Martin's Math Page

Hello and welcome to my Math page!

My name is Martin, and I am pleased to invite to a little online tour through the world of mathematics.

A long time ago, when I was a student of math and computer science at the University of Bonn, friends and relatives often asked me:

What do mathematicians do all day?

What Math is not!

Mathematics is many interesting things, and certainly it is not only about numbers and calculations. So let us quickly move those common misconceptions out of the way!

In primary school and later in high school, you have mainly learned arithmetic, some geometry, and how to apply numerous formulas to real world situations. It's all too easy to assume that this continues at university, and many people assume that mathematicians spend their day with numbers, calculations and perhaps statistics.

There is only one thing to say about this: nothing could be further from the truth!

Of course, mathematicians occasionally deal with calculations from time to time, but that does not reflect what mathematicians study and research. The everyday activities of mathematicians usually revolve much less around numbers and calculations than you might think.

Now, what is mathematics? Let's continue with a short summary of what mathematics is all about!


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What is Math? Proofs!

To best understand what mathematics is about, let us have a look what mathematicians actually do.

Students and researchers of mathematics spend their time with logical conclusions: we rigorously prove statements based on prior statements. Given statements are known or assumed to be true ("The sun is shining"), mathematicians derive new statements ("Your ice cream is gonna melt!").

The heart of mathematics is synthesizing new statements from old statements. Mathematicians start with as few basic assumptions as possible, which are the so-called “axioms”. and using logical reasoning they prove more complicated statements, which are so-called “theorems”. They combine already proven facts into increasingly more complex new facts. Starting from simple assumptions, mathematicians develop far-reaching abstract formalisms.

Of course, the objects of mathematical study are primarily related to notions that you typically associate with mathematics: numbers, geometry, and logic (rather than, say, cooking recipes). Even though many research topics have a background in the sciences, some research is of purely mathematical interest: mathematics for the sake of mathematics. For example, calculating the volume of a sphere in 256 dimensions might have little to no practical application, but it is a mathematically interesting nevertheless as it connects to surprisingly many different phenomena in math.

What stands out is the very high level of abstraction: mathematicians like to focus on general "structures" rather than specific examples. It is far more rewarding to prove abstract statements or principles, which apply across many special cases and examples at once.

This might sound more complicated than it is: Consider the Cartesian coordinates in two dimensions. They display the important feature that you can add two coordinate pairs (translation) and that you can multiply each coordinate pair with a number (scaling). The same features are present in three dimensions, and you easily set up this formalism for any higher dimension. Taking an even more general perspective, translation and scaling can be found in other context, such as function spaces.

Mathematicians thus recognized that the fundamental properties of Cartesian coordinates also appear in other context, and thus the abstract notion of vector space emerged. Recognizing that all common general features are present in these situations, mathematicians developed the notion of vector space. Let us not delve too much into the details of what vector spaces are, let me just say that vector spaces support translation and scaling just like Cartesian coordinates do. Here, we see how generalization is beneficial in the long run: once a theorem is proven for vector spaces, using only the abstract formalism, it applies to all specific instances of vector spaces that you might encounter.

Vector spaces are an example of a mathematical structure, and many more mathematical structures are the outcome of abstraction, recognizing the relevant features of a specific example or situation and working out the governing abstract principles.

Once a theorem is proven for a certain class of structures, the theorem applies to every special case of that general structure. Thus, a single proof gives you a statement that applies in many different instances.


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What is Math? Science!

How does the science of mathematics relate to other sciences, such as physics and engineering?

Everyone has a pretty good idea of what physicists study and research at university: their understanding of fascinating natural phenomena such as gravity, electricity, light, and the Big Bang, is developed via the tandem of experiments and theory.

Physical models of our world can be improved and superseded. For example, classical Newtonian mechanics was superseded by Einstein's theory of relativity. Of course, Newtonian mechanics is still a helpful model in our everyday life: e.g., you don't need to study the curvature of spacetime to understand whether a house is stable or not. Yet some physical theories are refuted by experiments and completely replaced.

This already highlights an important difference between physics and mathematics: mathematical statements that have been proven to be correct remain correct indefinitely.

