Topic: Finite Dimensional [latex]C^*[/latex]-Algebras
Suggested Background: MATH 2600: Linear Algebra

Topic: Quantum Mechanics
Suggested Background: MATH 2600: Linear Algebra
Description: When people think of physics, they often think of physical phenomena at the macroscopic scale (the momentum of a wrecking ball or the velocity of a car for instance). However, the theories of classical physics don’t work when trying to capture the behavior of things at the atomic scale. Quantum mechanics was developed to fill this hole in our understanding! The main mathematical tool to model quantum mechanical systems are Complex Hilbert Spaces (see above). This project would focus on exploring the connection between Hilbert spaces and Quantum mechanics.

Topic: Markov Processes
Suggested Background: MATH 2600: Linear Algebra
Description: A Markov process is a stochastic process that can make predictions using only the current state of a system. Importantly, the past plays nol role in the prediction. As it turns out, these processes are very useful when it comes to modeling real world phenomena as well as being heavily used in various data analysis and machine learning algorithms.

Topic: Perron-Frobenius Theory
Suggested Background: MATH 2600: Linear Algebra

Topic: Birkhoff’s Theorem
Suggested Text: Universal Algebra – Fundamentals and Selected Topics, Clifford Bergman
Suggested Background: MATH 3300 (Abstract Algebra)
Description: Universal Algebra generalizes many of the common structures encountered in abstract algebra (groups, rings, etc) to talk algebraic structures in general, including general direct products, homomorphisms, subalgebras, and congruences. The last, congruences, generalize normal subgroups and ideals and provide a general treatment of the Isomorphism Theorems. By constructing free algebras, Birkhoff’s Theorem can be proven, showing that classes of algebras defined by equations coincide exactly with classes of algebras which are closed undertaking direct products, homomorphic images, and subalgebras.

Topic: The Sylow Theorems
Suggested Text: Algebra, Michael Artin
Suggested Background: MATH 3300 (Abstract Algebra), or some familiarity with groups
Description: The famous Lagrange’s Theorem says that if [latex]G[/latex] is a finite group and [latex]H[/latex] is a subgroup of [latex]G[/latex], then [latex]|H|[/latex] divides [latex]|G|[/latex]. As a consequence, every element of [latex]G[/latex] has order which divides [latex]|G|[/latex]. The Sylow Theorems provides partial corollaries to Lagrange’s Theorem, showing that if [latex]p^n[/latex] is the largest power of a prime [latex]p[/latex] which divides [latex]|G|[/latex], then there exists a subgroup [latex]P[/latex] of [latex]G[/latex] of order [latex]p^n[/latex], as well as further properties of such Sylow [latex]p[/latex]-subgroups and conditions on the number of such subgroups. These end up providing useful tools for determining the number of isomorphism classes of groups of a given finite size.

Topic: Geometric Group Theory
Book: Office Hours with a Geometric Group Theorist, Matt Clay & Dan Margalit
Suggested Background: MATH 3200 (Introduction to Topology), MATH 3300 (Abstract Algebra)
Description: After learning the basics of group theory, a student can delve in to group actions and the interplay between geometry and groups. The Cayley graph associated to a group is a fundamental geometric object associated to a group, and by understanding the group action that arises on this space and on other spaces, one can begin to see many of the surprising properties of infinite groups and many of the fundamental interactions between geometric group theory and other fields of study.

Topic: [latex]p[/latex]-adic Analysis
Suggested Text: [latex]p[/latex]-adic Numbers – An Introduction, Fernando Q. Gouvea
Suggested Background: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
Description: [latex]p[/latex]-adic analysis represents a different approach to correcting the failings of the field of rational numbers, with a resulting theory that looks wildly different from the classical analysis of the real line. The [latex]p[/latex]-adic reals (for each prime [latex]p[/latex]) are nevertheless rich objects of study, both from an analytic point of view, as well as an algebraic one.

