The Abel Prize stands among the most esteemed awards in the field of mathematics. Bestowed annually by the Norwegian Academy of Science and Letters, it comes with a monetary reward exceeding £550,000.

Masaki Kashiwara’s contributions, much like those of other eminent mathematicians, span across multiple mathematical disciplines. While school curricula often segment mathematics into separate domains such as algebra, geometry, or calculus, the reality is that these branches are deeply interconnected. True innovation frequently occurs when insights or tools from one area are applied to another, creating breakthroughs and new directions in mathematical research.

Understanding Symmetries

To appreciate some of the areas Kashiwara has connected, consider the concept of symmetry. We typically associate symmetry with visual balance — transformations that leave an object unchanged in appearance.

Take a rectangle, for instance. It remains visually identical if reflected across its vertical or horizontal axis or rotated by 180 degrees. These transformations, including the act of doing nothing at all, are examples of the rectangle’s symmetries.

Visualizing the symmetries of a rectangle: it can be flipped across a vertical (red) or horizontal (green) axis or turned halfway around. These three actions, along with doing nothing, constitute the complete set of symmetrical transformations for a rectangle.

All these transformations form a cohesive structure. When you apply one symmetry and then another, the outcome is still a symmetry. For example, if you reflect the rectangle horizontally and then vertically, the result is equivalent to rotating it 180 degrees. Every symmetry also has an inverse — repeating a symmetry operation often returns the shape to its original state.

This system of operations aligns with a mathematical concept known as a group — a collection governed by specific rules regarding how elements combine and interact (more about groups can be found).

While keeping track of consecutive transformations can get complicated, a helpful tool is a composition table. For example, by labelling each symmetry (reflections and rotation) as shown in a diagram, you can determine the result of performing one after another. The intersection of the row for f and the column for g reveals the combined transformation, such as h.

Each letter in this table stands for a specific symmetry of the rectangle. The symbol e represents the identity symmetry — doing nothing.

Abstract Thinking in Mathematics

Here’s where things get more abstract. You might encounter mathematical entities that don’t obviously relate to shapes, yet still obey the same structural rules. One example is matrices — number grids that, when multiplied, produce another matrix within the same set. When you label these matrices and build a table like the one for rectangle symmetries, the two may end up identical in structure.

This opens the door to abstraction: we can understand groups not by the specific elements they contain but by how those elements relate to one another. The essence of a group lies in its internal structure, not in the nature of its components.

What you’re doing here is algebra — studying the frameworks of abstract collections. Other structures also exist, such as rings, which involve two operations (like addition and multiplication) instead of one.

This level of abstraction is incredibly powerful. Once you’ve understood the structure of an abstract group, ring, or other system, that knowledge applies to any real-world case that fits the same framework — whether it’s symmetry transformations, matrices, or something else entirely.

Mathematics of Change

Returning from abstraction to real-world applications, we observe that our environment is characterized by constant change — objects move, systems evolve, and quantities fluctuate. Mathematics enables us to model and predict such changes. Suppose your bank account grows by £1000 every month. You can project that after one year, your balance will increase by £12,000.

That’s a simple scenario with monthly intervals. But changes often happen continuously — like in the stock market. Mathematically, such continuous change is captured by differential equations, which describe how a quantity varies over time, space, or another parameter.

Solving these equations allows us to forecast future values of changing quantities. For example, a solution can tell you how much money you’ll have at a specific time, given how quickly it grows.

Differential equations are ubiquitous in real-world applications — from physics and biology to economics and engineering. Unfortunately, they can be incredibly complex and difficult to solve. (A basic introduction to differential equations can be found.)

For mathematicians, these challenges make differential equations a subject of intrinsic interest. Researchers explore them not only in applied contexts but also in purely theoretical ones. The analytical tools needed to tackle them include ideas from calculus — dealing with limits, infinitesimals, and continuous change.

The branch of mathematics that rigorously develops these tools is known as analysis.

The Fusion of Algebra and Analysis

At first glance, algebra and analysis may seem like entirely separate realms — algebra being clean and abstract, analysis more delicate and complex. But beneath the surface, they share deep connections.

