Students have completed a projects on a wide variety of topics throughout the years. See the below ideas of some topics that might help you to determine your project focus.<\/p>\n
Topic<\/strong>: Finite Dimensional [latex]C^*[\/latex]-Algebras Topic<\/strong>: Quantum Mechanics Topic<\/strong>: Markov Processes Topic<\/strong>: Perron-Frobenius Theory <\/p><\/div><\/div><\/div>\n\n\n\n\n\n\n\n Topic<\/strong>: Birkhoff’s Theorem Topic<\/strong>: The Sylow Theorems Topic<\/strong>: Geometric Group Theory Topic<\/strong>: [latex]p[\/latex]-adic Analysis Topic<\/strong>: Elliptic Curves Topic<\/strong>: Hilbert Spaces Topic<\/strong>: Irrationality and Transcendence Topic<\/strong>: Space-Filling Curves Topic<\/strong>: Infinitesimals and Hyperreals Topic<\/strong>: [latex]p[\/latex]-adic Analysis Topic<\/strong>: Smooth Manifolds Topic<\/strong>: Symplectic Geometry Topic<\/strong>: de Rham Cohomology
\r\nSuggested Background<\/strong>: MATH 2600: Linear Algebra<\/p>\r\n\r\n
\r\nSuggested Background<\/strong>: MATH 2600: Linear Algebra
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Background<\/strong>: MATH 2600: Linear Algebra
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Background<\/strong>: MATH 2600: Linear Algebra<\/p>\r\n\r\n<\/i>Algebra<\/a><\/h4><\/div>
\r\nSuggested Text<\/strong>: Universal Algebra – Fundamentals and Selected Topics, Clifford Bergman
\r\nSuggested Background<\/strong>: MATH 3300 (Abstract Algebra)
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: Algebra, Michael Artin
\r\nSuggested Background<\/strong>: MATH 3300 (Abstract Algebra), or some familiarity with groups
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nBook<\/strong>: Office Hours with a Geometric Group Theorist, Matt Clay & Dan Margalit
\r\nSuggested Background<\/strong>: MATH 3200 (Introduction to Topology), MATH 3300 (Abstract Algebra)
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: [latex]p[\/latex]-adic Numbers – An Introduction, Fernando Q. Gouvea
\r\nSuggested Background<\/strong>: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
\r\nDescription<\/strong>: [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.<\/p>\r\n\r\n
\r\nBook<\/strong>: Rational Points on Elliptic Curves, Joseph H. Silverman & John Tate
\r\nSuggested Background<\/strong>: MATH 3300 (Abstract Algebra), MATH 3800 (Theory of Numbers)
\r\nDescription<\/strong>: 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.<\/p><\/div><\/div><\/div>\n\n\n\n\n\n\n\n<\/i>Analysis<\/a><\/h4><\/div>
\r\nSuggested Text<\/strong>: Hilbert Space: Compact Operators and the Trace Theorem, J. R. Retherford
\r\nSuggested Background<\/strong>: MATH 3100 (Introduction to Analysis), MATH 2600 (Linear Algebra)
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: An Introduction to Number Theory, Ivan Niven, Herbert Zuckerman, & Hugh Montgomery, and Transcendental Numbers, M. Ram Murty & Purusottam Rath
\r\nSuggested Background<\/strong>: MATH 3300 (Abstract Algebra) is a prerequisite for the more advanced transcendence theory material
\r\nDescription<\/strong>: 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!<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: Space-Filling Curves, Hans Sagan
\r\nSuggested Background<\/strong>: MATH 3200 (Introduction to Topology)
\r\nDescription<\/strong>: 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<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: Foundations of Infinitesimal Calculus, H. Jerome Keisler
\r\nSuggested Background<\/strong>: MATH 3100 (Introduction to Analysis)
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: [latex]p[\/latex]-adic Numbers – An Introduction, Fernando Q. Gouvea
\r\nSuggested Background<\/strong>: MATH 3100 (Introduction to Analysis) or MATH 3300 (Abstract Algebra)
\r\nDescription<\/strong>: [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.<\/p><\/div><\/div><\/div>\n\n\n\n\n\n\n\n<\/i>Geometry and Topology<\/a><\/h4><\/div>
\r\nSuggested Text<\/strong>: Introduction to Smooth Manifolds, John M Lee (text may vary based on specific subject matter)
\r\nSuggested Background<\/strong>: MATH 3200 (Introduction to Topology) and some Linear Algebra are useful
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: Introduction to Symplectic Topology, McDuff and Salamon OR Lectures on Symplectic Geometry, Ana Cannas de Silva (pdf)
\r\nSuggested Background<\/strong>: MATH 3200 (Introduction to Topology) and MATH 2600 (Linear Algebra)
\r\nDescription<\/strong>: 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.<\/p>\r\n\r\n
\r\nSuggested Text<\/strong>: From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Ib Madsen & Jxrgen Tornehave
\r\nSuggested Background<\/strong>: MATH 2300 (Multivariable Calculus) and MATH 2600 (Linear Algebra), though MATH 3100 (Introduction to Analysis) and MATH 3200 (Introduction to Topology) may be helpful
\r\nDescription<\/strong>: “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.” —\u200aTerence Tao, Differential Forms and Integration<\/p><\/div><\/div><\/div>\n\n\n\n\n\n\n\n