The UCD Mathematics Department is located in the newly-completed Mathematical Sciences Building. The department is headed by Chairperson Joel Hass. The Chair of the Graduate Group in Applied Mathematics (GGAM) is Naoki Saito. The official office for departmental matters is MSB 1130. The staff are extremely helpful.
The department has been, and continues to be, home to many brilliant researchers. For a number of years Bill Thurston, Fields Medalist c/o his Geometrization Conjecture, was a member of the faculty. Mikhail Khovanov was in the department when he developed the famous homology theory that bears his name. Wayne Rosing is a senior fellow involved with the Large Synoptic Survey Telescope. Roger Wets, a professor in the department, is a managing editor of the Journal of Convex Analysis. A Mathematics ArXiv front and the Journal of Mathematical Physics are also run from within the department. Also of note, Craig Tracy has the famous Tracy-Widom distribution named for him.
The Math department offers A.B. and B.S. degrees, as well as a minor. The difference between the A.B. and B.S. is, as usual, the A.B. degree has less strict guidelines for courses as well as having less courses required (usually). The A.B. degree is probably better suited for students pursuing another major or those who wish to tailor a more unique upper division program. Do not forget to fulfill the three quarter foreign language requirement if you plan on getting an A.B. The minor is simply 20 units in upper division courses.
Within the B.S., students must choose a track within the major that decides the course of upper-division study. The tracks are "General Mathematics", "Mathematical and Scientific Computation", "Applied Mathematics", and "Mathematics for Secondary Teaching". Basically, if you are interested in pure mathematics, the general track is the most appropriate. While some people use to think that the pure track provides the best preparation for graduate school in other mathematical disciplines (i.e. theoretical physics), the applied track is much more rigorous than it use to be. The only difference between the two is that in the pure track you take three quarters of abstract algebra, as opposed to some abstract algebra and coursework in mathematical applications. The Computational track with a computing emphasis seems to attract a lot of students from computer science who wish to double-major in Mathematics. However, the computational track also has a biology emphasis option. This particular emphasis attracts people who like the interface between computers, mathematics and biology. The other option for those who are interested in mathematical biology is taking any of the three graduate research oriented tracks (general, applied, or scientific computation with a biology emphasis) and minor in Quantitative biology and bioinformatics and/or do research in the CLIMB program. The applied track focuses more on the theory behind solving problems in the specialty field of your interest (biology, physics, engineering, economics...), while the computational track emphasizes the computer programing used to solve such problems.
The Teaching track is, well, for those who wish to become High School/lower teachers. The program doesn't incorporate getting teaching credentials, but that's something you have to do on your own. However, if you are interested in teaching you should consider the MAST program or working as a group tutor at the learning skills center, to give you some experience.
The General Track requires completion of the 125 series, the 150 series, 135A, and 185A. Computational/Applied tracks require the 125 series, 150A, and 135A. 185A is also required for the applied track, but not the computational track. Both the applied and computational track require the completion of 128AB (computational must also complete 128C). Teaching requires the 125 series, 111, 115A, 135A, 141, and 150A. After taking these required classes for the specific track, one may choose between several restricted classes for the completion of their major.
There are two main graduate programs: Pure Mathematics (M.A. and Ph.D.), Applied Mathematics (M.S. and Ph.D.). There used to be a Teaching Mathematics (M.A.T.), but the math department phased the M.A.T. program out. All students must take the MAT 201 series in their first year. Those in the pure track must also take MAT 250AB. Those in the applied track used to take MAT 119A, an undergraduate class, but now get to take the new MAT 207 series in applied methods.
- Mathematics Department website — http://math.ucdavis.edu
- UC Davis General Catalog — http://registrar.ucdavis.edu/UCDWebCatalog/programs/MAT/MATfac.html
- Official Mathematics Department Facebook page — https://www.facebook.com/DepartmentOfMathematicsUcDavis
- One of the best professors I have ever had. Very enthusiastic and passionate about mathematics and especially so in his research areas. At first, I thought he was really intimidating, but as time has passed, he's actually one of the most personable faculty in the mathematics department. SachinSalgaonkar
- I had her for Math 127B and liked her a lot, her tests were kinda on the easy side though :) BryanBell
- Clever man. I took 22B and 131 with him, and he went extremely in depth with those courses, using real-world examples from engineering and physics. It was slightly terrifying. The 131 class went full-bore with derivations, clearly a learn-or-die situation, requiring a blue book for the final. But when we got there for the final showdown, and he passed out the test, we saw that it was just 20 multiple-choice questions, no written section at all. Easy... too easy. Turns out he had been watching the TAs threatening to strike that week, and he didn't want to risk having to grade all those written tests himself. — EricTalevich
So, everyone has to take Calculus. We have three Calculus series: 21, 17, and 16. 21 is for people who will take more Mathematics courses after the series, 17 is the newly added "Calculus for Biosci majors" (basically 16 with examples drawn from bioscience), and 16 is for those who need only Calculus for their further work. This means Science, Engineering, and Mathematics type majors have to take 21, most bio take 17, and Social Science / some Biology take 16. What you take after this depends highly on your major.
