological phases of matter): if this character is nontrivial, and not one of the six nontrivial Kervaire invariants, then the invertible phase is a topological insulator: a material all of whose boundary conditions have gapless conducting modes. This is joint work in progress with David Reutter.
With six exceptions, nontrivial characters of spheres do not admit gappable boundary conditions Sponsored by the Meyer Fund
Tue, Oct. 13 1:30pm (Math350)
Math Phys
Theo Johnson-Freyd (Perimeter Institute for Theoretical Physics and Dalhousie University)
X
Abstract: Kitaev has argued that the spaces of n-dimensional invertible phases of matter, as n varies, fit together into the structure of an Omega-spectrum. It follows that each n-dimensional phase determines a (possibly trivial) character on the nth stable homotopy group of spheres. I will present an enforced-gaplessness theorem (conditional on some reasonable semisimplicity assumptions about the category of topological phases of matter): if this character is nontrivial, and not one of the six nontrivial Kervaire invariants, then the invertible phase is a topological insulator: a material all of whose boundary conditions have gapless conducting modes. This is joint work in progress with David Reutter.
With six exceptions, nontrivial characters of spheres do not admit gappable boundary conditions
Tue, Oct. 13 2:30pm (MATH 3…
Lie Theory
Kelsey Brown (Colorado State University)
X
In the 1990's, Proctor classified the d-complete posets, which correspond to λ-minuscule elements of Weyl groups of simply laced Kac-Moody algebras. These have the Jeu de Taquin property for the natural generalization of standard Young tableaux theory to posets.
We define a notion of highest weight tableaux for several infinite families of d-complete posets. These constructions allow us to develop local algorithms for coswitching pairs of tableaux on these families, a coplactic version of tableau switching that, in the known setting of types A and B, connects naturally to Schubert calculus, crystals, and promotion and evacuation.
Semistandard tableaux and (co)switching on d-complete posets
Holomorphy over real algebras