PhD in Mathematics & Statistics in United States

Active Areas of Mathematics Research

Stony Brook University's Department of Mathematics offers opportunities for advanced study and research across a wide spectrum of mathematical disciplines. The program focuses on fostering in-depth understanding and contributing to the forefront of mathematical knowledge. Students engage with faculty whose research spans numerous active areas, encouraging intellectual growth and the development of strong analytical skills. The department is committed to providing a rigorous academic environment for aspiring mathematicians.

Critical metrics for the L^2-norm of the curvature tensor

This dissertation explores the concept of $\mathcal{R}$-critical metrics, which are metrics that are critical points for the $L^2$-norm of the curvature tensor functional on a manifold. The functional is defined as the integral of the squared magnitude of the Riemann curvature tensor over the manifold. The research investigates the properties of these metrics, particularly in lower dimensions (three and four), and their relationship to Einstein metrics. The work presents a partial converse to the statement that Einstein metrics are $\mathcal{R}$-critical. Specifically, it shows that in four dimensions, if a metric has non-positive sectional curvature and is $\mathcal{R}$-critical, then it must be an Einstein metric. Furthermore, the dissertation provides classifications of $\mathcal{R}$-critical homogeneous spaces in dimensions three and four. This includes detailed analysis of left-invariant metrics on Lie groups, with specific results for unimodular groups and those with a non-trivial center.

Doctor of Philosophy (Ph.D.) in Mathematics

The Mathematics Department at Princeton University offers a Ph.D. program designed for students passionate about advancing the field of mathematics. The program focuses on rigorous training and research, preparing graduates for academic and research careers. The curriculum is structured over approximately four years, with the possibility of a fifth year based on satisfactory progress and continued need for study. The department encourages applications from students with diverse academic backgrounds and has no rigid course prerequisites, offering a wide range of graduate-level courses to support varied academic directions. International students are encouraged to apply.

Doctor of Philosophy (PhD) in Computational Applied Mathematics and Operations Research

The Doctor of Philosophy (PhD) in Computational Applied Mathematics and Operations Research is designed for students who want to pursue advanced research in applied mathematics. The program aims to provide a strong foundation in graduate-level computational and applied mathematics across various sub-fields. Graduates will be equipped to conduct original research, effectively communicate their findings, and contribute to the field. While students are not typically admitted directly for a Master of Arts (MA) degree, it can be earned along the way to the PhD. The program requires a total of 90 credit hours for the PhD, including coursework and research, and culminates in a dissertation and public oral examination.

Doctor of Philosophy in Mathematics

The University of Michigan offers a Doctor of Philosophy (PhD) program in Mathematics designed to train students for research careers in academia and industry. The program focuses on advanced coursework and original research, preparing graduates to contribute to the field of mathematics. Students will engage with faculty who are leaders in various areas of mathematics, including commutative algebra, algebraic geometry, and representation theory. The curriculum is rigorous, emphasizing theoretical understanding and problem-solving skills. The department provides a supportive environment for students to develop their research interests and collaborate with peers and faculty. Graduates of the program are equipped with advanced analytical and quantitative skills, making them well-suited for a variety of career paths in research, higher education, and specialized industries.

Doctor of Philosophy in Mathematics

The University of California, San Diego (UCSD) offers a Doctor of Philosophy (Ph.D.) program in Mathematics. This program is designed to provide advanced academic training and research opportunities for students pursuing a career in mathematics or related fields. The Ph.D. program typically involves rigorous coursework, comprehensive examinations, and the completion of an original research dissertation. Financial support is a key component for Ph.D. students, with most admitted students receiving financial assistance for up to five academic years. This support commonly includes tuition and fee coverage, along with a salary, often through Teaching Assistant (TA) appointments. Additional funding may be available through Graduate Student Researcher (GSR) positions or fellowships, depending on research projects and funding availability. Both TA and GSR roles provide a salary and full payment of tuition and fees.

Doctor of Philosophy in Statistics

The Doctor of Philosophy in Statistics program at Texas A&M University provides comprehensive training in both statistical methods and theoretical statistics. The program focuses on developing your ability to identify relevant statistical approaches for solving real-world problems and to create new methodologies when necessary. You will gain a robust understanding of current statistical theory and cultivate the mathematical skills required for advancements in statistical methodology. The curriculum is designed to be flexible, allowing you to tailor your studies to your specific academic and research interests.

