Upcoming Events
Mon, Apr 20
Algebra Seminar
Douglas Rizzolo, University of Delaware
Ordered Chinese Restaurant Process up-down chains
4:00 PM, 250 Mathematics Building
Up-Down chains on branching graphs provide an interesting link between the algebraic structure of branching graphs and stochastic processes on the boundaries of these graphs. Up-down chains on branching graphs whose vertices are given by partitions of an integer are well understood, but up-down chains on branching graphs whose vertices are compositions of an integer have only begun to be studied recently. In this talk we will discuss up-down chains on graphs of compositions whose up-steps are based on the Ordered Chinese Restaurant Process. We will show how these can be used to construct diffusions on the boundary of this graph whose generators have simple expressions in terms of quasi-symmetric functions. We will give examples showing what sorts of new statistics can be understood based on the order structure, with an emphasis on a connection to phylogenetics where the order structure can be interpreted as the relative ages of alleles in a population.
Thu, Apr 23
Colloquium
Brandis Whitfield (University of Wisconsin-Madison/SLMath)
The geometries of subgroups of the mapping class group
4:00 PM, 250 Mathematics Building
Mapping class groups of surfaces play a prominent role in understanding the geometry and topology of three-manifolds. These connections have inspired decades-long research programs toward a notion of geometric finiteness for mapping class subgroups. In this talk, we’ll introduce the mapping class group, discuss methods of embedding different groups into it and how topological properties of a mapping class determine geometric data of its associated 3-manifold.
Fri, Apr 24
Topology and Geometry Seminar
Brandis Whitfield (University of Wisconsin-Madison/SLMath)
Constructing reducibly geometrically finite subgroups of the mapping class group
4:00 PM, 122 Mathematics Building
A longstanding goal within low-dimensional geometry and topology is to establish parallels between the theory of Kleinian groups and subgroups of the mapping class group. Convex-cocompact subgroups of the mapping class group have been well-studied in the past few decades; current research aims to extend this analogy to the more general class of geometric finiteness. Dowdall–Durham–Leininger–Sisto introduce parabolically geometrically finite (PGF) subgroups of the mapping class group whose coned-off Cayley graphs, with respect to a collection of twist subgroups, quasi-isometrically embed into the curve complex. In joint work with Aougab, Bray, Dowdall, Hoganson and Maloni, we extend this definition to allow the coned-off subgroups to be more generally reducible, and produce plentiful examples of this behavior. Our examples include free products of reducible subgroups and the beloved right-angled Artin constructions of Clay–Leininger–Mangahas.
Mon, Apr 27
Algebra Seminar
Michael Brannan, University of Waterloo
Quantum symmetries of Hadamard matrices
4:00 PM, Mathematics Building
A Hadamard matrix is a square matrix of complexnumbers whose entries have modulus 1, and whose columns are pairwiseorthogonal. In this talk I will discuss various notions of quantum symmetriesand equivalences of Hadamard matrices using ideas from quantum group theory.These symmetries, in the case of finite order (i.e., Butson-type) Hadamardmatrices, turn out to have interesting applications in the theory of non-localgames from quantum information theory. This is joint work with Daniel Gromada,Roberto Hernandez-Palomares, and Nicky Priebe.
Thu, Apr 30
Colloquium
Daniel Litt, University of Toronto
Braid groups and 2x2 matrices
4:00 PM, 250 Mathematics Building
Let \(X_n\) be the set of conjugacy classes of \(n\)-tuples of 2\(\times\)2 matrices, whose product is the identity matrix. There is a natural braid group action on \(X_n\), whose study goes back to work of Mark off in the late 19th century. The most basic question one can ask about this action, which dates to work of Painlevé, Fuchs, Schlesinger, and Garnier in the beginning of the 20th century, is: what are its finite orbits? I’ll explain the history of this question, as well as some recent work (combining results of Bronstein-Maret with results obtained jointly with Josh Lam and Aaron Landesman), which answers it completely. Time permitting, I’ll discuss "higher genus" variants of this question, whose answer relies on non-abelian Hodge theory and the Langlands program, and resolves conjectures of Esnault-Kerz, Budur-Wang, Kisin, and Whang.