22A - Linear Algebra. Starts from the very beginning: Matrix addition, dot products, cross products, inverses, Gaussian elimination, discriminants, eigenvalues and eigenvectors. An additional 1-unit section in MATLAB (22AL) is often taken concurrently, in order to show beginning students the utter futility of the grunt work they do by hand.
- It's my opinion that this is the most confused course in the department. It starts off with very basic operations on matrices, which are used to solve basic linear equations. This seems to encompass the majority of the class no matter how it's taught. Then the instructor must also use this class to prepare advancing students for material in upper division mathematics courses, and must focus a lot on vector spaces, change of basis, and linear operators (with their eigenvalues, eigenvectors). This leads to a tedious, calculation-based course (for the majority) and then a half-assed effort at a real linear algebra course. The 167 course tries to remedy this but fails due to the amount of material it must make up for. —PhilipNeustrom
- I've seen similar problems at other universities. Few schools seem to be able to put together a coherent set of courses dealing with linear algebra. —RoyWright
- "It's my understanding that the FRC is trying to restructure some of the mid-level classes to better prepare advancing students in this regard." —MarianneWaage
- "This class requires concurrent enrollment in 22AL, or alternatively Engineering 6. —DavidPoole
- Concurrent enrollment in Engineering 6 does not excuse you from the MATLAB requirement; you will need to take 22AL too. Why they do that is a mystery. —HarrisonM
But if you already have taken ENG 6, you are excused from 22AL. Or if you already have knowledge of MATLAB. Here is the course description from the catalog:
22A. Linear Algebra (3)
Lecture—3 hours. Prerequisite: nine units of college mathematics and Engineering 6 or knowledge of Matlab
or course 22AL (to be taken concurrently). Matrices and linear transformations, determinants, eigenvalues,
eigenvectors, diagonalization, factorization.
Not open for credit to students who have completed course 67.—I, II, III. (I, II, III.)
22B - Differential Equations. You learn to solve and classify basic ordinary differential equations. There may or may not be very complex applications shown, depending on the professor.
- taught by dr. john hunter, this may have been my favorite lower division class
23 - Introduction to Numerology. Covers the early history of numerology, the basics of theoretical numerology, and a few important applications. Sometimes BarnabasTruman is the TA for this class. Offered whenever the year in the Erisian reckoning of time doesn't relate to the number five in any way whatsoever.
25 - Advanced Calculus. A course that gives the fundamentals necessary for Real Analysis. Topics cover: sets, induction, infimum, supremum, sequences, series, proof writing, and some more properties of real numbers. This is a proof oriented class, that leads into upper division math. (replaced course 127A as of F2006)
67 - Modern Linear Algebra. Similar to 22a but more theory oriented. Expect to see plenty of proofs on your tests, as this class is meant to lead into upper division mathematics
108 - Introduction to Abstract Mathematics. This is perhaps the most talked-about course in the department. This is a "proof class" in the sense that you're supposed to learn how to give a good, rigorous mathematical argument. You can learn a lot in this class, but most people have bad experiences overall. There is a certain understandable level of anality (this isn't a word) that's given to the students after this class, but it is sometimes hard for students to learn that not every argument is given in strict propositional and existential quantifier terms. Back when this class was required for all math majors, it was considered to be the weeder class. When taught by some professors the mean grade can be as low as a C-. All Computer Science majors up until 2008 had to take this before they graduated, hence most of the people in the course were Computer Science majors.
111 - History of Mathematics Don't be fooled by the title. If you are a history major with no background in math do not take this course, there is actually a lot of math involved, just not as much as other upper division math courses. Students learn some ancient cultures math and then the history of western math through the later half of the millennium.
