This thesis develops the theory of weighted quadrature domains in parallel with the Faber transform as a tool for their analysis and explicit construction. Under this framework,e obtain a number of existence, uniqueness, and classification results for classical and weighted quadrature domains.
\r\n\r\nIn the classical setting, we derive explicit formulae relating the Riemann map of a simply connected quadrature domain to its quadrature function via the Faber transform. This reduces the inverse and direct problems to a problem of solving finitely many algebraic equations. Applying these results, along with tools from logarithmic potential theory, we obtain a complete classification of one-point quadrature domains with complex charge.
\r\n\r\nWe then introduce power-weighted quadrature domains (PQDs) - domains admitting a quadrature identity with respect to the weight \u03c1a(w) = \u2223w\u22232(a-1) for some a>0. A central structural result is that a simply connected domain is a PQD if and only if the ath power of the outer factor of its Riemann map extends to a rational function. This characterization yields Faber transform formulae analogous to the classical case, which we apply to partially classify one-point and monomial PQD families. Novel boundary phenomena - including the formation of boundary \"corners\" at the origin with angles that are integer multiples of \u03c0/a - are exhibited.
\r\n\r\nNext, we develop the theory of log-weighted quadrature domains (LQDs) - domains admitting a quadrature identity with respect to the weight \u03c10 = \u2223w\u2223-2, the limiting case a\u21920\u207a of \u03c1a |w|2(a-1). The non-integrable singularity at the origin introduces new phenomena: when the domain contains the origin, the quadrature function is no longer uniquely determined, but only up to the addition of a point charge q/w. Despite this loss of uniqueness, we show that a simply connected domain is an LQD if and only if the outer factor of its Riemann map extends to the exponential of a rational function. Classification results for null, monomial, and one-point LQD families are obtained, and the connection to classical quadrature domains via the exponential map is established.
\r\n\r\nFinally, we introduce algebraic quadrature domains (AQDs), defined with respect to weights of the form \u03c1R = |R'|\u00b2, where R is a non-constant rational function. This class subsumes both classical quadrature domains (R(w)=w) and integer power-weighted quadrature domains (R(w)=wn/n). We derive representation formulae in terms of the Faber transform and present several examples.
", "doi": "10.7907/tgqd-x974", "publication_date": "2026", "thesis_type": "phd", "thesis_year": "2026" }, { "id": "thesis:17252", "collection": "thesis", "collection_id": "17252", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05202025-052235420", "primary_object_url": { "basename": "zhang_jiaxin_2025_thesis.pdf", "content": "final", "filesize": 10784352, "license": "other", "mime_type": "application/pdf", "url": "/17252/1/zhang_jiaxin_2025_thesis.pdf", "version": "v6.0.0" }, "type": "thesis", "title": "On Multiple SLE Systems and their Deterministic Limits", "author": [ { "family_name": "Zhang", "given_name": "Jiaxin", "orcid": "0000-0002-5647-8949", "clpid": "Zhang-Jiaxin" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Hutchcroft", "given_name": "Thomas", "orcid": "0000-0003-0061-593X", "clpid": "Hutchcroft-Tom" }, { "family_name": "Yu", "given_name": "Tony Yue", "orcid": "0000-0002-6019-8552", "clpid": "Yu-Tony-Yue" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "In this thesis, we study multiple radial SLE(k) systems -- a family of random multi-curve systems in a simply-connected domain \u03a9, with marked boundary points z\u2081....z\u2099 \\in \u2202\u03a9 and a marked interior point q, where parameter k > 0 measures the randomness of the system. We also study the multiple radial SLE(0) systems as the deterministic limit of multiple radial SLE(k) systems.
\r\n\r\nAs a consequence of domain Markov property and conformal invariance, we derive that a multiple radial SLE(k) system is characterized by a conformally covariant partition function satisfying the null vector equations--a second-order PDE system. On the other hand, using the Coulomb gas method inspired by conformal field theory, we construct four types of solutions to the null vector equations, which can be classified according to topological link patterns.
