Maths Beyond the Desks 2024

The Maths Beyond the Desks Day 2024 was held on 19/04/2024. The event programme, seminar abstracts, flyers and photos can be found below.

Programme:

9 am reception of participants (room U2-08b)

9:30 am team competition and award ceremony (room U2-08b)

12 am lunch (Galleria della Scienza south side)

1 pm maths samples (Science Gallery)

3 pm mathematician's work (room U2-07)

5:30 pm final greetings (room U2-07)

Samples of mathematics

What is studied in a maths degree: previews and information

The Secrets of Mathematics - Ciphering and Deciphering Messages

Probably Strange - How probability theory helps us against misleading intuitions

Let's flatten the Earth - Mysteries and pitfalls in geographical maps

Plane tessellations - From Escher to the Einstein dowel

Floating-point arithmetic - How to kill your friends with a computer code

In search of prime numbers - How many prime numbers there are and how (not) to find them: from Eratosthenes' sieve to Conway's PRIMEGAME.

The mathematician's work

What does a mathematician do for work?

Dr. Luca Sabatini (Queen's U. Belfast) - "From equations to modern algebra: a brief history of Group Theory"

Abstract: Paris, 29 May 1832. In the middle of the night, on the eve of a duel, Évariste Galois, in his early twenties, picks up for the last time a manuscript full of equations and theorems that he had written four years earlier. "I have no time! I have no time!" he writes frantically at the margin. Beyond solving a famous problem that has obsessed the minds of great mathematicians for centuries, that manuscript marked the beginning of Group Theory, one of the most important disciplines of abstract algebra. Today, Galois' ideas are studied all over the world, and have applications that even he probably would not have imagined. Modern mathematicians have more time than he did, but many of their most important questions remain mysteries.

Prof. Annalisa Massaccesi (UniPd) - “Soap bubbles and minimal networks: shape optimisation problems (for humans, cetaceans and penguins)”

Abstract: In this seminar we will examine some famous shape optimisation problems (a field of mathematical analysis on the borderline between the calculus of variations and geometric analysis), starting with the isoperimetric problem, also known as Didone's problem, and then moving on to Plateau's problem and the "one-dimensional" problem of minimal networks, of which Steiner's problem is a particular case. If time and the classroom permit, I will show some simple experiments with soapy films.

Maths Beyond the Desks 2024 Flyer

Mathematician's Work Maths Beyond the Desks 2024

Event photos

Maths Beyond the Desks 2023

The Mathematics Beyond the Desks 2023 day was held on 27/4/23, as part of the events for the 25th anniversary of the University of Milano-Bicocca. Below is the text of the team competition, abstracts of the two lectures, links to photos of the event and videos of the two lectures.

The proposed maths samples were as follows:

Maths secrets - Ciphering and deciphering messages

An invite to topology - Strings that coil, deform, but don't break!

Probably Strange - How probability theory helps us against misleading intuitions

Let's flatten the Earth - Mysteries and pitfalls in geographical maps

Beziér curves - From points, segments, meshes.... to design and computer graphics

Topology and curvature of polyhedra - A discrete approach to some aspects of surface theory: Euler's constant and Cartesio's lost theorem.

Puzzles and abstract algebra: an unexpected encounter - The algebraic structures behind some classic puzzles, such as the Rubik's cube or the game of 15

The two lectures on the mathematician's work were given by:

• prof. Marco Abate (Unipi) - Fibonacci's sunflower
• prof. Alessandra Caraceni (SNS) - Percolation in two dimensions: Escape from the infinite labyrinth

Gérard Besson: Institute Fourier - Université Grenoble Alpes

On 21 and 22 May 2025, the following meetings were held at the University of Milano-Bicocca buildings:

#1

Title: On the intrinsic geometry of horospheres in negative curvature

Abstract: There are classical results showing that if a negatively closed manifold has its horospheres of constant mean curvature then it is locally symmetric. Here we shall present a rigidity result involving the intrinsic Riemannian structure of these horospheres. More precisely if one of them is flat than the closed manifold is locally real hyperbolic.

#2

Title: Some curious open manifolds

Abstract: We shall describe the construction of an open $3$-manifold called the Whitehead Manifold. The construction uses a knot in $S^3$ and variations on it produce a non countable family of strange objects. Very little is known on the Riemannian geometry and the analysis that they carry and we may mention some questions. They also play an important role in dimension 4. The construction will be presented with easy tools.

Camillo DE LELLIS: Institute for Advanced Study - Princeton. Stampacchia Medal (2009), Fermat Prize (2013), Caccioppoli Prize (2014), Maryam Mirzakhani Prize in Mathematics (2022).

On 24 and 27 February 2025, the following meetings were held at the University of Milano-Bicocca buildings:

#1

Title: Area-minimizing integral currents: singularities and structure

Abstract: Area-minimizing integral currents were introduced by De Giorgi, Federer, and Fleming to build a successful existence theory for the {\em oriented} Plateau problem. While celebrated examples of singular minimizers were discovered soon after, a first theorem which summarizes the work of several mathematicians in the 60es and 70es (De Giorgi, Fleming, Almgren, Simons, and Federer) and a second theorem of Almgren from 1980 give general dimension bounds for the singular set which match the one of the examples, in codimension 1 and in general codimension respectively.