114 - Convex Geometry. A convex region is one where all of the line segment between any two points in the region is in the region. So (the interior) of a triangle is convex, as is a circle, but a star is not. We will study the geometry of convex regions in 2, 3, 4, and many dimensions, with a lot of help from linear algebra, covering both general theory and various interesting examples and constructions of convex sets. Exact topics may depend to some extent on the interests of the students. —AlexanderWoo, who is teaching this Winter 2006
115A - Number Theory. You learn basic properties of congruences, prime numbers, diophantine equations, and learn some interesting functions (such as Euler's function and the Moebius function).
115BC are a continuation of A and topics are the choice of the instructor.
116 - Differential Geometry. A slight continuation of 21D. You study curves and surfaces and their curvature properties using vector analysis and differential geometry.
The philosophical implications of the course are profound. Differential geometry provides a context in which one can study many kinds of (non-Euclidean) geometries through the lens of calculus. Geometries will often be weird and interesting. For example, if let's look at the geometry on a sphere. A differential equation tells us that the great circles (equators of the sphere) are the analogs of straight lines. Thus, in a small enough region, the straight lines will be the paths that minimize distance between points. That's the same as in Euclidean geometry. On the other hand, two different equators will always intersect, so there are no "parallel" lines in the geometry of the sphere. Because the heavy machinery of differential calculus is used, such geometric calculations can be made in an efficient manner. It is much easier to see what goes on, than if the elegant, though cumbersome, axiomatic method of Euclid were the main tool.
This class studies curves and surfaces, but that is only the beginning of differential geometry. When we speak about a curve, we are considering a 1-dimensional object embedded in our 3-dimensional space. When we talk about a surface, we are considering something 2-dimensional embedded in 3-dimensional space. But we only put the 3-dimensional space in because we perceive ourselves to live in 3-D space, so that seems the most natural surrounding for our curve or surface.
In the nineteenth century, Riemann first realized that we may divorce the curve and surface from the ambient 3-D space. Furthermore, why limit ourselves to 1- or 2-dimensional objects? We can consider 4, 5, or N dimensions—as many as we want for a purpose at hand. Riemann called these N-dimensional surfaces, liberated from any surrounding space, manifolds. Einstein's theory of General Relativity considers the space-time continuum to be a 4-dimensional manifold. Gravity is curvature in the manifold, and particles under the influence of gravity follow the straight lines in the non-Euclidean geometry.
118A - Partial Differential Equations. You learn some methods of solving basic, special-case, PDEs including separation of variables. A major emphasis is placed on understanding the wave and diffusion/heat equations with various boundary conditions. The class also covers classical Fourier series and its application to solving PDEs on finite intervals for the last couple weeks of class.
118B - Partial Differential Equations. You learn about Green's Functions, a very useful way to solve some linear PDEs. Unfortunately, the Green's Function is usually quite difficult to determine. It depends on the geometry of the boundary conditions, and involves quite a bit of work even for simple geometry. For more complicated geometry, the problem becomes intractable. Also, Fourier series solutions are studied in more depth. The sines and cosines in a Fourier series can be thought of as a "basis" for the space of functions, in the sense of linear algebra. You learn about several different kinds of "bases" in that spirit. Selected topics from math 118C may be covered if 118C is not being offered the following quarter, such as distributions, Fourier transforms, and calculus of variations. The difficulty level tends to be stepped up a notch in B and C because the material is less intuitive.
118C - Partial Differential Equations. Hands down, the most thrilling of the 118 series. Like a cathedral among chapels. You learn the theory of distributions—a distribution being a mysterious "generalized function". For example, the delta function is a distribution that is 0 everywhere but at x = 0, and has an integral of 1. But this cannot be a function, it must be something more general! It turns out that the delta function and Green's Function are intimately related. Moreover, you learn about an extremely useful operation called convolution. You learn about Laplace and Fourier Transforms. The power of these combined techniques in essence allow you to solve any linear PDE.
119A - Ordinary Differential Equations. ODE's are a vital part of most sciences. This is sometimes referred to as "the Phase-plane class." It focuses on phase planes, stability of fixed points, bifurcations, classification of singularities, and various other forms of analyzing ODE's. About half of the class is generally composed of bored applied math graduate students.
119B - Ordinary Differential Equations. This class focuses on chaotic differential equations, maps, and fractals (some professors add a variety of topics that may have been skimmed over in 119A). Even though this course holds the majority of the interesting material on dynamical systems, most grad students do not continue on to 119B (mainly because it is not required). As a result the class is often small and intimate. Depending on the professor this course may contain some programing, although no previous programing experience is expected.
