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for events the day of Tuesday, September 18, 2018.

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Tuesday, September 18, 2018

11:00 am in 345 Altgeld Hall,Tuesday, September 18, 2018

Unstable $v_1$-periodic Homotopy Groups through Goodwillie Calculus

Jens Kjær (Notre Dame Math)

Abstract: It is a classical result that the rational homotopy groups, $\pi_*(X)\otimes \mathbb{Q}$, as a Lie-algebra can be computed in terms of indecomposable elements of the rational cochains on $X$. The closest we can get to a similar statement for general homotopy groups is the Goodwillie spectral sequence, which computes the homotopy group of a space from its "spectral Lie algebra". Unfortunately both input and differentials are hard to get at. We therefore simplify the homotopy groups by taking the unstable $\nu_h$-periodic homotopy groups, $\nu_h^{-1}\pi_*(\ )$ (note $h=0$ recovers rational homotopy groups). For $h=1$ we are able to compute the $K$-theory based $\nu_1$-periodic Goodwillie spectral sequence in terms of derived indecomposables. This allows us to compute $\nu_1^{-1}\pi_*SU(d)$ in a very different way from the original computation by Davis.

1:00 pm in 345 Altgeld Hall,Tuesday, September 18, 2018

Descriptive locale theory

Ruiyuan Chen (UIUC)

Abstract: A locale is, informally, a topological space without an underlying set of points, with only an abstract lattice of ``open sets''. Various results in the literature suggest that locale theory behaves in many ways like a generalization of descriptive set theory with countability restrictions removed. This talk will introduce locale theory from a descriptive set-theoretic point of view, and survey some known and new results which are common to both contexts.

2:00 pm in 243 Altgeld Hall,Tuesday, September 18, 2018

Typical structure of Gallai colorings

Lina Li (Illinois Math)

Abstract: An edge coloring of a graph G is a Gallai coloring if it contains no rainbow triangle. Like many of other extremal problems, it is interesting to study how many Gallai colorings are there and what is the typical structure of the Gallai colorings. We show that almost all the Gallai r-colorings of complete graphs are 2-colorings. We also study Gallai 3-colorings of non-complete graphs. This is joint work with Jozsef Balogh.

4:00 pm in 245 Altgeld Hall ,Tuesday, September 18, 2018

Möbius kaleidocycles: a new class of everting ring linkages

Eliot Fried (Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University)

Abstract: Many of Escher’s works have become mainstays of popular culture. Famous examples include his Möbius bands and kaleidocycles. Made from six identical regular tetrahedra joined by revolute hinges, a kaleidocycle possesses a single internal degree-of-freedom that is manifested by a cyclic everting motion that brings different faces of the tetrahedra into view. We will describe "Möbius Kaleidocycles," a previously undiscovered class ring linkages made from seven or more identical links joined by revolute hinges. For each number of links, there exists a specific twist angle between neighboring hinges for which the associated Möbius Kaleidocycle possesses only a single internal degree-of-freedom, allowing for cyclic eversion, and the hinge orientations induce a nonorientable topology equivalent to that of a 3π twist Möbius band. Apart from technological applications, including perhaps the design of new organic ring molecules with peculiar electronic properties, Möbius kaleidocycles generate a myriad of intriguing questions in geometry and topology, some of which will be addressed in this talk. This is joint work with postdoctoral scholar Johannes Schönke.