Browsing by Author "Hardt, Robert"

Now showing 1 - 6 of 6
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    A Mass Minimizing Flow for Real-Valued Flat Chains with Applications to Transport Networks
    (2017-04-21) Downes, Carol Ann; Hardt, Robert
    An oriented transportation network can be modeled by a 1-dimensional chain whose boundary is the difference between the demand and supply distributions, represented by weighted sums of point masses. To accommodate efficiencies of scale into the model, one uses a suitable Mα norm for transportation cost. One then finds that the minimal cost network has a branching structure since the norm favors higher multiplicity edges, representing shared transport. In this thesis, we construct a continuous flow that evolves some initial such network to reduce transport cost without altering its supply and demand distributions. Instead of limiting our scope to transport networks, we construct this M^α mass reducing flow for real-valued flat chains by finding a real current of locally finite mass with the property that its restrictions are flat chains; the slices of such a restriction dictate the flow. Keeping the boundary fixed, this flow reduces the M^α mass of the initial chain and is Lipschitz continuous under the flat-α norm. To complete the thesis, we apply this flow to transportation networks, showing that the flow indeed evolves branching transport networks to be more cost efficient.
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    Global Regularity for Euler Vortex Patch in Bounded Smooth Domains
    (2018-04-18) Li, Chao; Kiselev, Alexander; Hardt, Robert
    It is well known that the Euler vortex patch in two dimensional plane will remain regular if it is regular enough initially. In bounded domains, the regularity theory for patch solutions is less complete. In this thesis, I study the Euler vortex patch in a general smooth bounded domain. I prove global in time regularity by providing the upper bound of the growth on curvature of the patch boundary. For a special symmetric scenario, I construct an example of double exponential curvature growth, showing that our upper bound is qualitatively sharp.
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    Hitchin Components, Riemannian Metrics and Asymptotics
    (2014-12-04) Li, Qiongling; Wolf, Michael; Hardt, Robert; Gillman, Adrianna
    Higher Teichm\"uller spaces are deformation spaces arising from subsets of the space of representations of a surface group into a general Lie group, e.g., $$PSL(n,\RR)$$, which share some of the properties of classical Teichmueller space. By the non-abelian Hodge theory, such representation spaces correspond to the space of Higgs bundles. We focus on two aspects on the Higher Teichm\"uller space: Riemannian geometry and dynamics. First, we construct a new Riemannian metric on the deformation space for $$PSL(3,\RR)$$, and then prove Teichmueller space endowed with Weil-Petersson metric is totally geodesic in deformation space for $$PSL(3,\RR)$$ with the new metric. Secondly, in a joint work with Brian Collier, we are able to obtain asymptotic behaviors and related properties of representations for certain families of Higgs bundles of rank n.
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    Mathematical Results for Michell Trusses
    (2022-04-21) Chen, Yikai; Hardt, Robert
    Given an equilibrated vector force system F of finite mass and bounded support, we investigate the possibility and properties of a cost minimizing structure of given materials that balances F. Our work generalizes and reinterprets results of Michell and Gangbo where the given equilibrated force system occurs on a finite set of points and the balancing structure consists of finitely many stressed bars joining these points. Such a stressed bar corresponds to an interval [a,b] in Rn having a multiplicity lambda in R where |lambda| indicates the stress density on the bar and sgn(lambda) indicates whether it is being compressed or extended. While there exists a finite bar system to balance any given equilibrated finite force system, Michell already observed that a finite cost-minimizing one may not exist. In this thesis, we introduce two new mathematical representations of Michell trusses based on one- dimensional finite mass varifolds and flat Rn-chains. Here one may use a one-dimensional signed varifold to model the balancing structure so that the internal force of the positive (or compressed) part coincides with its first variation of mass while the internal force of the negative (or extended) part coincides with its negative first variation. For the chain model, we use the subspace of structural flat Rn chains in which the coefficient vectors are a.e. colinear with the orientation vectors. The net force then becomes simply the Rn chain boundary and so cost-minimization becomes precisely the mass-minimizing Plateau problem for structural chains. For either model, a known compactness theorem leads to existence of optimal cost-minimizers as well as time-continuous cost-decreasing flows.