- this is by far the highlight of the two part series. A lot of what you learn in 119A is applied to more interesting systems. Maps are introduced and explored. This class tends to be small and hence it is often more project/exploratory oriented. If you are an applied math major this may be one of the more important classes you take, because if you get the right professor, you will get to do a project related to your field of application. In spring 2007 Dr. Biello did a wonderful job of engaging this class in the material. If you didn't hate 119A, i urge you to continue, because it got a lot more fun during the second quarter.
124 - Mathematical Biology. This course focuses on the mathematics used to model and analyze many biological systems. Topics tend to include a lot of differential equations and linear algebra, with applications to cell biology, neuroscience, and ecology.
125AB - Real Analysis (replace courses 127BC as of F2006). This is a series in elementary real analysis. This means that a lot of the material will be familiar to you from your previous courses but will be set in a much more rigorous framework and worked with from there. 125A will relate to material covered in 21A and 125B will relate to material covered in 21BCD and will also use tools learned in 22A/67. After covering basic 1D integration, the B course focuses on developing more advanced topics in elementary analysis such as the total derivative (or Frechet derivative), the implicit/inverse function theorems, jordan regions, multivariate integration, and change of variables.
128ABC - Numerical Analysis. This is not a "series" and can be taken in any order. The topics differ but all focus on developing algorithmic methods of numerically solving or approximating mathematical problems. Examples include spline interpolation and numerical differentiation and integration. Involves a fair amount of programming in MATLAB.
129 - Fourier Analysis (new course as of F2006). This course fills the gap for those who need to learn tools like Fourier series and Fourier transforms but cannot afford the time to take 119 and 118.
133 - Mathematical Finance (new course as of F2006). Will first be offered W2007.
135A (131 prior to F2006) - Probability Theory. This is an introductory course in probability. Covers events, sample spaces, random variables, expectation, mean (and other moments), density, mass, and distribution functions (along with various examples of popular distributions), moment-generating and characteristic functions, and various limit theorems. There may be fairly complex/advanced examples presented. Now modified to differ from Statistics 131 (details pending).
- When I took the course last quarter (W2012), the course ended at sums of expectations. Moment-generating functions, bounds such as the Markov and Chebyshev Inequalities, were all reserved for 135B. SachinSalgaonkar
135B (132A prior to F2006) - Stochastic Processes. A continuation of 135A in the direction of stochastic processes, i.e. those that change randomly with time. Covers branching processes, Markov chains.
141 - Euclidean Geometry. This course has typically been taught to be an axiomatic, slow, and thorough treatment of Euclidean geometry. In recent years it has incorporated much (if not most of the course) time to discussion of alternative geometries such as spherical and hyperbolic. the course starts with axiomatic rules and systems and then moves on to comparing and contrasting the axioms, theorems, and ideas in hyperbolic, euclidean, and spherical surfaces. However, sometimes the course is less rigorous and more exploratory (ie building hyperbolic planes out of paper). It is a good course to take after 108 when you are still getting use to proofs. This is because the proofs will be less abstract (than lets say algebra or topology, even analysis) and deal with familiar concepts.
145 - Combinatorics. This is supposed to be a fun class. You learn basic counting methods, and learn about generating functions and recurrence relations. Generally the last half of the class is spent on graph theory. You learn basic concepts about graphs, trees, optimum spanning trees, colorings, and bipartite graphs.
146 - Algebraic Combinatorics (149A prior to F2006).
147 - Topology. This is a basic course on point-set and combinatorial topology. Topology generalizes the important ideas from analysis/calculus into a more abstract setting. Instead of having a notion of exact distance, you now only have a notion of "closeness". Intuitively, the closer two points are together, the more "open sets" contain both of them.
You describe a topological space by defining what its "open sets" are. Intuitively, these correspond to open intervals like (0,1) and (5, 15) in the real line. The open sets have to obey certain rules, and this makes their definition into a formal game. Sometimes, you don't have to define all the open sets. You just define a subcollection or "basis" that generates the open sets. These correspond to open balls in analysis.
A "closed set" is the complement of an open set. Intuitively, a closed set contains its boundary (if the boundary exists). Closed subsets of the real line are [0,1] and [5,15], for example. There's more to closed sets than just being the complement of open sets. If you take any convergent sequence in the closed set, the limit must also be in the closed set. This is an alternate way to define closed sets.