Doctoral Program in Biostatistics

The Doctoral Program in Biostatistics at Brown University aims to train students to conduct independent research in the theory, methodology, and application of statistics to critical problems in biomedical research, including biology, public health, and clinical medicine. The program emphasizes developing new quantitative methods and their underlying theory, applying innovative statistical approaches to complex scientific problems, and fostering leadership in interdisciplinary research across public health, medicine, biology, and social sciences. Students are required to demonstrate mastery of advanced biostatistical methods through coursework and examinations. This includes a strong foundation in statistical inference, general biostatistical methods, specialized domain areas, and public health principles. Additionally, students must develop competency in a substantive field of application, such as genetics, economics, or epidemiology, by completing relevant graduate-level coursework in another department.

Equivariant Lagrangian Floer Theory on Compact Toric Manifolds

This dissertation explores the field of Equivariant Lagrangian Floer Theory, specifically applied to compact symplectic toric manifolds with subtorus actions. The research introduces an equivariant version of Lagrangian Floer theory and demonstrates its application in proving that the set of Lagrangian torus fibers, along with weak bounding cochain data and non-vanishing equivariant Lagrangian Floer cohomology, forms a rigid analytic space. The work also shows how tropical geometry can be used to locate these Lagrangian torus fibers within the moment polytope. Furthermore, the study applies equivariant theory to establish that moment Lagrangian correspondences, induced by symplectic reduction, are unobstructed after bulk deformation, provided certain equivariant Kuranishi structures and compatible equivariant CF-perturbations exist.

Local Lagrangian and Fixed-Point Floer (Co)homologies

This dissertation presents two related research projects in mathematics. The first project develops a general framework for calculating specific types of invariants in Lagrangian Floer theory using local data. This work extends previous research by introducing a spectral sequence that helps compute these invariants, with applications demonstrated in the study of affine varieties. The second project focuses on a different kind of invariant related to symplectomorphisms, specifically examining families of singularities in Milnor fibrations. It offers a new proof for a conjecture by Zariski concerning the constancy of certain properties in these singularity families, and also recovers a result by Varchenko. While these two projects explore different areas, they are connected by the inspiration for the first project's definitions, which stem from singularities of algebraic varieties. Future work aims to use the developed spectral sequence for further computations. The dissertation includes foundational concepts relevant to both projects before delving into the specifics of each part, which can be read independently.

Many Faces of Renormalization

This program brings together mathematicians and physicists to explore the diverse applications and interconnections of renormalization in dynamical systems. Originally a key tool in Quantum Field Theory and Statistical Mechanics for understanding phase transitions and critical phenomena, renormalization group ideas have since been applied to dynamics, particularly in the study of universality and small-scale structures. The program aims to foster a unified approach to renormalization, examining its manifestations across various fields. It will delve into conformal aspects, including the small-scale properties of conformal dynamical systems and related geometric problems, as well as applications in areas like quasi-periodic Schrödinger operators. Furthermore, the program will investigate renormalization in physical contexts such as Quantum Field Theory, fluid dynamics, and statistical mechanics, focusing on the underlying stochastic mechanisms that drive statistical scaling invariance.

Mathematics

Stony Brook University's Department of Mathematics offers advanced graduate studies leading to a PhD. The program focuses on rigorous theoretical training and research, preparing students for academic and research careers. Doctoral candidates engage in in-depth study and contribute to the field through original research, culminating in a dissertation defense.

Mathematics Ph.D. (Ithaca)

The Doctor of Philosophy (Ph.D.) program in Mathematics at Cornell University is designed for students who aim to become professional mathematicians. The program typically takes five to six years to complete, and students are admitted directly to the Ph.D. program, not for a Master of Science (M.S.) degree. The faculty's broad range of research interests, spanning areas like algebra, analysis, geometry, number theory, and topology, makes the program particularly attractive. Cornell also maintains strong collaborative ties with graduate programs in applied mathematics, computer science, operations research, and statistics and data science. The curriculum emphasizes creating and communicating mathematics, culminating in a dissertation that represents an original and substantial contribution to the field. Students are encouraged to explore various mathematical areas before selecting a research focus and advisor. The program also includes a mandatory teaching assistant training component, preparing students for careers in academia, business, and government.