\r\n\r\nWe construct the multiple radial SLE(0) systems from stationary relations by heuristically taking the classical limit of partition functions as k > 0. By constructing the field integrals of motion for the Loewner dynamics, we show that the traces of multiple radial SLE(0) systems are the horizontal trajectories of an equivalence class of quadratic differentials. These trajectories have limiting ends at the growth points and form a radial link pattern.
\r\n\r\nThe stationary relations connect the classification of multiple radial SLE(0) systems to the enumeration of critical points of the master function of trigonometric Knizhnik-Zamolodchikov (KZ) equations.
\r\n\r\nFor k > 0$, the partition functions of multiple radial SLE(k) systems correspond to eigenstates of the quantum Calogero-Sutherland (CS) Hamiltonian beyond the fermionic states. In the deterministic case of k=0, we show that the Loewner dynamics with a common parametrization of capacity form a special class of classical CS systems, restricted to a submanifold of phase space defined by the Lax matrix.
", "doi": "10.7907/spf7-9j65", "publication_date": "2025-06-13", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:8975", "collection": "thesis", "collection_id": "8975", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06032015-011726054", "primary_object_url": { "basename": "linghu_daiqi_2015_thesis.pdf", "content": "final", "filesize": 1214103, "license": "other", "mime_type": "application/pdf", "url": "/8975/1/linghu_daiqi_2015_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Chains of Non-Regular de Branges Spaces", "author": [ { "family_name": "Linghu", "given_name": "Daiqi", "clpid": "Linghu-Daiqi" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Frank", "given_name": "Rupert L.", "clpid": "Frank-R-L" }, { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" }, { "family_name": "Silva", "given_name": "Elwadura Prabath S.", "clpid": "Silva-E-P-S" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We consider canonical systems with singular left endpoints, and discuss the concept of a scalar\r\nspectral measure and the corresponding generalized Fourier transform associated with a canonical\r\nsystem with a singular left endpoint. We use the framework of de Branges\u2019 theory of Hilbert spaces\r\nof entire functions to study the correspondence between chains of non-regular de Branges\r\nspaces, canonical systems with singular left endpoints, and spectral measures.
\r\n\r\nWe find sufficient integrability conditions on a Hamiltonian H which ensure the existence of a\r\nchain of de Branges functions in the first generalized P\u00f3lya class with Hamiltonian H. This result\r\ngeneralizes de Branges\u2019 Theorem 41, which showed the sufficiency of stronger integrability\r\nconditions on H for the existence of a chain in the P\u00f3lya class. We show the conditions that de\r\nBranges came up with are also necessary. In the case of Krein\u2019s strings, namely when the Hamiltonian\r\nis diagonal, we show our proposed conditions are also necessary.
\r\n\r\nWe also investigate the asymptotic conditions on chains of de Branges functions as t approaches\r\nits left endpoint. We show there is a one-to-one correspondence between chains of de Branges\r\nfunctions satisfying certain asymptotic conditions and chains in the P\u00f3lya class. In the case of\r\nKrein\u2019s strings, we also establish the one-to-one correspondence between chains satisfying certain\r\nasymptotic conditions and chains in the generalized P\u00f3lya class.