Topology allows us to define connectedness. A topological space is "connected" if its only subsets that are both closed and open are the whole space and the empty set.
One central question of topology is, "When can a topological space be made into a metric space?" That is, when can we place a notion of distance on it? We can try to answer this question by examining the behavior of the real line (and other metric spaces). For instance, in a metric space, take two closed sets that are disjoint (i.e., their intersection is void). It is possible to surround each of them by an open set, so that the open sets are also disjoint. We call this "separating" the closed sets. This leads us to define separation axioms for arbitrary topological spaces—what kind of sets can be separated by open sets? The axiom just described is called "normality".
The weakest separation axiom one can demand is that points be closed. Slightly stronger is the Hausdorff axiom—that one can separate points. Spaces are for the most part useless if they are not Hausdorff. The man Hausdorff was a great mathematician. He was Jewish, and when the Holocaust came about in Germany, he and his wife committed suicide to avoid being sent to a concentration camp.
The next strongest axiom one can impose is "regularity". This means that one can separate points from closed sets. Urysohn's Metrization Theorem says that if a space is regular, and if it has a basis that can be arranged in a sequence (countable basis), then the space can be endowed with a metric. Urysohn was a brilliant mathematician whose spark was taken from us in a drowning accident.
Maps between topological spaces that preserve topological structure are called "continuous functions". In fact, topology can be described as a study of continuity. Continuous maps that are continuously invertible are the equivalences between topological spaces. These are called "homeomorphisms". It is too ambitious to attempt to classify all topological spaces. A more tractable question is to try to classify spaces up to homeomorphic equivalence.
It is rather straightforward to show that two spaces are homeomorphic. Simply construct a homeomorphism. The inverse question of showing that two spaces are not homeomorphic is much more difficult. Topological spaces are typically too complicated to allow direct proofs that two of them are not homeomorphic. One tries to reduce this to a problem in another field of mathematics, like abstract algebra. Poincare's fundamental group, defined for any topological space, is a first step in this direction. But that takes us away from point-set topology to the realm of algebraic topology.
- When I took it, it was entirely point-set and we focused just on the first chapters of Munkres without covering anything algebraic.-PhilipNeustrom
- I just finished it and we covered up to section 31 in Munkres so it was all point-set. I recommend it for all math majors it made clear all the topology we covered in 127A and was a good complement to 127B since we covered convergence in a more general way. I think more material could be and should be covered in the class. BryanBell
148 - Discrete Mathematics (149B prior to F2006).
150ABC - Modern Algebra. This is the standard abstract algebra series. The difference about Davis is that instead of being merely one or two courses, it's three. This allows for a lot of time to carefully develop the ideas and theories. You learn, basically, groups, fields, and rings. It's a whole lot more than just that, though.
Abstract algebra is the study of sets with operations on them that behave roughly like our usual addition (+), subtraction (-), multiplication (*), and division (/). Thus, we study algebra in an abstract setting, not just the ring of integers or the field of real numbers—hence the name. From now on, we'll speak about algebra, leaving out the word abstract. This is appropriate, because to understand our usual algebra in a structured manner, it is necessary to study abstract algebra.
An algebraic structure you have already studied is the vector space from linear algebra. These are very rigid structures algebraically. For instance, every vector space has a basis. In general algebra this is not true. Structures with a basis are called "free", and they are difficult to classify.
In 150 you study "groups", which have + and - and 0. The addition is associative, but it might not be commutative (A + B doesn't equal B + A in general). Groups are important because they act—as rotations, as reflections, as rigid motions. In general they act as some kind of mapping. In fact, any element in a group acts as a permutation of that group's elements.
One studies groups by understanding its "subgroups"—smaller groups embedded into the mother group. One can also observe transformations between groups. It turns out that these points of view are equivalent.
Later on you study rings. These are groups where the addition commutes. Furthermore there is multiplication (*) but not necessarily division. Also (AB = BA) might not be true. Therefore, square matrices form a ring. The quintessential ring is the ring of integers. In fact, ring theory is like a generalized, organized version of number theory.
Finally, in 150 C you study fields. These are rings with commutative multiplication and division (/) except for division by 0. Because they carry so much structure, fields are very rigid. Vector spaces are built over a field, that is why they are so inflexible. (Modules, on the other hand, are just built over rings. Their structure is more flexible, their study more rich.)