It would be incorrect to say that the difference between mathematics and physics is that physics examines existing phenomena whereas mathematics does not. Rather, mathematicians explore the general principles that underlie many natural phenomena and study these ideas within an abstract formalism. By contrast, physics aims to model reality as accurately as possible using mathematical models that aim to be precise representations of physical phenomena.

Obviously, mathematics plays an important role in many scientific fields. For instance, the models of physics are expressed in the language of mathematics, so close is the relationship between physics and mathematics that they would have been considered the same centuries ago. The rapid development of physics has driven new groundbreaking developments in mathematics because new mathematical tools were required to advance physical models. In that light, a significant part of physics is mathematics, and physics drives much of modern mathematics.

Physics and mathematics are strongly connected. For example, the theory of general relativity, one of the central pillars of modern physics, was developed in conjunction with the mathematical field of differential geometry. Mathematics has similar connections with every quantitative science, such as economics, computer science and all engineering sciences.

Mathematics is an indispensible tool in many disciplines of science and engineering, and be it only via a modicum of statistics. However, it would not do mathematics justice to just perceive it as a "problem solvers" in service of other sciences: mathematics is not only advanced via its exchange with other scientific disciplines but also pioneers new directions by itself.


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Why Math?

What is exciting about studying mathematics?

Mathematics is challenging but also very rewarding. It should have become clear now that doing mathematics largely consists of precise logical thinking and reasoning, as well as the study of very complex and abstract structures. The pursuit of mathematics trains abstract thinking and comprehension skills in a way that appeals requires you to grasp and evaluate complex relationships.

Apart of the logical thinking, it is rewarding to find non-obvious solutions to problems. Good mathematicians need to think outside the box and sometimes several steps ahead, while keeping in mind the original exact assumptions and the exact result to be achieved. As a mathematician, you are therefore train yourself to quickly understand, model and simplify complex tasks.

Mathematics is a subject that cannot be studied just "on the side". Studying this science requires a lot of discipline and mental fitness: mathematician must analyze complicated relationships and shed light on the most important features.

Lastly, you cannot study mathematics if you don't have fun with intellectual challenges!


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Mathematics and its branches

Mathematics is a vast science that encompasses many different sub-branches, each of which has its own unique concepts, methods and applications. These sub-branches sometimes overlap or interrelate, but each comes with its own unique charm and challenges.

Let us take a closer look at the variety of the fascinating world of mathematics and what makes some of its sub-branches so special. Whether you're a math enthusiast who wants to learn more about your favorite subject or you're just curious about what mathematics has to offer, let us continue this little tour!


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Analysis

Mathematical analysis is the advanced and more rigorous version of calculus. Its typical subjects of study are sequences of numbers and functions in one or several variables. Particular interest in questions related to differentiation and integration.

A very typical question demonstrates what mathematicians study in the domain of analysis: what happens if add an infinite series of ever smaller numbers? For example, any first-semester math student will need to figure out what result you get if you keep successively adding the numbers \[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \] This is a sum of infinitely many numbers. But the result is perhaps surprising: even though we add up an infinite sequence of numbers, the sum has a finite value. In this specific case this means that, as we keep adding this sequence of numbers, we get closer to \(2\). On the other hand, such a pleasant outcome does not occur if we keep adding \[ 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \cdots \] Even though we keep adding smaller and smaller numbers, the sum will keep growing to infinity. No matter how large a threshold we choose, summing up the above sequence of numbers will sooner or later (rather later) surpass that number.




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Algebra

As mentioned earlier, mathematicians study structures that generalize common notions and examples. Algebra is the branch of mathematics that specifically dedicated to such structures.

A strong but helpful simplification is to say that algebra is the study of symbols and the rules for manipulating those symbols. A typical starting point into algebra are polynomial equations: we would like to their solutions. For example, in high school we learned about quadratic equations such as \[ ax^2 + bx + c = 0 \] and how to solve them with a solution formula. That being said, finding solutions is more complicated once we consider cubic equations, \[ ax^3 + bx^2 + cx + d = 0 \] or quartic equations \[ ax^4 + bx^3 + cx^2 + dx + e = 0 \] Mathematicians have discovered solution formulas even for those equations.

But let's increase the challenge: what the more general case? Can we hope to find neat solution formulas for polynomial equations of degree \(n\)? \[ a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0 \] Mathematicians have long been wondering whether equations of higher degree admit any such general-purpose solution formulas. This is where so-called Galois theory comes into play, whose main result is that there is no solution formulas for equations of degree 5 and above. This is an astonishing revelation whose proof requires considerable work with abstract formalisms.

