In this talk, I will sketch a surprising proof due to Gyorgy Elekes on a non-trivial lower bound for the sums-versus-product problem in combinatorial number theory.

Representations of KLR (quiver Hecke) algebras categorify the positive part of the quantum group associated to any symmetrizable Cartan matrix. This categorical perspective makes certain symmetries more natural to study. For example, the induction and restriction functors between categories of KLR algebra modules play an important role in the theory. A closer investigation of these functors reveals surprising new symmetries. In this talk, I will explain how the induction and restriction functors for KLR algebras can be used to obtain a 2-representation of the corresponding affine positive part in type A. I will also describe a new categorification of a closely related algebra, the q-boson algebra, in all symmetrizable Kac-Moody types.

Over the past few years, the theory of quantum graphs has emerged as a field of growing interest. In 2022, Matsuda and Gromada have given concrete examples by classifying the undirected quantum graphs on the quantum space M2. Based on the solid theory of directed quantum graphs developed in 2024, it became possible to complete the classification of quantum graphs on M2 also in the directed case. We observe that there is a far bigger range of directed quantum graphs than of undirected quantum graphs on M2. This talk is based on a joint work with Björn Schäfer.

Boltzmann samplers of random discrete structures typically only facilitate approximate-size sampling in linear time. We construct enriched-trees samplers which facilitate linear time exact-size sampling, providing the fastest known samplers for subcritical classes of graphs and maps, as well as substitution-closed classes of permutations. Joint work with K. Panagiotou and L. Ramzews.

According to G. H. Hardy, the 'real' mathematics of the greats like Fermat and Euler is 'useless,' and thus the work of mathematicians should not be judged on its applicability to real-world problems. Yet, mysteriously, much of mathematics used in modern science and technology was derived from this 'useless' mathematics. Mobile phone technology is based on trig functions, which were invented centuries ago. Newton observed that the Earth's orbit is an ellipse, a curve discovered by ancient Greeks in their futile attempt to double the cube. It is like some magic hand had guided the ancient mathematicians so their formulas were perfectly fitted for the sophisticated technology of today. Using anecdotes and witty storytelling, this book explores that mystery. Through a series of fascinating stories of mathematical effectiveness, including Planck's discovery of quanta, mathematically curious people will get a sense of how mathematicians develop their concepts.

After briefly recalling the history of the augmented Lagrangian approach to constrained optimization problems,  the solution of the distributed control problem for the steady, incompressible Navier-Stokes equations is addressed via inexact Newton linearization of the optimality conditions. Upon discretization by a finite element scheme, a sequence of large symmetric linear systems of saddle-point type is obtained. An equivalent augmented Lagrangian formulation is  solved by the flexible GMRES method used in combination with a block triangular preconditioner. The preconditioner is applied inexactly via a suitable multigrid solver. Numerical experiments indicate that the resulting solver appears to be fairly robust with respect to viscosity, mesh size, and the choice of regularization parameter when applied to 2D problems.  This is joint work with Santolo Leveque (Houston) and Patrick Farrell (Oxford).

Functions of matrices (more generally, operators) have long attracted the interest of mathematicians and arise frequently in physics  and other fields.  An interesting property of (smooth) functions of large and sparse matrices is that they tend to be strongly localized, i.e, most of the non-negligible entries are concentrated in certain locations; for example, if A is a banded Hermitian matrix, the entries of exp(A) decay super-exponentially in magnitude moving away from the main diagonal.   This property is shared to some extent by more general matrix types and functions, with the precise rate of decay depending on the regularity of the function and on the distance between possible singularities and the spectrum (or numerical range) of the matrix. In my talk I will give an account of recent results on localization for matrix functions and describe some applications of the theory.

After training a neural network, what has it learned? This talk will present the analysis of two methods that address this question. First, we will discuss neural network descrambling, an explainability algorithm that was proposed by Amey et. al in 2021 for understanding the latent transformations in the weight matrices of individual neural network layers. We will show that the explanations provided by descrambling can be characterized via the singular vectors of neural network weights, and in turn these singular vectors can help explain the actions of the affine transformations within neural network layers. Second, we will discuss neural collapse--the phenomenon where a classifier's terminal features and weights converge to the vertices of a regular simplex--and study this phenomenon in the orthoplex regime where there are more classes than feature dimensions. In this case, spherical codes will play a key role in characterizing the arrangements produced under neural collapse and the emergence of a "goldilocks" region where the temperature in the cross entropy loss promotes certain spherical codes over others.

Foliations are important and connect many areas of mathematics including algebraic geometry, complex analysis, topology, etc. Due to the close connection between foliations and the open conjectures in birational geometry, people have been studying the birational geometry of foliations for a long time. The birational geometry of foliations is rich, but also differs strikingly in many aspects from the birational geometry of varieties. On one hand, foliations, as a powerful tool, can unveil the properties of varieties themselves surprisingly and deeply. For example, foliations play a crucial role in the proof of several key cases of the abundance conjecture for threefolds and the characterization of uniruled compact Kähler manifolds. On the other hand, many essential difficulties have emerged during the study of foliations. For example, abundance conjecture, effective birationality, Bertini-type theorems and many more important results in the birational geometry of varieties fail for foliations, and there is no hope to remedy this situation for just foliations. Two extremely essential and important divergences are the failure of finite generation and the failure of boundedness for foliations.

I will discuss some joint work with Paolo Cascini, Jingjun Han, Jihao Liu, Calum Spicer, Roberto Svaldi and Lingyao Xie where the notion of adjoint foliated structures is studied to overcome the aforementioned pathological behavior and difficulties of the birational geometry of foliations, and this methodology works. In particular, we prove the finite generation of the canonical rings and boundedness results for algebraically integrable adjoint foliated structures which can be viewed as analogues of the classical results for varieties.