Topic: Elliptic Curves
Book: Rational Points on Elliptic Curves, Joseph H. Silverman & John Tate
Suggested Background: MATH 3300 (Abstract Algebra), MATH 3800 (Theory of Numbers)
Description: Elliptic curves have become one of the most exciting fields of study in recent years. Fundamentally, elliptic curves are simply the solution set of [latex]y^2 = x^3 + ax + b[/latex], which would appear to not be much more difficult to understand than conic sections. However, it turns out that they contain a breadth of number theoretic information, being fundamental to Andrew Wiles’s proof of Fermat’s Last Theorem. Additionally, they have proved useful in a variety of other areas, such as cryptography. A reading program in this area would entail learning the basics, giving the student an understanding of both the difficulty and depth of this area, and allowing them to see why so many mathematicians have become fascinated with these objects.

Topic: Hilbert Spaces
Suggested Text: Hilbert Space: Compact Operators and the Trace Theorem, J. R. Retherford
Suggested Background: MATH 3100 (Introduction to Analysis), MATH 2600 (Linear Algebra)
Description: In a few words, Hilbert spaces are complete inner product spaces. In the same way that the theory of vector spaces is extremely nice (the dimension of a vector space determines that vector space up to linear isomorphism), the theory of Hilbert spaces is similarly nice. Likewise, Hilbert spaces have diverse applications in both mathematics and physics.

Topic: Irrationality and Transcendence
Suggested Text: An Introduction to Number Theory, Ivan Niven, Herbert Zuckerman, & Hugh Montgomery, and Transcendental Numbers, M. Ram Murty & Purusottam Rath
Suggested Background: MATH 3300 (Abstract Algebra) is a prerequisite for the more advanced transcendence theory material
Description: Irrational numbers are real numbers which cannot be expressed as the quotient [latex]n/m[/latex] of two integers (where [latex]m[/latex] is non-zero). Transcendental numbers are real numbers which are not the roots of any polynomial of a single-variable whose coefficients are rational. Famously, both [latex]e[/latex] and [latex]pi[/latex] are transcendental, but proving this is extremely non-trivial. In general, transcendentality is difficult to prove (it is known that at least one of [latex]e+pi[/latex] and [latex]ecdot pi[/latex] are transcendental, but neither has been proven so). However, it ends up that almost all real numbers are transcendental!

Topic: Space-Filling Curves
Suggested Text: Space-Filling Curves, Hans Sagan
Suggested Background: MATH 3200 (Introduction to Topology)
Description: In the 1800s, Georg Cantor showed that the unit interval and unit square are in a one-to-one correspondence, astounding the mathematical community. Such a bijection, however, cannot be continuous. Weakening the hypothesis that such a function from the unit interval to the unit square is only surjective, there are in fact continuous such functions. These are ‘space-filling curves’. More generally, the Hahn-Mazurkiewicz Theorem states that a non-empty Hausdorff topological space is a continuous image of the unit interval if and only if it is compact, connected, locally-connected, and second-countable

Topic: Infinitesimals and Hyperreals
Suggested Text: Foundations of Infinitesimal Calculus, H. Jerome Keisler
Suggested Background: MATH 3100 (Introduction to Analysis)
Description: In its infancy, calculus was described in terms of ‘infinitesimals’, which were non-zero quantities which were smaller in magnitude than any real number. Using this intuition, limits, continuity, differentiation, and integration were developed and studied. As time wore on, a distrust of such notions grew and eventually was replaced with the modern approach ushered in by Weierstrass and Cauchy. In the mid 1900s, however, Abraham Robinson showed that infinitesimals could be placed on a rigorous foundation, bringing back the old motivations in the form of non-standard analysis.

Topic: [latex]p[/latex]-adic Analysis
Suggested Text: [latex]p[/latex]-adic Numbers – An Introduction, Fernando Q. Gouvea
Suggested Background: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
Description: [latex]p[/latex]-adic analysis represents a different approach to correcting the failings of the field of rational numbers, with a resulting theory that looks wildly different from the classical analysis of the real line. The [latex]p[/latex]-adic reals (for each prime [latex]p[/latex]) are nevertheless rich objects of study, both from an analytic point of view, as well as an algebraic one.

