For over 150 years, the study of phase transitions—such as water freezing into ice or magnets losing their magnetism—has been a cornerstone of statistical physics. In this talk, we explore the critical behavior of two-dimensional percolation models, which use random graphs to model the behavior of porous media. At the critical point, remarkable symmetries and emergent properties arise, providing precise insights into the nature of these systems and enriching our understanding of phase transitions. The presentation is designed to be accessible and does not assume any prior background in percolation theory.

About the Speaker: 

Hugo Duminil-Copin is is a French mathematician recognised for his groundbreaking work in probability theory and mathematical physics. He was appointed full professor at the University of Geneva in 2014 and since 2016 has also been a permanent professor at the Institut des Hautes Études Scientifiques (IHES) in France. In 2022 he was awarded the Fields Medal, the highest distinction in mathematics.

Inversion problems, such as full waveform inversion (FWI), based on wave propagation, are computationally costly optimization processes used in many applications, ranging from seismic imaging to brain tomography. In most of these uses, high-order methods are required for both accuracy and computational efficiency. Within finite element methods (FEM), using high(er)-order can provide accuracy and the usage of flexible meshes. However, FEM are rarely employed in connection with unstructured simplicial meshes because of the computational cost and complexity of code implementation. They are used frequently with quadrilateral or hexahedral spectral finite elements, but the mesh adaptivity on those elements has not yet been fully explored. In this work, we address these challenges by developing software that leverages accurate higher-order mass-lumped simplicial elements with a mesh-adaption parameter, allowing us to take advantage of the computational efficiency of newer mass-lumped simplicial elements together with waveform-adapted meshes and the accuracy of higher-order function spaces. We also calculate these mesh-related parameters and develop software for high-order spectral element methods, allowing mesh flexibility. We will also discuss future developments. The open-source code was implemented using the Firedrake framework and the Unified Form Language (UFL), a mathematical-based domain specific language, allowing flexibility in a wide range of wave-based problems. 

Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on newly introduced mixed characteristic counterparts of Du Bois and log canonical singularities and discuss their properties. 

This is joint work with Bhargav Bhatt, Linquan Ma, Zsolt Patakfalvi, Karl Schwede, Kevin Tucker and Jakub Witaszek. 

Derived algebraic geometry allows us to study formal moduli problems via their tangent Lie algebras. After briefly reviewing this general paradigm, I will explain how it sheds light on deformations of Calabi-Yau varieties. 
In joint work with Taelman, we prove a mixed characteristic analogue of the Bogomolov–Tian–Todorov theorem, which asserts that Calabi-Yau varieties in characteristic $0$ are unobstructed. Moreover, we show that ordinary Calabi–Yau varieties in characteristic $p$ admit canonical (and algebraisable) lifts to characteristic $0$, generalising results of Serre-Tate for abelian varieties and Deligne-Nygaard for K3 surfaces. 
If time permits, I will conclude by discussing some intriguing questions related to our canonical lifts.  
 

Let X be an integral projective variety of degree at least 2 defined over Q, and let B>0 an integer. The dimension growth conjecture, now proven in almost all cases following works of Browning, Heath-Brown, and Salberger, provides a certain uniform upper bound on the number of rational points of height at most B lying on X. 

Shifting to the geometric setting (where X may be defined over C(t)), the collection of C(t)-rational points lying on X of degree at most B naturally has the structure of an algebraic variety, which we denote by X(B). In ongoing work with Tijs Buggenhout and Floris Vermeulen, we uniformly bound the dimension and, when the degree of X is at least 6, the number of irreducible components  of X(B) of largest possible dimension​ analogously to dimension growth bounds. We do this by developing a geometric determinant method, and by using results on rational points on curves over function fields. 

Joint with Tijs Buggenhout and Floris Vermeulen.