Mathematics PhD Program

Stony Brook University's Mathematics Department, part of the Institute for Mathematical Sciences, offers a Doctor of Philosophy (PhD) program for students seeking advanced study and research in mathematics. The program focuses on developing rigorous mathematical understanding and research capabilities. Students engage with faculty on cutting-edge research, contributing to the field through original work. This PhD program prepares graduates for careers in academia, research institutions, and advanced roles in industry where strong analytical and problem-solving skills are required.

Nonlinear Optimization

This graduate-level course provides a comprehensive analytical and computational approach to nonlinear optimization problems. It covers a wide range of methods for both unconstrained and constrained optimization, including gradient-based techniques, Newton's method, interior point methods, and Lagrange multiplier methods. The curriculum delves into theoretical aspects such as convex analysis, duality theory, and optimality conditions, often using a geometric perspective. Applications are drawn from diverse fields including control systems, communications, machine learning, and resource allocation.

One parameter families of Schwarz reflection maps arising from Shabat-Belyi maps

This dissertation explores the field of complex dynamics, focusing on anti-holomorphic actions and the iteration of rational maps. It delves into Schwarz reflection maps, a class of anti-holomorphic dynamics, building upon previous work in the field. The research adapts the classical Douady-Hubbard straightening theorem to develop a semi-local theory for the dynamics of these Schwarz reflections, particularly those with parabolic behavior at the cusp point. The study also investigates one-parameter families of these maps, which arise from Shabat polynomials and are indexed by rooted planar trees. The findings demonstrate that the parameter spaces and escape loci for these families are connected and simply connected, implying that the connectedness loci are themselves connected. The research identifies similarities to the Mandelbrot set, showing the presence of many renormalizable parameters that lead to smaller copies of Multibrot and Multicorn sets within the connectedness loci. Furthermore, it is shown that non-renormalizable parameters are combinatorially rigid.

Ph.D. in Statistics

The Ph.D. in Statistics at the University of Michigan offers a flexible curriculum allowing students to specialize in areas such as statistical methodology, interdisciplinary research, theoretical statistics, or probability theory. The program typically involves coursework in the initial stages, followed by a transition to research that culminates in a dissertation. This dissertation is considered the most critical component of the doctoral program.

Ph.D. Program in Biostatistics

The Ph.D. Program in Biostatistics at UC Davis focuses on advanced statistical modeling and inference, applicable to a wide range of fields. These include bioinformatics, biological sciences, veterinary medicine, and traditional areas like medicine, epidemiology, and public health. This interdisciplinary approach leverages UC Davis's strengths in the life sciences, with faculty bringing diverse backgrounds and methodological expertise.

Ph.D. Program in Mathematics

The Ph.D. program in Mathematics at UC Davis is a rigorous graduate program designed to prepare students for advanced research and academic careers in mathematics. The program requires 48 units of graduate coursework, including 24 units of core courses and 24 units of electives. Students have the flexibility to substitute up to three courses outside of mathematics if they are relevant to their specialization. Advancement to candidacy involves fulfilling all program requirements, maintaining a minimum GPA of 3.0, passing preliminary and qualifying examinations, and completing a dissertation with an exit seminar. Core coursework covers essential areas of mathematics, including analysis, algebra, and topology. Students can also take elective courses at the 200-level and engage in research through MAT 299 units. The program also includes requirements for English language proficiency, teaching assistantship experience, and a dissertation demonstrating original research contributions.

PhD in Applied Mathematics

Northwestern University's PhD program in Applied Mathematics offers a comprehensive curriculum combining advanced coursework, in-depth research, and professional development. The program is designed to train scholars and researchers capable of pushing the boundaries of interdisciplinary knowledge. The coursework integrates core applied mathematics topics with specialized courses tailored to students' research interests. Students typically focus on coursework and preliminary exams in the first year, followed by choosing research advisors and commencing dissertation research in the second year. Many students also gain valuable teaching experience as teaching assistants. The program is fully funded for incoming PhD students, including tuition and a monthly stipend. While specific details are being updated for the 2025-2026 academic year, the program has historically involved core courses in differential equations, mathematical modeling, asymptotic analysis, and numerical methods, along with research experience and examinations leading to a dissertation defense. The average completion time for the PhD is approximately five years. The program encourages students from diverse backgrounds with a strong interest in interdisciplinary research, particularly those looking to bridge technological expertise with social sciences and humanities.