Towards accomplishments like this, algebra introduces many mathematical structures, far more abstract than vector spaces. These structures usually consist of a set, rules on how to operate with elements of this set. For example, polynomials are instance of a mathematical structure known as "rings".

In some sense, algebra develops an abstract notion of what it means to calculate.


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Complex Analysis

As we all learned in high school, the quadratic equation \[ x^2 + 1 = 0 \] does not admit any solutions, right? However, there is more nuance to it. We can solve such equations if we introduce the imaginary unit \(i\), which satisfies the equation \(i^2 = -1\). No real number has this property, but if we allow ourselves to formally use the imaginary unit, then we handle a powerful mathematical tool. We introduce the complex numbers such as \[ 1 + 3i, \quad 2 + 4i, \quad -2 - 7i, \quad, \dots \] Even though this seems like an exercise in pure abstraction, using complex numbers has surprising applications in mathematics and the sciences.

In some sense, we can solve real problems by (temporarily) using numbers that do not exist.

Complex analysis is an area of mathematics that studies functions defined in complex numbers. It extends the concepts of calculus to functions that act on complex numbers. Complex analysis provides new surprising possibilities to handle mathematical problems in physics and engineering, such as computing many integrals. It also allows us to address problems in purely abstract mathematics, such as number theory.


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Differential Equations

The study of differential equations is among the core disciplines of modern mathematics: it is said that most of research mathematics nowadays is directly or indirectly related to differential equations.

Differential equations are of paramount importance for modeling phenomena in science and engineering, providing the mathematical description of heat diffusion, wave propagation, fluid mechanics, quantum mechanics, and more. For example, the heat equation \[ \partial_t u(x,t) = \partial_{xx}^2 u(x,t). \] models the diffusion of heat. Here, \(u(x,t)\) is a function that describes the temperature at position \(x\) and time \(t\). The heat equation models a basic theory of heat diffusion where the rate of temperature change is proportional to second-derivative of the heat distribution. Once the model is derived on the basis of reasonable physical principles, mathematicians take on to study its properties.

Mathematicians want to find a function that satisfies the heat equation (or whatever differential equation they are interested in). In the example of the heat equation, the solution would obviously provide the heat distribution. Even though we can explicit write down formulas for differential equations that are sufficiently simple, explicit solution formulas are beyond our possibilities. Nevertheless, mathematics can still reveal interesting features of even very complicated differential equations, such as mass or energy conservation.

Every differential equation is its own little world that requires different mathematical techniques. Differential equations are very challenging but also very potent as a tool for any scientist and engineer, and they have been crucial for understanding the world around us.

As already mentioned, it is often beyond our possibilities to find explicit solution formulas for differential equations. Yet more often than not we are content with having a good approximation to the solution. That is where numerical simulations take the center stage.


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Numerical Simulations

Numerical simulations are of critical importance for research and development in science and engineering. While mathematicians usually enjoy pure abstraction, numerical mathematics is were finally get back to computation!

That is, we let the computer do the computation for us.

Many mathematical problems that arise in physics and other natural sciences can solved exactly in theory, but in practice we are neither able to find the exact solution, nor we actually need to. Sufficiently good approximations serve our purpuse just as well in practice. Numerical mathematics is about computer algorithms that provide sufficiently accurate solutions for such problems in analysis.

Consider a linear system of equations in 100 unknowns. In theory, we could solve such systems using Gaussian elimination, familiar from school, and much of numerical mathematics is about implementing such algorithms on a computer.

But what if we have 1'000'000 unknowns instead? Gaussian elimination would take an unacceptable time to finish, But a sufficiently accurate solution, correct within a few orders of magnitude, may serve our practical needs while being much easier to find. Following on that principle, iterative solution techniques achieve good approximations to the true solution within acceptable time.

Numerical mathematics is used extensively for differential equations: whereas explicitly calculating the solution to a differential equation might be difficult or even impossible, there is a huge variety of algorithms that approximate the solution. Considerable research efforts are directed into developing advanced algorithms that extract the last bit of efficiency on the computer for the purpose of solving differential equations.

