Mats Stensrud

On algorithm-assisted human decision making

 March 26, 2024    16:30 - 17:20
 LH 2.02, Learning Center, Campus Belval

Abstract. Systems for algorithmic decision-making are on the rise. Self-driving cars have been a classical example, but such systems are also used to individualize decision rules in many other domains. In particular, the current focus on precision medicine reflects the interest in individualized decision rules, adapted to a patient's characteristics. In this talk, I will introduce theory and methods for finding optimal decision rules. I will then present an apparent paradox in the optimal regimes literature: in plausible decision settings, there is no formal guarantee that conventional optimal regimes, learned algorithmically from data, will outperform human decision-makers, like medical doctors. Next, I will introduce superoptimal decision rules, which resolve this ostensible shortcoming. I will further discuss how the superoptimal rules can be identified in data fusion contexts, where experimental data and (possibly confounded) observational data are available. The results will be illustrated by examples from medicine and economics.

Erika Roldan

Topology and Geometry of Random Cubical Complexes

 November 14, 2023    16:30 - 17:20
 MSA 4.510, Campus Belval

Abstract. In this talk, we will explore the expected topology (measured via homology) and local geometry of two different models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also compare the expected topology that these average structures exhibit with the topology of the extremal structures that it is possible to obtain in the entire set of these cubical complexes. You can look at some of these random structures here and start making some guesses about their topological behavior.

Emily Riehl

Path Induction and the indiscernibility of Identicals

 June 20, 2023    16:30 - 17:20
 MSA 4520, Campus Belval

Abstract. Mathematics students learn a powerful technique for proving theorems about an arbitrary natural number: the principle of mathematical induction. This talk introduces a closely related proof technique called "path induction," which can be thought of as an expression of Leibniz's "indiscernibility of identicals": if x and y are identified, then they must have the same properties, and conversely. What makes this interesting is that the notion of identification referenced here is given by Per Martin-Löf's intensional identity types, which encode a more flexible notion of sameness than the traditional equality predicate in that an identification can carry data, for instance of an explicit isomorphism or equivalence. The nickname "path induction" for the elimination rule for identity types derives from a new homotopical interpretation of type theory, in which the terms of a type define the points of a space and identifications correspond to paths. In this homotopical context, indiscernibility of identicals is a consequence of the path lifting property of fibrations. Path induction is then justified by the fact that based path spaces are contractible.

Ara Basmajian

Counting problems on the modular surface

 April 18, 2023    16:30 - 17:20
 MNO 1.020, Campus Belval

Abstract. The modular surface X is the punctured sphere with cone points of orders 2 and 3; equivalently X is the quotient of the upper half-plane by the modular group, PSL(2,Z). In this talk, after setting up the basics, we'll focus on various classes of geodesics and their growth rates (with respect to word length and geometric length) leading to several counting problems. These counting problems are part of more general phenomena that intertwine the geometry and topology of curves on surfaces with number theoretic considerations.

Olga Paris-Romaskevich

Several stories around tiling billiards

 September 27, 2022    16:00 - 16:50
 MNO 1.050, Campus Belval

Abstract. Can one fold a paper in such a way that a polygon drawn on it can be cut out in one scissor cut ? What is a difference in the dynamics of an interval exchange transformation and that of an interval exchange transformation composed with an involution? What is a behavior of foliations defined by the restriction of a 1-form with constant coefficients on a 3-torus to a null-homologous surface ? What is a geometric object behind a substitution on 3 letters? And what is a link between these four questions? Come to my talk if you want to get some answers!