PhD in Computational Applied Mathematics

The PhD program in Computational Applied Mathematics at Rice University focuses on developing students' understanding of physical phenomena and their ability to make complex decisions. The program combines intensive, small-class coursework with close interaction with professors. Students have immediate opportunities to participate in seminars and engage in research projects from the outset. Research within the program covers a broad spectrum of mathematical fields and computational methods. Faculty expertise includes numerical analysis, operations research, optimization, differential equations, and scientific computation. Students can conduct high-impact research in application areas such as energy, sustainability, hazards, mathematical biology, and health care delivery, working closely with CMOR faculty.

PhD in Mathematics

This PhD program focuses on pure mathematics and research. It is designed for students who wish to pursue advanced studies and contribute to the field of theoretical mathematics. The department emphasizes a strong research component and expects students to engage deeply with mathematical inquiry. Applicants with primary interests in applied mathematics or computer science are advised to consider other departments at Stony Brook University, as they have separate admissions processes. Admission is typically for the Fall semester only. The application process involves submitting several required documents, including a statement of purpose, transcripts, and recommendation letters. English proficiency test scores are required for non-native English speakers, though this requirement may be waived if prior degrees were earned in English. GRE scores are not considered for admission.

PhD in Mathematics

The PhD program in Mathematics at New York University is housed within the Courant Institute of Mathematical Sciences, a globally recognized center for mathematical research and education. Ranked among the top departments in the US and worldwide, the program offers a rigorous curriculum and extensive research opportunities. Graduates are prepared for diverse careers in academia, research, finance, technology, and government, leveraging advanced mathematical skills to solve complex problems. The program emphasizes both theoretical understanding and practical application, fostering innovation and critical thinking in the field of mathematics.

PhD in Mathematics

The Department of Mathematical Sciences at Carnegie Mellon University offers a PhD program in Mathematics. This program aims to equip students with advanced knowledge and research skills in mathematics. Admission to the program requires applicants to demonstrate a level of competence equivalent to graduating from a recognized U.S. four-year college, university, or institute of technology. The program is designed for students seeking to pursue in-depth study and contribute to the field of mathematics through research.

PhD in Mathematics (Complex Hessian Equations and Poincar�e type Extremal K�ahler Metrics)

PhD in Mathematics (Degeneration techniques in complex geometry)

PhD in Mathematics (Generalized ALG Calabi-Yau Manifolds in Higher Dimensions)

PhD in Mathematics (Monoidal Structures on Fukaya Categories)

PhD in Mathematics (Some regularity results of complex MongeAmpre type equations)

PhD in Mathematics (Stabilized measures of association)

PhD in Mathematics (The Fukaya Algebra over The Integers)

PhD in Mathematics (The Stablilty of the Penrose Inequality in Spherical Symmetry)

PhD in Mathematics (Type Theory and Higher Structures)

Quantitative Methods Doctoral Program

This Ph.D. program in Quantitative Methods, housed within the Department of Educational Psychology, is designed for students with a strong interest in advanced quantitative analysis and research methodology. It equips you to become a professional educational researcher specializing in modern statistical techniques, measurement theory, and research design. The program emphasizes developing sophisticated analytical skills to address complex educational and social questions through both applied and methodological research. You will gain experience in creating, assessing, and refining quantitative models, fostering a rigorous scholarly environment focused on methodological innovation and precision in research. The curriculum involves close collaboration with faculty mentors whose research aligns with your interests. You will complete foundational courses in educational psychology, core quantitative methods competency courses, specialized electives, and coursework outside your specialization, culminating in a qualifying process and dissertation. The program prepares graduates for careers in academia, testing organizations, educational research agencies, and other institutions requiring advanced quantitative expertise.