Besides, the numerical simulation of differential equations typically involves the solution of large linear systems of equations as a subroutine. This is a nice illustration how achievements in mathematics incrementally build on successive results.


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Probability & Statistics

Probability and statistics are two closely related branches of mathematics which deal with data and and uncertainty.

Probability theory addresses the mathematical modeling and analysis of random events and processes. If we flip a perfectly balanced coin ten times, what are the odds of it always showing heads? What about the probability that a molecule travels from corner of the room to the opposite corner? Probability theory provides solid mathematical answers to such questions. Mathematically, probability theory builds upon analysis, most famously measure theory and integration. In turn, probability provides the conceptual foundations for the rich field of statistics.

Statistics is well-known to virtually anyone working within the quantitative sciences. Statisticians' work centers around collecting and analyzing empirical data. They use the methods of probability theory to draw conclusions from the data set. Sound mathematical foundations are very important here to draw the correct conclusions from data: we wish to recognize valuable information even in the presence of measurement errors but we also want to distinguish real signals in the data from spurious mistakes. Statistics permeates all sciences and technologies and is essential in extracting information from data, recognizing patterns, and making predictions about the future from past observations.


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Topology

Topology studies properties of geometric bodies from a qualitative and abstract point-of-view.

Topologists study so-called topological spaces, which are sets that are equipped with an abstract concept of “closeness”. It turns out that topological spaces have qualitative properties that do not change under deformations such as stretching and bending, and these are the focus of topology.

Topology therefore explores abstract qualitative properties that are retained when objects are deformed, bent or stretched. Albeit very abstract, topology is everywhere throughout mathematics: it has long surpassed its humble origins in geometry and provides the foundations for much of modern mathematics. Topology has influenced analysis, algebra, and geometry, and has even found applications in physics and computer science.


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Number Theory

Number theory is the study of integers and the relationships between them. Mathematics all started with numbers, and so it is only fitting that the field of "number theory" is dedicated solely to them.

An amazing feature of number theory is that many of its important questions are easy to state even though the mathematics behind them is among the most challenging. For example, take the famous "Fermat's last theorem", which states that the equation \[ a^n + b^n = c^n \] has no integer solutions with \( a, b, c > 0 \) und \( n > 2 \).

The statement is easy to understand, but the proof is far from easy: Fermat wrote down that equation in his notebook in the 17th century. Though Fermat claimed that he had found an easy proof and merely lack the space to write it down, mathematicians spent centuries trying to determine whether Fermat's claim was really true.

It was only until 1993, more than 350 years later, that Wiles and Taylor were able to prove this statement in a comprehensive proof of 100 pages. Their proof utilized many new results from algebraic number theory that could never have been known to Fermat at that time, and only the most capable number theorists today have full understanding of the proof.

Even though number theory has remained a purely artisanal endeavor of dedicated enthusiasts for centuries, the last decades have catapulted it to the forefront of information technology: all of modern cryptography and data security are essentially the offspring of number theory, whose mathematics had been millennia in the making.


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Logic

Logic is essential for mathematics, but it is also a mathematical research area on its own!

Like no other science, mathematics develops networks of logical conclusions. Starting with the initial axioms, mathematicians create ever more complex theories. It is thus a very important questions for mathematicians which mathematical statements can be inferred by which mathematical tools from which mathematical axioms.

Surprisingly, modern mathematics only began analyzing its logical foundations late in the 19th century. Mathematicians soon encountered the "problem" of not being able to prove itself: showing that a system of proofs emerging from certain axioms is correct would in turn require proving the axioms themselves, which would again require another set of axioms. Throughout the 20th century, mathematical research consolidated a set of axioms that are now generally accepted as the foundations of mathematics: the Zermelo-Fraenkel set theory.

However, other axioms still remained controversial. While the laws of logic apply to all mathematicians the same, some mathematicians are reluctant to use certain axioms such as the axiom of choice. Another important but controversial axiom is known as continuum's hypothesis: roughly speaking, the continuum's hypothesis states that there is no intermediate infinity between the infinity of the natural numbers and the infinity of real numbers. Whether to use that axiom remains indeed at any mathematician's discretion, though this will hardly ever apply to any "real-life" mathematics.