#2

Title: When reason produces monsters while it is wide awake

Abstract: I turn with terror and horror from this lamentable scourge”. This sentence was uttered by a very famous mathematician towards the end of the XIX century, while referring to the work of another very famous colleague. The object which generated such virulent reaction is actually nowadays rather well accepted, in fact it is often mentioned in basic textbooks on differential calculus. In this lecture I will argue that it is just the first of a long series of counterintuitive mathematical constructions, which all share some common aspects. I will touch upon famous examples of the fifties, sixties, eighties and nineties of the last century, all looking like bizarre games of mathematicians made to defy common sense. However I will finally turn to some discoveries of the last few years, which confirm an old hypothesis of a theoretical physicist, recipient of the Nobel prize in chemistry!

Michael COWLING: University of New South Wales

On 10 and 13 June 2024, the following meetings were held at the University of Milano-Bicocca buildings:

#1

Title: Admissibility of uniformly bounded representations of $SL(2,R)$ on Hilbert spaces

#2

Title: What else do we know about uniformly bounded representations on Hilbert spaces?

Abstract: These presentations are based on work in collaboration with Francesca Astengo (Genova) and Bianca di Blasio (Milano Bicocca). This work will be integrated into its historical and mathematical context. From about 1950, there has been considerable study of the representations $\pi$ of the group $G := SL(2,R)$, andmore generally those of semisimple Lie groups, on Hilbert spaces $\cH$. Some of these representations areisometric, that is, $\| \pi(g)\xi \| = \| \xi \|$ for all $g \in G$ and $\xi \in \cH$. Others are uniformly bounded, that is, theoperator norms of the $\pi(g)$ are uniformly bounded in $G$. An early question (1950) of Dixmier was whether every uniformly bounded representation of a locally compact groupis similar to a unitary representation; a counterexample was found by Ehrenpreis and Mautner in 1955. During thissame period, Harish-Chandra introduced the concept of admissible representation (which means essentiallyaccessible with the tools of algebra) and showed that irreducible unitary representations are admissible. In 1988 Soergel constructed an example of a nonadmissible isometric representation of $G$ on a Banach space, using thenegative solution of the so-called invariant subspace problem in Banach spaces (due to Enflo and Read). Recentlytwo proposed solutions of the same problem in Hilbert spaces (by Enflo and by Neville have appeared; these bothclaim that the invariant subspace problem has a positive solution.

In the first talk, we show that if the proposed solutions to the invariant subspace problem in Hilbert spaces are valid, then uniformly bounded representations of $SL(2,R)$ are admissible.

In the second, we ask what uniformly bounded representations are good for. We present a summary of known resultsabout them and mention a few of their applications (one of which is still conjectural).

Aaron Naber: New Horizons in Mathematics prize for 2018 and Fermat prize for 2023 F. Burgess Professor of Mathematics at Northwestern University.

On 29 February and 1st March 2024, the following meetings were held at the University of Milano-Bicocca buildings:

#1

Title: Structure and Regularity of Nonlinear Harmonic Maps

Abstract: We will consider harmonic maps between Riemannian manifolds u:M->N .  The first part of the talk will discuss and explain the known regularity of such mappings, in particular joint work with Daniele Valtorta on the size and rectifiability of the singular sets.  The second part of the talk will focus on sequences of such mappings u_j:M->N, where it is known that blow-up can occur on a m-2 dimensional subset.  This blow-up is characterized by the so-called defect measure, which we will review and discuss.  In recent joint work with Valtorta we have proved the energy identity, a conjectured explicit description of the defect measure in terms of bubble energy counting.

#2

Title: Ricci Curvature, Fundamental Group and the Milnor Conjecture

Abstract: Crossings between geometry, algebra and analysis. In 1968 Milnor conjectured that there is a powerful link between Ricci curvature and the fundamental group of a manifold. After 50 years, we discuss a counterexample, because math never stops being surprising. In particular, it was conjectured by Milnor in 1968 that the fundamental group of a complete manifold with nonnegative Ricci curvature is finitely generated.  In this talk, we will discuss a counterexample, which provides an example M^7 with Ric>= 0 such that \pi_1(M)=Q/Z is infinitely generated.  The work is joint with Elia Brue and Daniele Semola. There are several new points behind the result.  The first is a new topological construction for building manifolds with infinitely generated fundamental groups, which can be interpreted as a smooth version of the fractal snowflake.  The ability to build such a fractal structure will rely on a very twisted glueing mechanism.  Thus the other new point is a careful analysis of the mapping class group \pi_0Diff(S^3\times S^3) and its relationship to Ricci curvature.  In particular, a key point will be to show that the action of \pi_0Diff(S^3\times S^3) on the standard metric g_{S^3\times S^3} lives in a path connected component of the space of metrics with Ric>0.