Quasisymmetries of the Feigenbaum Julia set and transcendental dynamics

This dissertation investigates the group of quasisymmetries of the Feigenbaum Julia set, denoted as Jc. Quasisymmetry is a type of geometric mapping that preserves the qualitative structure of sets under distortion. The Feigenbaum Julia set is a non-hyperbolic quadratic Julia set, presenting a different challenge compared to hyperbolic sets. The research describes the group of topologically extendable quasisymmetric self-maps of Jc, providing the first example for an infinitely renormalizable map. The study relies on the topology and geometry of the transcendental dynamics of a related renormalization fixed point to fully determine this group.

Revisiting Localization, Periodicity and Galois Symmetry

This dissertation explores the relationship between algebraic geometry and topology, specifically how the Galois group of the algebraic closure of rational numbers (Q) affects the underlying manifold structures of complex algebraic varieties defined over Q. While algebraic isomorphisms can exist between varieties, their topological structures (like homeomorphism type) might differ. The research uses concepts like 'profinite completion' to understand these changes, as the Galois action preserves this 'profinite homotopy type'. The study investigates what algebraic-topological data is sufficient to define a manifold within a given homotopy type, how to construct these manifolds algebraically ('étale construction'), and how the Galois action can be expressed using this data.

Stability for the Positive Mass Theorem and the Penrose Inequality

This dissertation investigates the stability of two fundamental inequalities in general relativity and differential geometry: the Positive Mass Theorem and the Riemannian Penrose Inequality. The Positive Mass Theorem states that an asymptotically flat 3-manifold with non-negative scalar curvature must have non-negative mass, and the mass is zero only for Euclidean space. The Penrose Inequality, concerning asymptotically flat 3-manifolds with an outermost minimal boundary, posits a relationship between the mass and the area of the boundary, with equality holding for the Schwarzschild manifold. This work focuses on the stability aspects of these theorems, examining how manifolds behave when their mass is close to the critical values defined by these inequalities. Specifically, it demonstrates that under certain conditions, such manifolds are topologically close to Euclidean space or the Schwarzschild manifold, modulo minor perturbations. The research explores the geometric properties of these manifolds, employing concepts like the pointed measured Gromov-Hausdorff topology. It analyzes scenarios where the mass is almost zero, showing that the manifold approximates Euclidean 3-space. Additionally, it investigates the case where the mass is close to the lower bound given by the Penrose inequality, demonstrating proximity to the Schwarzschild manifold. This stability analysis is crucial for understanding the robustness of these geometric and physical principles.

Statistics, PhD

This program is a Doctor of Philosophy (PhD) in Statistics, offered as a named option within the broader Statistics PhD. It is designed for students with a strong background in mathematics, typically with a bachelor's degree in a natural science, social science, or engineering field. The program emphasizes a strong mathematical foundation for advanced statistical study and research. Applicants should have consistently high undergraduate grades in mathematics to be considered for graduate work in statistics. The program is delivered face-to-face on the UW-Madison campus.

Stony Brook University PhD in Mathematics

Stony Brook University's PhD program in Mathematics provides advanced training for students aspiring to careers in research, academia, or specialized industry roles. The program focuses on developing a deep understanding of mathematical theories and their applications. Students engage in rigorous coursework, independent research, and scholarly activities under the guidance of faculty experts. The program aims to equip graduates with the analytical and problem-solving skills necessary to contribute to the field of mathematics.

The Stability of the Spacetime Penrose Inequality in Spherical Symmetry

This dissertation explores the stability of the Spacetime Penrose Inequality within the framework of general relativity, focusing on spherically symmetric, asymptotically flat initial data that adheres to the dominant energy condition. The research investigates the scenario where the ADM mass is closely aligned with the half area radius of the outermost apparent horizon. Employing the generalized Jang equation approach, this work demonstrates that under these conditions, the initial data must closely resemble an isometric embedding into a static spacetime. Specifically, the time-slice is shown to be near a Schwarzschild time-slice in terms of volume-preserving intrinsic flat distance, static potentials are close in L², and the initial data's extrinsic curvature approximates the second fundamental form of the embedding in L². The Penrose inequality itself posits that the mass of a universe model is bounded below by a function of the area of its boundary black holes. This inequality is crucial in general relativity for understanding the cosmic censorship conjecture, which suggests that singularities (points where spacetime is ill-defined) are hidden behind black holes. If the inequality were false, it could imply unguarded singularities, violating this conjecture. The rigidity aspect of the inequality further suggests that equality implies the universe is a time-slice of Schwarzschild spacetime, the simplest black hole model.