", "doi": "10.7907/Z9C24TD4", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:8915", "collection": "thesis", "collection_id": "8915", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05292015-072640484", "primary_object_url": { "basename": "BSkinnerFinalThesis.pdf", "content": "final", "filesize": 455400, "license": "other", "mime_type": "application/pdf", "url": "/8915/1/BSkinnerFinalThesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Logarithmic Potential Theory on Riemann Surfaces", "author": [ { "family_name": "Skinner", "given_name": "Brian Paul", "clpid": "Skinner-Brian-Paul" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Marcolli", "given_name": "Matilde", "orcid": "0000-0002-2045-2907", "clpid": "Marcolli-M" }, { "family_name": "Alberts", "given_name": "Tom", "clpid": "Alberts-T" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We develop a logarithmic potential theory on Riemann surfaces which generalizes logarithmic potential theory on the complex plane. We show the existence of an equilibrium measure and examine its structure. This leads to a formula for the structure of the equilibrium measure which is new even in the plane. We then use our results to study quadrature domains, Laplacian growth, and Coulomb gas ensembles on Riemann surfaces. We prove that the complement of the support of the equilibrium measure satisfies a quadrature identity. Furthermore, our setup allows us to naturally realize weak solutions of Laplacian growth (for a general time-dependent source) as an evolution of the support of equilibrium measures. When applied to the Riemann sphere this approach unifies the known methods for generating interior and exterior Laplacian growth. We later narrow our focus to a special class of quadrature domains which we call Algebraic Quadrature Domains. We show that many of the properties of quadrature domains generalize to this setting. In particular, the boundary of an Algebraic Quadrature Domain is the inverse image of a planar algebraic curve under a meromorphic function. This makes the study of the topology of Algebraic Quadrature Domains an interesting problem. We briefly investigate this problem and then narrow our focus to the study of the topology of classical quadrature domains. We extend the results of Lee and Makarov and prove (for n ≥ 3) c ≤ 5n-5, where c and n denote the connectivity and degree of a (classical) quadrature domain. At the same time we obtain a new upper bound on the number of isolated points of the algebraic curve corresponding to the boundary and thus a new upper bound on the number of special points. In the final chapter we study Coulomb gas ensembles on Riemann surfaces.", "doi": "10.7907/Z9Q52MK8", "publication_date": "2015", "thesis_type": "phd", "thesis_year": "2015" }, { "id": "thesis:7252", "collection": "thesis", "collection_id": "7252", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11032012-213910304", "primary_object_url": { "basename": "Chang-Sha-2013.pdf", "content": "final", "filesize": 666640, "license": "other", "mime_type": "application/pdf", "url": "/7252/1/Chang-Sha-2013.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Hele-Shaw Flow Near Cusp Singularities", "author": [ { "family_name": "Chang", "given_name": "Sha", "clpid": "Chang-Sha" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lee", "given_name": "Seung-Yeop", "clpid": "Lee-S-Y" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis discusses the radial version of the Hele-Shaw problem. Different from the channel version, traveling-wave solutions do not exist in this version. Under algebraic potentials, in the case that the droplets expand, in finite time, cusps will appear on the boundary and classical solutions may not exist afterwards. Physicists have suggested that for (2p+1,2)-cusps, that near cusp singularities of Hele-Shaw flow, after scaling X, Y by some powers of time t respectively, the main part of Y(X, t) is a one-parameter family and does not depend on time t. They have also suggested that the solutions of the Hele-Shaw problem are connected with dispersionless KdV (dKdV) hierarchy. In this study, we rigorously proved that this is the case for (3,2)-cusps when the droplets are simply connected and the external potentials are algebraic. We gave exact solutions and showed that the main parts of the exact solutions are some special solutions of the dispersionless string equation. More over, borrowed from the physical paper$\\cite{Teo}$ with a little more details, we showed the arguments of how these special solutions are related to dKdV hierarchy.", "doi": "10.7907/YJEK-W376", "publication_date": "2013", "thesis_type": "phd", "thesis_year": "2013" }, { "id": "thesis:3079", "collection": "thesis", "collection_id": "3079", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-08102004-142550", "primary_object_url": { "basename": "D_Zhan_thesis.pdf", "content": "final", "filesize": 648843, "license": "other", "mime_type": "application/pdf", "url": "/3079/1/D_Zhan_thesis.