Sabetta Matsumoto

The Inside View: Raymarching and the Thurston geometries

 October 12, 2021    16:00 - 16:50
 Zoom meeting

Abstract. The properties of euclidean space seem natural and obvious to us, to the point that it took mathematicians over two thousand years to see an alternative to Euclid’s parallel postulate. The eventual discovery of hyperbolic geometry in the 19th century shook our assumptions, revealing just how strongly our native experience of the world blinded us from consistent alternatives, even in a field that many see as purely theoretical. Non-euclidean spaces are still seen as unintuitive and exotic, but with direct immersive experiences we can get a better intuitive feel for them. The latest wave of virtual reality hardware, in particular the HTC Vive, tracks both the orientation and the position of the headset within a room-sized volume, allowing for such an experience. We create realtime rendering to explore the three-dimensional geometries of the Thurston/Perelman geometrization theorem. In this talk, we use the “inside view” of each manifold to try to understand its geometry and what life might be like on the inside. Joint work with Rémi Coulon, Henry Segerman and Steve Trettel.

Lisa Sauermann

On the cap-set problem and the slice rank polynomial method

 June 15, 2021    16:00 - 16:50
 Zoom meeting

Abstract. In 2016, Ellenberg and Gijswijt made a breakthrough on the famous cap-set problem, which asks about the maximum size of a subset of \(\mathbb{F}_3^n\) not containing a three-term arithmetic progression. They proved that any such set has size at most \(2.756^n\). Their proof was later reformulated by Tao, introducing what is now called the slice rank polynomial method. This talk will discuss Tao's proof of the Ellenberg-Gijswijt bound for the cap-set problem, as well as further applications of Tao's slice rank polynomial method.

Alison Henrich

Knotty Games

 May 11, 2021    16:00 - 16:50
 Zoom meeting

Abstract. Far too many people in this world are under the mistaken impression that math can't be fun. The aim of much of my research with student and faculty collaborators, in part, is to provide yet another counterexample to this claim. Our work combines a delightfully visual mathematical subject, knot theory, with one of the most common sources of fun: games. Motivated by the fascinating work of Ayaka Shimizu on an unknotting operation called the region crossing change and research of Ryo Hanaki on unusual types of knot diagrams called pseudodiagrams, we have invented and explored several knot games. In this talk, we will play these games on knot diagrams, developing both our spatial intuition and our understanding of the structure of knots along the way.

De la recherche en mathématiques au lycée? L'exposé se poursuivra par une table ronde d'une heure (en français) consacrée à l'introduction de projets de recherche en mathématiques avec les élèves et étudiant·e·s. Cette table ronde est organisée dans le cadre d'une formation IFEN.

Persi Diaconis

Adding numbers and shuffling cards

 March 23, 2021    16:00 - 16:50
 Zoom meeting

Abstract. When numbers are added in the usual way, ``carries'' appear along the way. If you are me (and hopefully you after this talk) it's natural to ask: ``How do the carries go? How many are there and, if you just had a carry, is it more or less likely that the next sum has a carry?'' Strangely, carries form a Markov chain with an ``amazing'' transition matrix. More strangely, this same matrix appears in the analysis of the usual way of riffle shuffling cards. I will explain the connection, tell you about the ``seven shuffles theorem'' and attempt to do it ``in English''.

Olga Paris-Romaskevich

Triangle tiling billiards, interval exchange transformations with flips and movies

 May 12, 2020  Postponed
 MNO 1.020, Campus Belval

Abstract. T.B.A.

Diana Davis

Periodic paths on the pentagon

 June 18, 2019    16:30 - 17:20
 MNO 1.020, Campus Belval

Abstract. Mathematicians have long understood periodic trajectories on the square billiard table, which occur when the slope of the trajectory is rational. In this talk, I'll explain my joint work with Samuel Lelièvre on periodic trajectories on the regular pentagon, describing their geometry, symbolic dynamics, and group structure. The periodic trajectories are very beautiful, and some of them exhibit a surprising "dense but not equidistributed" behavior.

Béatrice de Tilière

The Z-invariant Ising model via dimers

 April 1, 2019    16:00 - 16:50
 MNO 1.020, Campus Belval

Abstract. The Ising model is one of the most classical models of statistical mechanics. We shall present its relation to dimers, its Z-invariant version and some results obtained with C. Boutillier and K. Raschel.