# Browsing by Author "Stern, Mark A"

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Item Open Access Analytic Torsion, the Eta Invariant, and Closed Differential Forms on Spaces of Metrics(2016) Andreae, PhillipThe central idea of this dissertation is to interpret certain invariants constructed from Laplace spectral data on a compact Riemannian manifold as regularized integrals of closed differential forms on the space of Riemannian metrics, or more generally on a space of metrics on a vector bundle. We apply this idea to both the Ray-Singer analytic torsion

and the eta invariant, explaining their dependence on the metric used to define them with a Stokes' theorem argument. We also introduce analytic multi-torsion, a generalization of analytic torsion, in the context of certain manifolds with local product structure; we prove that it is metric independent in a suitable sense.

Item Open Access Equivariant Nahm Transforms and Minimal Yang--Mills Connections(2020) Beckett, Matthew James PaulThis dissertation examines two different subjects within the study of instantons: the construction of Nahm transforms for instantons invariant under certain group actions; and a generalization of the proof that Yang--Mills minimizers are instantons.

The first Nahm transform examined is the ADHM construction for $S^1$-invariant instantons on $S^4$, which correspond to singular monopoles on $\RR^3$. In this case, there is a decomposition of the ADHM data in terms of $S^1$-subrepresentations of $\ker \Dir$. The moduli spaces of $S^1$-invariant $SU(2)$-instantons are given up to charge 3, and examples of ADHM data for instantons of charge $4$ are also provided.

The second Nahm transform considered is for instantons on a certain flat quotient of $\RR^4$ with nonabelian fundamental group. Equivalently, one can consider these to be $\ZZ_2$-invariant instantons on $T^4$, and the Nahm transform yields instantons invariant under a crystallographic action.

In our study of minimal Yang--Mills connections, we extend results of Bourguignon--Lawson--Simons and Stern, who showed that connections that minimize $\|F_\nabla\|^2$ on homogeneous manifolds must be instantons or have instanton subbundles. We extend the previous arguments by considering variations constructed using conformal vector fields, and also allow these vector fields to be incomplete. We prove a minimality result over a half-cylinder.

Item Open Access L2 Index Theory and D-Particle Binding in Type I' String Theory(2009) McCarthy, Janice MarieIn this work, we apply $L^2$-index theory to compute the index of a non-Fredholm elliptic operator. The operator arises in Type I' string theory, and the index is found to be non-zero, thus implying existence of bound states.

Item Open Access On the Betti Numbers of Compact Rank 2 Locally Symmetric Spaces(2024-04-23) Ong, NathanaelWe obtain upper bounds for the second Betti numbers of compact rank 2 locally symmetric spaces, namely $\Gamma\backslash SL(3)/SO(3)$, $\Gamma\backslash Sp(4)/U(2)$, and $\Gamma\backslash G_{2(2)}/SO(4)$, where $\Gamma$ is a cocompact, torsion free lattice. We use representation theory and directly apply the techniques of Di Cerbo and Stern in \cite{dicerbo2019price}. In the case of $\Gamma\backslash Sp(4)/U(2)$, we also use unitary holonomy, the complex structure operator that arises from it and the (p,q) decomposition of exterior powers to obtain stronger bounds. In particular, the bounds we provide on the Betti numbers of $\Gamma\backslash Sp(4)/U(2)$ and $\Gamma\backslash G_{2(2)}/SO(4)$ are exponential bounds involving injectivity radius. However, the bound we obtained for $\Gamma\backslash SL(3)/SO(3)$ is a weaker polynomial one.Item Open Access Ricci Yang-Mills Flow(2007-05-04T17:37:34Z) Streets, Jeffrey D.Let (Mn, g) be a Riemannian manifold. Say K ! E ! M is a principal K-bundle with connection A. We define a natural evolution equation for the pair (g,A) combining the Ricci flow for g and the Yang-Mills flow for A which we dub Ricci Yang-Mills flow. We show that these equations are, up to di eomorphism equivalence, the gradient flow equations for a Riemannian functional on M. Associated to this energy functional is an entropy functional which is monotonically increasing in areas close to a developing singularity. This entropy functional is used to prove a non-collapsing theorem for certain solutions to Ricci Yang-Mills flow. We show that these equations, after an appropriate change of gauge, are equivalent to a strictly parabolic system, and hence prove general unique short-time existence of solutions. Furthermore we prove derivative estimates of Bernstein-Shi type. These can be used to find a complete obstruction to long-time existence, as well as to prove a compactness theorem for Ricci Yang Mills flow solutions. Our main result is a fairly general long-time existence and convergence theorem for volume-normalized solutions to Ricci Yang-Mills flow. The limiting pair (g,A) satisfies equations coupling the Einstein and Yang-Mills conditions on g and A respectively. Roughly these conditions are that the associated curvature FA must be large, and satisfy a certain “stability” condition determined by a quadratic action of FA on symmetric two-tensors.Item Open Access Sigma Models with Repulsive Potentials(2017) Huang, JingxianMotivated by questions arising in the study of harmonic maps and Yang Mills theory, we study new techniques for producing optimal monotonicity relations for geometric partial differential equations. We apply these results to sharpen epsilon regularity results. As a sample application, we analyze energy minimizing maps from compact manifolds to the space of hermitian matrices, where the energy of the map includes the usual kinetic term and a singular potential designed to force the image of the map to lie in a set homotopic to a Grassmannian.

Item Open Access Solitons for the Closed G2 Laplacian Flow(2021-04-19) Dayaprema, AnukGeometric flows are partial differential equations on smooth manifolds which describe the time evolution of some geometric structure on the manifold, such as a Riemannian metric. In the setting of G2 geometry, which is specific to seven dimensions, a natural geometric flow to consider is the (closed) G2 Laplacian flow. Critical points of this flow correspond to torsion-free G2 structures, which satisfy a system of nonlinear partial differential equations. We are interested in such G2 structures since they give rise to Riemannian metrics with exceptional holonomy. Solitons for the Laplacian flow are self-similar solutions which we hope provide models for finite-time singularities of the flow. Since the soliton equation is a nonlinear system of PDEs, to get a feel for concrete solutions we consider a dimensional reduction of the equation by studying cohomogeneity-one solitons. We consider the cohomogeneity-one setting where the group G of isometries is either SU(3) or Sp(2). In these regimes, we show that the soliton equation is equivalent to a (nonlinear) first-order system of real-analytic ODEs. We then describe the space of G-invariant closed G2 cones, identifying it with a smooth surface/curve in R3. We then investigate the uniqueness of a complete shrinking Sp(2)-invariant soliton which possesses an asymptotically conical geometry, and conjecture that it is rigid to first order. We finally discuss ways to study the general closed Laplacian soliton equation, reviewing previous results by others in related directions, with an eye towards asymptotically conical shrinking solitons in particular.