It is too easy to believe that all theorems published in the leading math journals are too technical to understand. This is not always the case. In 2014, Yitang Zhang proved that there are infinitely many pairs of prime numbers $p$ and $q$ such that $0 < p-q \leq 70,000,000$. This is dramatic progress towards the twin prime conjecture, which claims that there are infinitely many prime numbers, $p$, $q$, such that $0 < p-q \leq 2$. Whilst this result is ground-breaking, it can also be understood by anyone who recalls the prime numbers from school. This is not a unique example. The book Theorems of the 21st Century by Bogdan Grechuk collects over 100 of major math theorems published between 2001-2010 whose statements can be explained to everyone with secondary-school maths knowledge.
Reading about such recent theorems is a great start, but even better is being able to ask your own questions about them. Would you like to hear from the authors of these theorems, and put your questions directly to them? If so, then the new series of Theorems of the 21st Century Online Seminars is your chance. This series will be completely free, available online via Teams and recorded to watch back whenever you want.
The first speaker, on 28 January 2026 (Wednesday) at 4PM (UK time), will be Boaz Klartag, a professor at the Weizmann Institute. He is an expert in high-dimensional geometry with a long track record of solving central open questions in this area. He will present a significant breakthrough in the classical problem of sphere packing. This problem originates from the desire to pack oranges (which are modelled as perfect spheres of the same size) in a box as densely as possible. First conjectured by astronomer Kepler, this formulation was open for centuries until it was solved by Thomas Hales in 2005.
Image: Engin Akyurt (Pexels)
In this 3D case, if the centre of ball $i$ has coordinates $(x_i,y_i,z_i)$, and radius $1/2$, then the distance between any two centres is at least $1$, hence $(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2 \geq 1$ for all distinct balls $i$ and $j$. These coordinates can be thought of as any sequence of three numbers. In recent years however, motivated by applications for error-correcting codes, mathematicians are interested in arranging sequences of $n$ numbers $(x_1, x_2, x_3, \dots, x_n) = x$ such that the distance between any sequences $x$ and $y$, defined as $d(x,y) = (x_1-y_1)^2+(x_2-y_2)^2+\dots +(x_n-y_n)^2$, is at least $1$. Geometrically, this corresponds to packing $n$-dimensional balls, whose centres have coordinates of the form $(x_1, x_2, \dots, x_n)$, as densely as possible.
Maryna Viazovska resolved this problem for $n=8$, and then, with co-authors, for $n=24$. For this work, she received the Fields Medal, maths’ highest award for young mathematicians, becoming only the second woman ever to do so.
In the above solved cases, the actual packing had been known for centuries, and the problem was to prove that it is optimal. However, for larger $n$, it is not even clear how to pack. In 1947, Rogers developed a (non-optimal) lattice to pack $n$ dimensional balls for all $n$, and, for many years, people were able to improve on this construction only by a small factor. Now, in 2025, Klartag found a way to pack balls much more densely than Rogers, with the number of balls per volume larger by a factor of approximately $n$. For example, if $n=1000$, the new construction fits, in the same volume, about 1000 times more balls than Rogers one. Boaz will explain this breakthrough in detail during the first seminar in the series. Join the talk via this link.
The second speaker, who will talk on 11 February 2026 (Wednesday) at 4PM (UK time) is Avi Wigderson, a mathematician and computer scientist specializing in algorithms and complexity theory. His fundamental contributions have been recognized with the highest awards, including the Abel Prize and the Turing Award. In this talk, he will discuss expander graphs, their constructions, and their applications.
Imagine a large number of computers which need to be connected into a single network. We could just connect any two by a direct cable, but this requires too many cables. Another option is to enumerate all computers from $1$ to $n$, and connect computers $i$ and $i+1$ in a line. This is much cheaper, with just $n-1$ cables sufficient, but if just one cable were broken, the network would break into two disconnected pieces. The quality of any network is high if a small number of broken cables cannot disconnect a large set of computers from the others.
The standard mathematical model for a computer network is a graph. A graph $G$ of size $n$ is a set of $n$ vertices, some pairs of which are connected by edges. In our example, vertices are computers, and edges are cables, but graphs have many other applications. For example, vertices may be cities and edges can represent high-quality highways built between them. For many applications, it is desirable to construct graphs such that (i) the number of edges is not too high, but (ii) we cannot disconnect the graph into two large pieces by removing a small number of edges. Graphs with properties (i) and (ii) exist; they are called expander graphs. They are fundamental for many applications in mathematics and computer sciences. Join the talk via this link.
You can find information about future talks at the Next talks page. Or to get all information as conveniently as possible you can join the seminar mailing list. You will receive one email per month with the schedule of upcoming talks and a list of the confirmed future speakers. To join the mailing list, please email the seminar organizer, Bogdan Grechuk, at bg83@leicester.ac.uk, or use this form to enter your name and e-mail.
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