Math Department
https://wp0.vanderbilt.edu/math
College of Arts and Science | Vanderbilt UniversityThu, 27 Feb 2020 22:10:44 +0000en-UShourly1https://wordpress.org/?v=4.9.8146940276End Sums of Open Manifolds
https://wp0.vanderbilt.edu/math/2020/02/end-sums-of-open-manifolds/
Thu, 27 Feb 2020 22:10:44 +0000https://wp0.vanderbilt.edu/math/?p=9864Connected sums and boundary connected sums play important roles in the study of closed manifolds and manifolds with boundary, respectively. When working with open manifolds, a third variety of connected sum—the “end sum”—is useful. Each of these operations involves a number of arbitrary choices, making well-definedness of the resulting manifold a significant question. With regards to the end sum operation, we will discuss familiar situations where all goes smoothly (the notion of semistability plays a role here) and others where significant problems arise. Our analysis leads naturally into the subtle and interesting theory of infinitely generated abelian groups. The work to be presented in this talk is joint with Jack Calcut and Patrick Haggerty.
]]>9864Incongruences for Modular Forms and Applications to Partition Functions
https://wp0.vanderbilt.edu/math/2020/02/talk-title-tba-335/
Thu, 27 Feb 2020 18:00:21 +0000https://wp0.vanderbilt.edu/math/?p=9512The study of arithmetic properties of coefficients of modular forms enjoys a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of S. Ahlgren, B. Kim, N. Andersen, and S. Loebrich have employed the q-expansion principle of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we extend the method to give additional results for a large class of modular forms, and investigate the consequences of that result. (Joint work with S. Garthwaite.)
]]>9512Free Products and Random Walks in Acylindrically Hyperbolic Groups
https://wp0.vanderbilt.edu/math/2020/02/talk-title-tba-344/
Wed, 26 Feb 2020 22:10:02 +0000https://wp0.vanderbilt.edu/math/?p=9629The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years. In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, Out(Fn), and right-angled Artin and Coxeter groups, among many others. I will discuss how a random element of such a group interacts with fixed subgroups, especially so-called hyperbolically embedded subgroups. In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.
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https://wp0.vanderbilt.edu/math/2020/02/9876/
Wed, 26 Feb 2020 21:30:16 +0000https://wp0.vanderbilt.edu/math/?p=98769876Optimal Sampling and Reconstruction on General Multivariate Domains
https://wp0.vanderbilt.edu/math/2020/02/optimal-sampling-and-reconstruction-on-general-multivariate-domains/
Tue, 25 Feb 2020 22:10:40 +0000https://wp0.vanderbilt.edu/math/?p=9868Motivated by non-intrusive approaches for high-dimensional parametric PDEs, we consider the general problem of approximating an unknown arbitrary function in any dimension from the data of point samples. The approximants are picked from given or adaptively chosen finite-dimensional spaces. One principal objective is to obtain an approximation which performs as good as the best possible using a sampling budget that is linear in the dimension of the approximating space. We will show that this objective can is met by taking a random sample distributed according to a well-chosen probability measure, and reconstructing by appropriate least-squares measures. We discuss these optimal sampling strategies in the adaptive context and for general non-tensor-product multivariate domains.
]]>9868The Baum-Connes Correspondence for the Pure Braid Group on Four Strands
https://wp0.vanderbilt.edu/math/2020/02/the-baum-connes-correspondence-for-the-pure-braid-group-on-four-strands/
Fri, 21 Feb 2020 22:10:42 +0000https://wp0.vanderbilt.edu/math/?p=9827We calculate the left-hand side and the right-hand side of the Baum-Connes correspondence for the pure braid group on four strands, each side relying on different techniques. Our long-term goal is to elucidate Schick’s abstract proof of the correspondence for braid groups by explicitly describing the assembly map. This is joint work with Sara Azzali, Sarah Browne, Maria Paula Gomez Aparicio, and Hang Wang.
]]>9827Improvability of the Dominant Energy Scalar and Bartnik’s Stationary Conjecture
https://wp0.vanderbilt.edu/math/2020/02/talk-title-tba-346/
Fri, 21 Feb 2020 22:10:07 +0000https://wp0.vanderbilt.edu/math/?p=9740We will introduce the concept of improvabilty of the dominant energy scalar and discuss strong consequences of non-improvability. This concept is important to find perturbations of initial data sets that preserve or reinstate the dominant energy condition. We introduce infinite-dimensional families of deformations of the modified Einstein constraint operator and show that, generically, their adjoint linearizations are either injective, or else one can prove that kernel elements satisfy a “null-vector equation”. Combined with a conformal argument, we make substantial progress toward Bartnik’s stationary conjecture. More specifically, we prove that a Bartnik minimizing initial data set can be developed into a spacetime that both satisfies the dominant energy condition and carries a global Killing field. We also show that this spacetime is vacuum near spatial infinity. This talk is based on the joint work with Dan Lee.
]]>9740Emergent Behavior in Collective Dynamics
https://wp0.vanderbilt.edu/math/2020/02/talk-title-tba-349/
Thu, 20 Feb 2020 22:10:43 +0000https://wp0.vanderbilt.edu/math/?p=9761A fascinating aspect of collective dynamics is the self-organization of small-scales and their emergence as higher-order patterns — clusters, flocks, tissues, parties. The emergence of different patterns can be described in terms of few fundamental “rules of interactions”. I will discuss recent results of the large-time, large-crowd dynamics, driven by anticipation that tend to align the crowd, while other pairwise interactions keep the crowd together and prevent over-crowding. In particular, I address the question how short-range interactions lead to the emergence of long-range patterns, comparing different rules of interactions based on geometric vs. topological neighborhoods.
]]>9761Boundaries and CAT(0) Cube Complexes
https://wp0.vanderbilt.edu/math/2020/02/talk-title-tba-341/
Wed, 19 Feb 2020 22:10:12 +0000https://wp0.vanderbilt.edu/math/?p=9586The universe of CAT(0) cube complexes is rich and diverse thanks to the ease by which they can be constructed and the many of natural metrics they admit. As a consequence, there are several associated boundaries, such as the visual boundary and the Roller boundary. In this talk we will discuss some relationships between these boundaries, together with the Furstenberg-Poisson boundary of a “nicely” acting group.
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https://wp0.vanderbilt.edu/math/2020/02/9860/
Wed, 19 Feb 2020 21:30:48 +0000https://wp0.vanderbilt.edu/math/?p=98609860