Thesis Defense: Strata of residueless meromorphic differentials

This document is a thesis defense announcement for Myeongjae Lee at Stony Brook University. The thesis is titled 'Strata of residueless meromorphic differentials' and focuses on the enumeration and classification of generalized strata of meromorphic differentials. These generalized strata are specific loci within the usual strata of differentials where the sum of certain residues equals zero. They play a role in the boundary of multi-scale compactifications of standard strata. The research presented investigates strata of residueless meromorphic differentials, which represent the most constrained cases regarding residue conditions. The thesis demonstrates that the connected components of these strata can be categorized by established topological invariants: hyperellipticity, spin parity, and rotation number. Furthermore, for one-dimensional projectivized strata, the research provides formulas to calculate the genus of these strata and the degree of their mapping to the moduli of elliptic curves, utilizing the cellular decomposition derived from the period coordinate.

Topics in Quantitative Rectifiability: Traveling Salesmen, Lipschitz Decompositions, Densities, and Big Pieces

This dissertation presents and proves a series of results in the field of quantitative rectifiability. It explores several key areas: the quantitative rectifiability of Jordan arcs in Hilbert spaces, with a focus on a traveling salesman beta number estimate; the existence of Lipschitz decompositions for domains with quantitatively flat boundaries; the regularity of Hausdorff measure on uniformly rectifiable metric spaces, proving the weak constant density condition; and the iteration of the big pieces operator in Ahlfors regular metric spaces, demonstrating stabilization after two iterations through a general extension theorem.

Topological Quantum Field Theories in Dimension Four

This dissertation explores Topological Quantum Field Theories (TQFTs) in four dimensions. It examines the Crane-Yetter theory through three distinct applications. First, it demonstrates the equivalence between Crane-Yetter theory and the Turaev shadow state sum as a 4-manifold invariant. Second, it investigates the values of Crane-Yetter theory for 2-manifolds, identifying these values as linear categories and describing their structures for oriented surfaces with at least one puncture. Finally, an application of Crane-Yetter theory to a problem within tensor categories is presented. The work builds upon previous research by the author and offers insights into representing mathematical objects and their relationships using advanced algebraic structures.

Tunnels, Wells, and Scalar Curvature

This dissertation explores advancements in Riemannian geometry, specifically focusing on how scalar curvature influences the shape and size of mathematical spaces. It introduces new examples related to stability conjectures concerning the characterization of spheres by scalar curvature. The work addresses the necessity of conditions that prevent 'bubbling' in sequences of manifolds and constructs examples that converge in a specific sense (volume-preserving intrinsic flat sense) but not in another (Gromov–Hausdorff sense). This is achieved by refining a technique called the 'tunnel construction' to allow for the attachment of 'tunnels' or 'wells' to manifolds while minimally affecting scalar curvature. The research also builds upon and extends previous work on constructing manifolds with specific properties, including limits with no geodesics and counterexamples to existing conjectures. The study delves into the theoretical aspects of scalar curvature and its implications for classifying geometric structures. It presents novel constructions that challenge existing conjectures and expand the understanding of manifold properties. The dissertation aims to provide a deeper insight into the relationship between curvature, convergence of manifolds, and their topological characteristics, offering non-perturbative counterexamples with complex topologies.

Understanding the Defining Ideal Through Cones

This dissertation investigates the properties of "cone polynomials" in algebraic geometry, focusing on when these polynomials can generate the defining ideal of a smooth projective variety. The primary area of study is the specific case where the variety X is a finite set of points in a 2-dimensional projective space (P²). The research explores the relationship between cone polynomials and the Castelnuovo-Mumford regularity of varieties. Cone polynomials offer a geometric method to derive equations that define a variety, and understanding their generating power helps in bounding the algebraic complexity of these varieties. The work builds upon previous research, particularly concerning generic points in P² and the ideal generated by cone polynomials. A key finding presented is the result for collinear points in P², which establishes that the ideal generated by cone polynomials matches the defining ideal of the points for degrees k greater than or equal to 2d-2, where d is the number of points. This result is extended to arbitrary finite sets of points in P² through deformation arguments, providing an upper bound for the saturation degree.