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    Properties of Shortest Length Curves inside Semi-Algebraic Sets and Problems related to an Erdos Conjecture concerning Lattice Cubes
    (2021-04-29) Yang, Chengcheng; Hardt, Robert
    Part I: Properties of Shortest Length Curves inside Semi-algebraic Sets The first part of the paper concerns an analytical stratification question of real algebraic and semi-algebraic sets. An algebraic set is defined by finitely many polynomial equations and the definition of the more general semi-algebraic set may also entail polynomial inequalities. In 1957 Whitney \cite{W} gave a stratification of real algebraic sets, it partitions a real algebraic set into partial algebraic manifolds. In 1975 Hironaka \cite{H} reproved that a real algebraic set is triangulable and also generalized it to sub-analytic sets, following the idea of Lojasiewicz's \cite{L} triangulation of semi-analytic sets in 1964. During the same year, Hardt \cite{H2} also proved the triangulation result for sub-analytic sets by inventing another method. Since any semi-algebraic set is also semi-analytic, thus is sub-analytic, both Hironaka and Hardt's results showed that any semi-algebraic set is homeomorphic to the polyhedron of some simplicial complex. Following their examples and wondering how geometry looks like locally for a semi-algebraic set, Part I of the paper tries to come up with a stratification, in particular a cell decomposition, such that it satisfies the following analytical property. Given an arbitrary semi-closed and connected semi-algebraic set $X$ in $\mathbb{R}^2$, any two points in $X$ may be joined by a continuous path $\gamma$ of shortest length. We will show that there exists a semi-algebraic cell decomposition $\mathcal{A}$ of $X$ such that for each $A \in \mathcal{A}$, each component of $\gamma \cap A$ is either a singleton or a real analytic geodesic segment in $A$; furthermore, $\gamma \cap A$ has at most finitely many components. An application of this property is that given any semi-algebraic sets $Y \subset X \subset \mathbb{R}^2$, any shortest length curve in $X$ intersects $Y$ at most finitely many components. We try to generalize to higher-dimensional semi-algebraic sets and the question is still open. The other analogous open question concerns the regularity of a rectifiable current minimizing mass in a semi-algebraic set. Part II: Problems related to an Erd{\"o}s conjecture concerning lattice cubes. The second part of the paper studies a problem of Erd\"{o}s concerning lattice cubes. Given an $n \times n \times n$ lattice cube, we want to find the maximum number of vertices one can select so that no eight corners of a rectangular box are chosen simultaneously. Erd\"{o}s conjectured that it has a sharp upper bound, which is $O(n^{11/4})$, but no example that large has been found yet. We start approaching this question for small $n$ using the method of exhaustion, and we find that as $n$ increases, the method becomes cumbersome and one reason is that the condition cannot be easily decoded into workable algebraic conditions. Next, we study an equivalent two-dimensional version of this problem and look for patterns that might be useful for generalizing to the three-dimensional case. Since an $n \times n$ grid is also an $n \times n$ matrix, we rephrase and generalize the original question to: what is the minimum number $\alpha(k,n)$ of vertices one can put in an $n \times n$ matrix with entries 0 and 1, such that every $k \times k$ minor contains at least one entry of 1, for $1 \leq k \leq n$? We discover some interesting formulae and asymptotic patterns that shed new light on the question. Then we examine many examples that succeed for $O(n^{8/3})$ but fail for $O(n^{11/4})$. Last we describe a new method which has the hope to prove that $O(n^{11/4})$ is the sharp upper bound for the maximum number. In the end, we extend from the discrete questions to nondiscrete questions, which become very interesting to study, because we will see topological manifolds, and linear groups.
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    Smooth minimal transport networks and non-orientable minimal surfaces in S3
    (2019-04-12) Wu, Jianqiu; Hardt, Robert
    In this paper we introduce a new optimal transport problem which involves roughly a finite system of simultaneous time-parametrized transport which favors merging paths for efficiency over various time intervals and involves continuously differentiable transitions at the mergings (as with train tracks). We will describe suitable spaces of parametrized networks, topologies, and functionals, and then give an existence and regularity theory. Along the way we obtain necessary and sufficient optimality conditions applicable at times of various mergings. Additionally we introduce the problem of finding minimal surfaces in S3. In particular, we are interested in whether a certain minimal Mobius band is the unique minimal nonorientable surface with boundary a great circle. As this problem is too hard to tackle directly, we studied a related problem in a different bilipschitz space, the boundary of the bi-cylinder D2*D2.