Posted March 16, 2025
Last modified April 2, 2025
Mathematical Physics and Representation Theory Seminar
2:30 pm – 3:20 pm Lockett 233
Justin Lanier, University of Sydney
Twisting cubic rabbits
A polynomial can be viewed as a branched cover of the sphere over itself that is compatible with a complex structure. If handed a topological branched cover of the sphere, we can ask whether it can arise from a polynomial, and if so, which one? In 2006, Bartholdi and Nekrashevych used group theoretic methods to explicitly solve this problem in certain special cases, including Hubbard’s twisted rabbit problem. Using a combinatorial topology approach that draws from the theory of mapping class groups, we solve an infinite family of twisted polynomial problems that are cubic generalizations of Hubbard’s twisted rabbit problem. This is joint work with Becca Winarski.
Posted March 31, 2025
Algebra and Number Theory Seminar Questions or comments?
2:30 pm – 3:30 pm Lockett 233 or click here to attend on Zoom
Be'eri Greenfeld , University of Washington
Complexity and Growth of Infinite Words and Algebraic Structures
Given an infinite word (for example, 01101001$\ldots$), its complexity function counts, for each n, the number of distinct subwords of length n. A longstanding open problem is the "inverse problem": Which functions $f:\mathbb N\to \mathbb N$ arise as complexity functions of infinite words? We resolve this problem asymptotically, showing that, apart from submultiplicativity and a classical obstruction found by Morse and Hedlund in 1938, there are essentially no further restrictions. We then explore parallels and contrasts with the theory of growth of algebras, drawing on noncommutative constructions associated with symbolic dynamical systems.
Posted February 3, 2025
Informal Geometry and Topology Seminar Questions or comments?
1:30 pm
Adithyan Pandikkadan, Louisiana State University
TBD
Posted March 8, 2025
Last modified March 9, 2025
Tomoyuki Kakehi, University of Tsukuba
Snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation
In this talk, we deal with snapshot problems for the wave equation and for the Euler-Poisson-Darboux equation. For simplicity, let us consider the wave equation $\partial_t^2 u - \Delta u =0$ on $\mathbb{R}^n$ with the condition $u|_{t=t_1} =f_1, \cdots, u|_{t=t_m} =f_m$. It is natural to ask when the above equation has a unique solution. We call the above problem the snapshot problem for the wave equation, and call the set of $m$ functions $\{ f_1, \cdots, f_m \}$ the snapshot data. Roughly speaking, one of our main results is as follows. {\bf Theorem.} We assume that $m=3$ and $(t_3-t_1)/(t_2 -t_1)$ is irrational and not a Liouville number. In addition, we assume a certain compatibility condition on the snapshot data $\{ f_1, f_2, f_3 \}$. Then the snapshot problem for the wave equation has a unique solution. We also consider a similar snapshot problem for the Euler-Poisson-Darboux equation. This is a joint work with Jens Christensen, Fulton Gonzalez, and Jue Wang.