At the same time, mathematical logic was haunted by the spectre of contradiction: mathematicians could not exclude the possibility that the generally accepted axioms of mathematicians would eventually lead to contradictory statements, which would signify a serious flaw in modern mathematics.

Kurt Gödel, one of the most famous logicians, showed a fascinating yet sobering result in mathematical logic: simply put, when we use any system of axioms, then either there are true statements whose truth can never be proven, or there are contradictions. This result of 1931 is now famously known as Gödel's incompleteness theorem.

As amazing as it may sound: there are (mathematical) truths that can never show to be true.


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Discrete Mathematics

Discrete mathematics, aptly named, focuses on "discrete" structures, such as finite sets and graphs. Discrete mathematics thus studies objects that can be clearly separated from each other, which is a contrast to analysis, and it can be described as an extension and generalization of combinatorics.

Speaking of combinatorics, you now at least some combinatorics from high school. Combinatorics deals with counting, rearrangements, and combinations of finitely many elements. A classical example of a combinatorial question is this: how many possibilities are there to rearrange \(n\) different objects in a row? The answer is \(n!\) (n-factorial), which is product of all positive integers up to \(n\).

The perhaps largest branch of discrete mathematics nowadays is graph theory. In the context of discrete mathematics, a "graph" can be understood as a huge "bubble diagram": we have discrete nodes and connections between those nodes, the latter also being called "edges". Graphs are interesting objects from a purely mathematical point of view, but they also model numerous real life structures: just think of computer networks or social networks. A typical question of graph theory is, for example, how to pass a message from one node in the network to another node such that the message passes only as few edges as possible.

Discrete mathematics, while not as famous as continuous mathematics, is central to modern technology and computer science. The foundations of computer science and data structures in programming build on top of graph theory. Among the most challenging and astonishing applications is chip design: discrete mathematics has prepared the groundwork for the modern construction of computer chips.


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Theoretical Computer Science

Theoretical computer science is at the intersection of computer science and abstract mathematics. Abstract mathematical formalisms have been tremendously successful in preparing the foundations of technological advancements and taming the complexity of modern computation. But the many fascinating aspects of theoretical computer science have also led to exciting new fields of mathematics.

At its very foundations is computability theory: what does computation mean and what problems can be solved by a computer algorithm? British mathematician Alan Turing introduced what is now known as Turing machine: an abstract model of computation that exactly captures the notion of what computer algorithm means. He thus consolidated the theoretical groundwork for our modern information age.

A famous problem in computability theory is the halting problem, which asks the following: is an algorithm that can decide whether a computer program will stop or run forever? Any programmer who has encountered an infinite loop bug will wish there'd be such an algorithm. But Alan Turing proved that no such algorithm can exist, which means that the halting problem cannot be solved by any computer program.

Advanced topics in theoretical computer science include the study of data structures and algorithms. In a certain sense, theoretical computer science prepares conceptual templates of algorithms that are implemented on different hardwares in different programming languages and for a wide range of different problems. For example, researchers study the sorting problem: given a list of numbers (or other items that can be ordered), what is the fastest way of ordering this list? You can easily imagine that alphabetically sorting a completely random list of 100'000'000 names is demanding task for any computer, so this seemingly simple problem has attracted considerable efforts throughout the years and researchers have been inventive in improving existing algorithms: not by fine-tuning specific implementations, but by a better understanding of the mathematics of the sorting problem.


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Thank you for visiting!

We have reached the end of our tour of mathematics. I hope you enjoyed exploring my website and that you gained some interesting insights into the world of mathematics (and some computer science too).

Mathematics remains a fascinating and active science, Mathematical research constantly raises new questions and sheds new light on old problems. Since ages, it has offered not only intellectual challenges but also the practical tools that shape technological progress.

Mathematics has made enormous contributions in today's world solely by the means of pure logic. My goal was to giving you a broad overview of modern mathematics in an understandable way and share some of the fascination that I experience.

If you have any questions, would like to give suggestions or just want to discuss a specific topic, I am happy to help. You can contact me at any time via email.


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About

My name is Martin and I was only 20 years old when I wrote these text.

Back then, I was a student of mathematics and computer science at the university of Bonn.

My excitement has continued ever since.

Feel free to message me: martin.licht@epfl.ch





The background photos are: 1) Bonn's Messdorfer Feld 2) some canyon in San Diego 3) once more Bonn!


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