Topic: Smooth Manifolds
Suggested Text: Introduction to Smooth Manifolds, John M Lee (text may vary based on specific subject matter)
Suggested Background: MATH 3200 (Introduction to Topology) and some Linear Algebra are useful
Description: The word manifold is thrown around a lot in math and physics, but what is a manifold? Informally, manifolds are spaces where at each point, it looks like Euclidean space. For instance, the Earth is a sphere, but to people standing on the surface, it looks flat like the plane. This notion can be made rigorous using ideas from topology, and it doesn’t stop there! You can then study manifolds with additional structures. This project would focus on topics surrounding smooth manifolds.

Topic: Symplectic Geometry
Suggested Text: Introduction to Symplectic Topology, McDuff and Salamon OR Lectures on Symplectic Geometry, Ana Cannas de Silva (pdf)
Suggested Background: MATH 3200 (Introduction to Topology) and MATH 2600 (Linear Algebra)
Description: One of the additional structures one could place on a smooth manifold is a symplectic structure. This consists of a smoothly varying symplectic form on the tangent space at the point. While it might not be immediately apparent, this structure gives rise to many interesting mathematical phenomena (the Principle of the Symplectic Camel for instance) as well as being adjacent to Hamiltonian mechanics. This project would explore basic properties of symplectic manifolds with the end goal of connecting symplectic manifolds to contact/Kahler manifolds or Hamiltonian dynamics.

Topic: de Rham Cohomology
Suggested Text: From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Ib Madsen & Jxrgen Tornehave
Suggested Background: MATH 2300 (Multivariable Calculus) and MATH 2600 (Linear Algebra), though MATH 3100 (Introduction to Analysis) and MATH 3200 (Introduction to Topology) may be helpful
Description: “The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.” — Terence Tao, Differential Forms and Integration

Topic: Intro to Relativistic Fluids
Description: What are relativistic fluids and how can we model them using PDEs? We can discuss standard PDE and geometric techniques that are used to show the existence of solutions, as well as applications.

Miscellaneous PDE Topics: Quasilinear wave equations, functional-analytic techniques for local well-posedness

Miscellaneous topics in Mathematical Physics: based on interest of the student, Special/General Relativity, Magneto-Hydrodynamics (MHD), astrophysics, etc.

Topic: Szemeredi’s Regularity Lemma and Applications.
Suggested Text: Graph Theory, written by Reinhard Diestel, Graduate Texts in Mathematics 173
Suggested Background: MATH 4710 Graph Theory
Description: In the course of the proof of a major result on the Ramsey properties of arithmetic progressions, Szemeredi developed a graph theoretical tool whose fundamental importance has been realized more and more in recent years. Roughly speaking, the lemma says that all graph can be approximated by random graph in the following sense: every graph can be partitioned, into a bounded equal parts, so that most of its edges run between different parts and the edges between any two parts are distributed fairly uniformly–just as we would expect it if they had been generated at random.

Topic: Linear Algebra in Graph Theory
Suggested Text: Algebraic Graph Theory, Norman Biggs
Suggested Background: MATH 2600 Linear Algebra
Description: In the study of algebraic graph theory, the aim is to translate properties of graphs into algebraic properties and then, using the results and methods of algebra, to deduce theorems about graphs. We first introduce the adjacency matrix of graph, which completely determines the graph, and its spectral properties are shown to be related to properties of the graph. Another matrix which completely describes a graph is the incidence matrix of the graph. This matrix represents a linear mapping which determines the homology of the graph.

Topic: Introduction to Random Graphs and Ramsey Theory
Suggested Text: Graph theory and Additive Combinatorics, Yufei Zhao; Random Graphs, Bela Bollobas
Suggested Background: MATH 3640 Probability
Description: The theory of random graphs was founded by Erdos and Renyi had discovered that probabilistic methods were often used in tackling extremal problems in graph theory. Although Erdos did not construct such graphs explicitly but showed that most graphs in a certain class could be altered slightly to give the required examples.