Galois Theory attempts to understand fields that extend a given field F, and are subfields of an extension field K. So they are wedged between F and K. The Fundamental Theorem says that one can reduce this to the study of groups. The group of field transformations from K to itself that leave elements of F unchanged. This is the original setting in which groups were studied. Galois, the originator of this theory, was shot in the stomach in a duel. He died at 20.
Algebra is crucial for the study of topology. Choose a point * in a topological space. The loops that contain * form a group, Poincare's Fundamental Group. Consider two loops the same if you can continuously deform one to another. On the sphere, every loop can be shrunk to a point. Thus the sphere has the trivial one-element fundamental group. The circle, on the other hand, has for its Fundamental Group, the free group on one generator. This counts how many times a loop winds about the circle.
This is not to say that topology does not contribute to algebra. The fundamental theorem of algebra can be proved by topology. So can the one says: subgroups of a free group are themselves free!
165 - Math and Computers. This is another fun one, and fairly new to the program. This course mostly avoids the numerical methods covered in the 128 series and most engineering courses, and instead picks a few interesting algorithms to analyze and discuss: B-rule algorithm (simplex with Bland's rule), fast Fourier transform, some geometry, all relatively new and exciting stuff. Basically the focus is the use of computers in checking and generating proofs. There might not be an accompanying textbook.
167 - Applied Linear Algebra (Advanced Linear Algebra prior to F2006). Picks up the slack 22A left and covers the rest of the foundations: Vector spaces, matrix transformations (similarity, diagonalization, change of basis, orthogonalization), types of matrices, and other things that are likely to reappear in other courses, especially numerical analysis.
- I'm taking it this summer(2005) and the first half of the course has been pretty much all review. The average on the midterm was 95/110 which is way too high, most boring class I've taken at davis. BryanBell
- i took it this summer (2006) and it seemed like he may have got a little harder. the midterm was easy, 1/3 the class at 90/108 or better. Watch out for the final though it kicked booty someone got 200/200, next best was 160/200 and it went down hill from there with 1/3 of the class below 95/200.
168 - Optimization. The first half of the course covers the simplex method and its applications. After that, interior-point methods are offered as an alternative or improvement for solving linear problems, and the last few weeks are spent on network flow problems, solved using network simplex methods. There are two programs assigned during the course, made easier by the fact that much of the necessary code is supplied by the textbook's author.
For students interested in business and optimization problems, 168 is very relevant, but not difficult. The methods covered are still active fields of research, particularly interior-point methods, and while the math this is based on isn't too high-level, the design of algorithms has its own draw.
The textbook is Linear Programming: Foundations and Extensions by Robert Vanderbei, and is used alongside a well-developed website.
180 - Special Topics. Past topics have included Fractals, Mathematical biology, Mathematical finance (before it was an official course), and String theory (generally one or two is offered every year, topics are almost never repeated and tend to be non-traditional)
185AB - Complex Analysis with Applications
189 - Advanced Problem Solving - Generally students learn to solve advanced problems in various areas of pure mathematics. Quite often former Putnum problems are used.
201ABC - Analysis. Standard first-year graduate analysis. Taught by different professors from year to year, and sometimes from quarter to quarter. The courses normally cover most of chapters 1-11 in the textbook, Applied Analysis, written by Professors Hunter and Nachtergaele and available for free on the web, as well as lecture notes on Differential Analysis. Lieb and Loss is normally assigned but little used in practice.
204 - Asymptotic Analysis. Is no longer offered. The material has been incorporated into the 207 series, description forthcoming.
205 - Complex Analysis. Splitting into two courses this year, 205A and 205B. Standard graduate complex analysis material with Stein and Shakarchi as the usual text.
206 - Measure Theory. A nice, careful investigation of the measure theory left out of the 201 series.
218AB (and sometimes C) - Partial Differential Equations. Taught by Prof. Shkoller, the department's PDE expert and funloving surfer, in odd years (eg: 2005-2006). An intense treatment of modern PDE theory in an arbitrary number of dimensions and shape of domain. In general, "modern theory" doesn't involve the actual solution of PDE's, but rather their analysis, meaning the determination of whether they have unique solutions.
227 - Mathematical Biology. Last taught by Prof. Mogilner in Spring 2005. It more closely resembles a seminar (albeit an unusually interesting one) than a lecture course, and grading is entirely based on homework problems. No background in biology is required, and the mathematics is not too difficult, either. There is no text. The department has a reputation for math bio and it shows in this course and the math bio seminar.