Analysis often works with functions — like $f(x) = 1000x$, which represents the monthly increase in your bank balance. Functions can be composed and manipulated much like symmetries. Sets of functions can form algebraic structures such as groups or rings. Similarly, differential operators, which relate functions to their rates of change, can also be studied algebraically.

This creates a natural bridge between algebra and analysis. The field that harnesses this connection to study differential equations through algebraic methods is called algebraic analysis. While such ideas have been around for some time, it was the Japanese mathematician Mikio Sato who laid the foundation for treating algebraic analysis as a formal discipline.

Kashiwara studied under Sato at the University of Tokyo during the 1970s. Under Sato’s mentorship, he wrote a groundbreaking Master’s thesis at the age of just 23. In it, he introduced the theory of D-modules — algebraic constructs deeply connected to differential equations. These became cornerstones of algebraic analysis and have profoundly influenced many mathematical domains. (For a technical dive, check out the translated version of Kashiwara’s thesis.)

One key example of the power of D-modules involves a famous challenge posed by David Hilbert in 1900 — the Riemann-Hilbert problem, which was the 21st of his 23 problems intended to guide mathematical progress in the 20th century. The problem asks whether certain differential equations can always be matched with specific criteria.

While the simplest form of the problem was resolved by others, Kashiwara extended the inquiry far beyond Hilbert’s original framing. He developed and proved results in much more general settings. You can learn more about this in , written in 2018 when Kashiwara was awarded the Chern Medal. His research continues in this field, and in collaboration with Andrea D’Agnolo, he has recently expanded on his earlier work related to the Riemann-Hilbert problem.

Capturing Abstract Concepts Through Representation

Kashiwara’s achievements in algebraic analysis also connect with representation theory — the second major area highlighted in the Abel Prize citation. Representation theory deals with ways to express abstract algebraic structures using more concrete mathematical objects, such as matrices or linear transformations.

Representation theory brings us full circle to where this discussion began — with algebraic groups. Initially, we observed that groups consist of abstract entities, and it’s not their specific identities that matter, but rather how they interact with each other.

While this level of abstraction is elegant, it can sometimes feel overly detached. In many situations, it’s beneficial to have a more concrete sense of what the group elements actually are. This becomes even more critical when comparing different groups — it’s far easier to draw parallels when you’re working with similar types of objects rather than entirely dissimilar ones.

Fortunately, the structure of mathematics often provides such clarity. A significant number of groups, particularly those with a finite number of elements, can be represented by matrices — those familiar grids of numbers mentioned earlier . For instance, the symmetries of geometric shapes, like the rotations and reflections that preserve a rectangle’s form, can be captured using 2×2 matrices. These matrices effectively describe transformations of the plane, as they operate on vector spaces.

In fact, given an abstract group, it’s often possible to find a corresponding set of matrices that mirrors its structure precisely. This is a major advantage, as mathematicians have developed a deep understanding of matrices over time.

At its core, representation theory examines how groups — and other algebraic constructs — can be expressed through matrices acting on vector spaces. This area of study plays a central role in modern mathematics and intersects with a wide array of other fields. A significant development in this context is Kashiwara’s groundbreaking theory of D-modules, which has become a powerful tool within representation theory.

Demonstrating yet another remarkable synthesis of ideas, and driven by challenges emerging from theoretical physics, Kashiwara also introduced the concept of crystal bases into the landscape of representation theory. These structures make it possible to explore complex algebraic questions through combinatorics — a branch of mathematics centered on counting and discrete structures. The impact of crystal bases extends beyond representation theory, with notable applications in number theory and statistical physics as well.

The discussion above merely touches on the foundations and far-reaching consequences of Kashiwara’s contributions. As noted in the Abel Prize citation,

“Kashiwara is an exceptionally prolific mathematician with more than seventy collaborators. For over fifty years he has reshaped and deeply enriched the fields of representation theory, in its numerous incarnations, and algebraic analysis. His work continues to be at the forefront of contemporary mathematics and to inspire generations of researchers.”

Kashiwara is set to receive the Abel Prize from the King of Norway on May 20, 2025, in Oslo. Our heartfelt congratulations go out to him!