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Random Loewner Chains in Riemann Surfaces", "author": [ { "family_name": "Zhan", "given_name": "Dapeng", "orcid": "0000-0001-5528-8142", "clpid": "Zhan-Dapeng" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Borodin", "given_name": "Alexei", "clpid": "Borodin-A" }, { "family_name": "Simon", "given_name": "Barry M.", "orcid": "0000-0003-2561-8539", "clpid": "Simon-B" }, { "family_name": "Berger", "given_name": "Noam", "clpid": "Berger-N" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "The thesis describes an extension of O. Schramm's SLE processes to complicated plane domains and Riemann surfaces. First, three kinds of new SLEs are defined for simple conformal types. They have properties similar to traditional SLEs. Then harmonic random Loewner chains (HRLC) are defined in finite Riemann surfaces. They are measures on the space of Loewner chains, which are increasing families of closed subsets satisfying certain properties. An HRLC is first defined on local charts using Loewner's equation. Since the definitions in different charts agree with each other, these local HRLCs can be put together to construct a global HRLC. An HRLC in a plane domain can be described by differential equations involving canonical plane domains. Those old and new SLEs are special cases of HRLCs. An HRLC is determined by a parameter K \u2265 0, a starting point and a target set. When K = 6, the HRLC satisfies the locality property. When K = 2, the HRLC preserves some observable that resembles the observable for the corresponding loop-erased random walk (LERW). So HRLC\u2082 should be the scaling limit of LERW. With reasonable assumptions, HRLC8/3 differs from a restriction measure by a conformally invariant density; for K \u2208 (0,8/3), HRLCK differs from a pre-restriction measure by a conformally invariant density. A restriction measure could be constructed from a pre-restriction measure by adding Brownian bubbles.", "doi": "10.7907/TK0N-3F57", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" }, { "id": "thesis:1772", "collection": "thesis", "collection_id": "1772", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05132004-233348", "primary_object_url": { "basename": "thesisfinal7.pdf", "content": "final", "filesize": 392765, "license": "other", "mime_type": "application/pdf", "url": "/1772/1/thesisfinal7.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Conformal Laminations", "author": [ { "family_name": "Gupta", "given_name": "Vineet", "clpid": "Gupta-Vineet" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Calegari", "given_name": "Danny C.", "orcid": "0009-0007-9304-2822", "clpid": "Calegari-D" }, { "family_name": "Hersonsky", "given_name": "Saar", "clpid": "Hersonsky-S" }, { "family_name": "Schlag", "given_name": "Wilhelm", "clpid": "Schlag-W" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A lamination on a circle is an equivalence relation on the points of the circle. Laminations can be induced on a circle by a map that is continuous on the closed disc and injective in the interior. Such laminations are characterized topologically, as being flat and closed. In this paper we investigate the conditions under which a closed, flat lamination is induced by a conformal mapping. We show that if the set of multiple points of the lamination form a Cantor set, whose end points are identified under the lamination, then the lamination is conformal. More generally, the union of such laminations is also conformal. We also show conjecture that any closed, flat lamination, such that the set of multiple points is of logarithmic capacity zero, is conformal.", "doi": "10.7907/MQAY-KR87", "publication_date": "2004", "thesis_type": "phd", "thesis_year": "2004" }, { "id": "thesis:3554", "collection": "thesis", "collection_id": "3554", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-09152006-144938", "primary_object_url": { "basename": "Smirnov_sk_1996.pdf", "content": "final", "filesize": 9203792, "license": "other", "mime_type": "application/pdf", "url": "/3554/1/Smirnov_sk_1996.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Spectral Analysis of Julia Sets", "author": [ { "family_name": "Smirnov", "given_name": "Stanislav K.", "clpid": "Smirnov-Stanislav-K" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" }, { "family_name": "Kahn", "given_name": "Jeremy", "clpid": "Kahn-Jeremy" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Luxemburg", "given_name": "W. A. J.", "clpid": "Luxemburg-W-A-J" } ], "local_group": [ { "literal": "Caltech Distinguished Alumni Award" }, { "literal": "div_pma" } ], "abstract": "We investigate different measures defined geometrically or dynamically on polynomial Julia sets and their scaling properties. Our main concern is the relationship between harmonic and Hausdorff measures.
\r\n\r\nWe prove that the fine structure of harmonic measure at the more exposed points of an arbitrary polynomial Julia set is regular, and dimension spectra or pressure for the corresponding (negative) values of parameter are real-analytic. However, there is a precisely described class of polynomials, where a set of preperiodic critical points can generate a unique very exposed tip, which manifests in the phase transition for some kinds of spectra.