228ABC - Numerical Solution of Differential Equations. A continuation of 128 series. Always taught by Prof. Puckett, and only offered in even years (eg: 2006-2007) at the moment. Puckett does a lot of work in gas dynamics, and has tilted the subject matter in that direction in past years. In 2004-2005, however, he spent most of the year building up to and dealing with the Navier-Stokes (fluid mechanics) equations. There was no text, largely because Puckett tries to keep the class as up-to-date as possible, using recent Ph.D. theses and such.
229A - Numerical Linear Algebra. Last taught by Prof. Strohmer in 2006. Most of the first half of the course builds up the theory of linear algebra. A great deal of this should be review to students familiar with the topic. The course mostly centers around the singular value decomposition, a vital tool with many applications, and similar manipulations. There is much discussion of image compression as an application of the various topics covered.
235ABC - Probability Theory. Taught in even years by statistics faculty, and in odd years by mathematics faculty. Rumored to be quite difficult.
258A - Numerical Optimization. Taught every year by Professor Wets, using a set of notes written by him and updated from year to year. Wets is a rather prominent figure in stochastic optimization, which should be the real title of the course, since it largely focuses on the theoretical foundations of stochastic optimization rather than numerical methods.
Some recommended courses for applied math majors, depending on interests:
Environmental Science 121 - Mathematical Ecology. Taught by Prof. Hastings, a Ph.D. in mathematics and member of the Graduate Group in Applied Math. This class is an undergraduate level course that introduces students to the concepts of mathematical ecology. This course seems more focused on teaching students how to be effective molders rather than mathematical methods. The pre-reqs for this course include math 16abc (or 17/21) and either bio 1b or 1c. I talked to the professor and it seems that the bio pre-reqs aren't that important if you are interested in the subject matter.
Wildlife Fish and Ecology 122 - Population Biology. Like esp 121 this course is undergraduate level and suited for both math and science majors. PDEs are used in this class but previous experience with them are not required.
Biology 132 - Dynamic Modeling in Biology. Similar to Math 124, but offered every year instead of alternate years. Covers models related to many fields of biology. Emphasizes differential equations, difference equations, linear algebra and bifurcations. Students write a term paper based on the scientific literature, instead of the traditional final. the 16/17 is the only prereq for the course.
Neurology Physiology and Behavior 163 - Information processing models in neuroscience. Covers basic modeling techniques used in neuroscience. Specific topics include differential equations, linear systems theory, Fourier transforms, neural networks, probabilistic inference, and information theory. Emphasis on understanding information processing in neural systems.
Physics 104 - Mathematical Physics. Focuses on the mathematical theory used in physics. Topics include ODEs, PDEs, Fourier transforms and many other things.
Civil Engineering 212A - Finite Element Procedures. Usually taught by Prof. Sukumar, a member of the Graduate Group in Applied Math. A fairly rigorous introduction to finite elements, a branch of numerical methods used widely in applications but lamentably not covered by any math course at Davis. Followed-up by 212B, which gets into gory details of solid mechanics that might not interest a math major.
Ecology 231 - Mathematical Methods in Population Biology. Taught by Prof. Hastings, a Ph.D. in mathematics and member of the Graduate Group in Applied Math. Much of the class deals with the subject matter of Math 119A and may be review for grad students in math, but the course also addresses difference equations, PDE's, and the relation of all of these topics to current work in Ecology. The course is followed-up in odd years by ECL 232, Theoretical Ecology, a much more mathematically interesting course. 232 is also taught by Hastings, and requires the student to write a large paper. You really must have a firm interest in population biology or ecology in order to do well in this course.
Economics 122 - Game Theory. This course focuses on the basics of game theory. Around a third of the class is devoted to nash equilibrium, another third about subgame perfect nash equilibrium and the final third is about incomplete information games. Economics 100 is listed as a prerequisite, although anyone with a background in mathematics (21 series) can do well in the course. If you have taken MAT 168, this course is an application of the material learned in 168, but extremely lay-man.
CLIMB - paid research ($15/hr) in the field of mathematical biology, although this program has been cancelled due to lack of funding.
REU - summer research in various mathematical fields, comes with a small stipend.
McNair Scholars Program - for students from disadvantaged and/or underrepresented backgrounds who want to pursue a PhD in any field. Comes with a stipend.