If you are interested in higher level mathematics, here are some of the topics you can look forward to learning.

Theoretical Computer Science– study of models of computation such as finite automata and Turing machines, and pushdown automata. Turing machines can simulate any modern day computer. TCS also studies computational complexity, where one analyzes the runtime of algorithms. TCS also studies computability, so whether or not something is even computable. One famous unsolved problem in theoretical computer science is P = NP. P is the class of all problems which can be solved with a deterministic Turing machine in polynomial time. NP is the class of all problems which can be solved with a nondeterministic Turing machine in polynomial time, or equivalently the class of all problems which can be checked by a deterministic turing machine in polynomial time. No one knows if P = NP. If this were true, then many cryptosystems previously thought to be secure, such as RSA, might prove to be actually insecure.

Mathematical Logic – study of mathematical logic. Two common quantifiers are “for all” or “for every” or “for each” and “there exists”

Linear Algebra– study of vector spaces and transformations between them. It’s important to understand the geometry of linear transformations as well as the abstract algebraic properties. One important theorem in linear algebra is the Cayley Hamilton theorem, which says that every matrix satisfies its own characteristic polynomial.

Abstract Algebra – study of groups, rings, fields, ideals, modules
Multivariable Calculus

Real Analysis-study of how we build the real numbers, how to prove theorems from calculus such as Taylor’s Theorem, study of sequences, and studies of properties of functions.

Complex Analysis-mathematical analysis but with complex variables

Probability Theory – axiomatizes the study of probability, studies probability from a firm axiomatic standpoint. One famous theorem in probability is the central limit theorem, which the limit of n i.i.d. continuous random variables tends to a normal distribution as n goes to infinity. Another foundational result in probability is the law of large numbers, which says that averages of repeated trials converge to the expected value (mean). Note: i.i.d. stands for independent and identically distributed.

Topology– studies properties of an object that are preserved under continuous deformations, like stretching, bending, or squishing. One famous theorem from topology is the Brouwer Fixed Point theorem, which states that if you take a continuous transformation f from a n-sphere to itself, there is a point fixed point, namely, a point x0 such that f(x0) = x0.

Cryptography– Cryptography is the study of how to send messages securely. One famous cryptosystem is RSA, the Rivest, Shamir, Adleman algorithm.

Information Theory – Information theory deals with how to send information securely, over noisy channels, and so on.

Graph Theory
A graph is a pair (V,E) where V is a set of objects called vertices and E is a set pairs in V x V called edges. Graph Theory is used to model many real world situations, from traffic flow to network flow to percolation theory. One can form the adjacency matrix of a graph and study the eigenvalues of this matrix. It turns out that the second and first largest eigenvalues have properties that turn out to be useful to measure the connectivity of a graph.

Combinatorics
Enumerative combinatorics– studies the enumerative side of combinatorics
Algebraic Combinatorics-studies combinatorics using group theory and linear algebra, and other tools of algebra
Analytic Combinatorics-studies the asymptotics of some combinatorial functions and objects. Studies combinatorics with tools from analysis.

Category Theory – Study of sets called Objects and morphisms between then. For instance, for Set, the category of sets, the Objects are set and the morphisms are functions between sets. For groups, the category is Grp, the objects are groups and the morphisms are homomorphisms between groups. For Rings, the category is Rng, the objects are rings and the morphisms are ring homomorphisms. You can even have a Category of Categories, where the objects are Categories, and the morphisms are called functors from one category to another.

Number Theory-study of the integers
Analytic NT-studies the asymptotics using complex analysis
Algebraic Number Theory-studies number theory from an algebraic side using tools from Abstract Algebra
Computational Number Theory – studies how to make certain number theoretic algorithms run faster.

Set Theory-The foundations of mathematics are built upon set theory. One can define the integers as follows: Let phi be the empty set. Then let 1 = the set containing the empty set, let 2 be the set containing the set containing the empty set, and so on. In this way, one can rigorously define the positive integers in terms of set theory.

Statistics – studies means, moments. Example: Given i.i.d Random variables X_1, X_2,… X_n, the kth order statistic is defined to be the kth smallest out of them. One can find a formula for X_1 and X_n, and X_j in general.

Here are some other topics and fields of study:

Differential equations-study of differential equations

Commutative Algebra – study of commutative rings

Analysis on Manifolds

Introduction to Proofs- It is good to learn proofs from a course on graph theory or a course on linear algebra. Learning proofs from a course in real analysis is harder. Common methods of proofs include proof by contradiction, proof by counterexample (to disprove something), a direct proof (describing a proof that does not proceed by contradiction), induction, and so on.

Foundations of Mathematics-One famous theorem is Godel’s Incompleteness theorem, which says that

Differential Geometry

Algebraic Topology

Tropical Geometry

Extremal Combinatorics-one question in extremal combinatorics is, What is the most number of edges a graph can have without containing a copy of a given H? Extremal combinatorics deals with the study of limits of certain objects, and questions about extrema (maxima and minima).

Game Theory-given a set of outcomes for every possible pairs of states (X_i,Y_k) of two players X and Y, what is the best move to make (what move optimizes my value function)? What if this game is played repeatedly?

Combinatorial Game Theory

Not all topics in math fall neatly under one field of study.