\r\n\r\nFor parabolic and subhyperbolic polynomials, and also semihyperbolic quadratics we analyze the spectra for the positive values of parameter, establishing the extent of their regularity.
\r\n\r\nResults are proved through spectral analysis of the transfer (Perron-Frobenius-Ruelle) operator.
", "doi": "10.7907/X37M-D376", "publication_date": "1996", "thesis_type": "phd", "thesis_year": "1996" }, { "id": "thesis:18509", "collection": "thesis", "collection_id": "18509", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:04242026-202608511", "primary_object_url": { "basename": "Leung_H-M_1996.pdf", "content": "final", "filesize": 14339283, "license": "other", "mime_type": "application/pdf", "url": "/18509/1/Leung_H-M_1996.pdf", "version": "v2.0.0" }, "type": "thesis", "title": "Conformal Laminations on the Circle", "author": [ { "family_name": "Leung", "given_name": "Hoi-Ming", "clpid": "Leung-Hoi-Ming" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "A lamination L on T is an equivalence relation on T. In this paper, we consider laminations\r\nL(\u03c6) induced by some continuous mapping \u03c6: D \u2192 C which is one-to-one\r\nin D, i.e \u03be L(\u03c6) \u019e if and only if \u03c6(\u03be) = \u03c6(\u019e) for any \u03be, \u019e \u2208 T . The laminations\r\narise as in above can be characterized topologically as continuous and flat laminations.\r\nOur major question is what conditions on L can ensure that L=L(\u03c6) for some\r\ncontinuous mapping \u03c6: D \u2192 C which is conformal in D. In this paper, we study\r\nvarious aspects of conformal laminations and get conditions for conformality in several\r\nconfigurations. We relate the conformal welding problem with the classical conformal\r\nse,ving problem. By the extremal length method, we obtain a generalization of\r\nOikawa's condition for conformal sewings to a sufficient condition for conformal laminations\r\nand also obtain necessary conditions for conformal laminations. We prove\r\nthat a continuous, flat lamination L on T is conformal if capML=0 and the quotient\r\nspace ML/L is a totally disconnected set where ML is the set of multiple points of\r\nL. Let E be a compact subset of T. Suppose that In = (an, bn) are the components\r\nof the set T\\E . We define L to be the lamination that identifies an and bn for each\r\nn. We prove that if capE > 0 and E is Dirichlet regular, then the lamination L as\r\ndescribed above is conformal. Furthermore, the quotient space E/L is homeomorphic\r\nto the unit circle. We also conjecture that: If cap ML = 0, then L is conformal.", "doi": "10.7907/vnh6-rp77", "publication_date": "1996", "thesis_type": "phd", "thesis_year": "1996" }, { "id": "thesis:4048", "collection": "thesis", "collection_id": "4048", "cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-10122007-080912", "primary_object_url": { "basename": "Poltoratski_a_1995.pdf", "content": "final", "filesize": 2639843, "license": "other", "mime_type": "application/pdf", "url": "/4048/1/Poltoratski_a_1995.pdf", "version": "v3.0.0" }, "type": "thesis", "title": "Boundary behavior of Cauchy integrals and rank one perturbations of operators", "author": [ { "family_name": "Poltoratski", "given_name": "Alexei G.", "clpid": "Poltoratski-A-G" } ], "thesis_advisor": [ { "family_name": "Makarov", "given_name": "Nikolai G.", "clpid": "Makarov-N-G" } ], "thesis_committee": [ { "family_name": "Unknown", "given_name": "Unknown" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "We develop new methods based on Rohlin-type decompositions of Lebesgue measure on \nthe unit circle and on the real line to study the boundary behavior of Cauchy integrals. We \nalso apply these methods to investigate the notion of Krein spectral shift of a self-adjoint \noperator. Using this notion we study the spectral properties of rank one perturbations of \noperators.\n", "doi": "10.7907/xm4x-r304", "publication_date": "1995", "thesis_type": "phd", "thesis_year": "1995" } ]