We present some basics of L-functions usually (to be) covered in the undergraduate and graduate entry level number theory classes.

Elliptic curves are fundamental objects in modern number theory, connecting algebraic geometry and arithmetic. In this talk, I will give an introductory overview of elliptic curves and their arithmetic, comparing the number field and function field settings and highlighting their similarities and differences. I will then present recent results on elliptic curves of analytic rank one and their ranks over certain ring class fields, based on joint work with Seokhyun Choi and Bo-Hae Im.

We discuss lower bounds on Lyapunov exponents for linear PDEs driven by random velocity fields, including the advection–diffusion equation and the linearized stochastic Navier–Stokes equations. In particular, we show that the exponential rate of decay of the L^2-norm for solutions to the advection–diffusion equation is optimal. A key ingredient in the proof is high-frequency stochastic instability, arising from the non-degeneracy of the driving noise. This talk is based on joint work with Martin Hairer, Tommaso Rosati, and Sam Punshon-Smith.

I discuss ‘almost counterexamples’ to Seymour’s second neighbourhood conjecture. In what we call Seymour-tight orientations, the size of the first neighbourhood of each vertex equals the size of its second neighbourhood. We give several examples and constructions. Specifically, we prove that the class of Seymour-tight orientations is closed under taking (generalized) lexicographic products. Moreover, the lexicographic product of a putative counterexample to Seymour’s second neighbourhood conjecture and a Seymour-tight orientation is again a counterexample. Using lexicographic products, we show that if the conjecture is false, then there exist counterexamples that are close to regular tournaments, and moreover that any digraph occurs as an induced subgraph of a counterexample. We then use this same machinery to construct special putative counterexamples to Sullivan’s conjecture. The inherent symmetry of these orientations give access to an algebraic perspective. Seymour-tight orientations that are also Cayley digraphs correspond to special pairs of critical sets in groups, which connects potentially to additive combinatorics. We use Kemperman’s theorem to characterize those Seymour-tight orientations that are the Cayley digraph of an abelian group.

A classic problem in topology is determining how many smooth structures a given manifold can admit. We say that a manifold admits exotic smooth structures if there are at least two distinct smooth structures on the manifold. Understanding these exotic phenomena has been a central topic in both geometric and algebraic topology. Ever since Milnor discovered exotic 7-spheres, there has been extensive research searching for exotic smooth structures on various manifolds. Furthermore, the work of Donaldson and Freedman opened a new world for exotic phenomena in dimension 4. In this talk, we will provide an overview of exotic smooth structures on topological manifolds, review their history, and explore cutting-edge developments in the field.

ML/DL method are widely used for prediction in biomedical research, but predictive performance alone often provides limited scientific insight. Explainable artificial intelligence (XAI) and visualization methods can help identify important variables, characterize complex relationships, and generate interpretable findings beyond prediction accuracy. In this seminar, I will introduce practical XAI and visualization approaches and demonstrate their applications using two biomedical examples: drug-related expression prediction in human and mouse liver data, and analysis of hypnotic medication timing and use patterns. These examples illustrate how predictive modeling can be extended toward interpretation, hypothesis generation, and data-driven scientific discovery.

Hadwiger famously conjectured that every $K_h$-minor-free graph is properly $(h-1)$-colourable. This talk will present the following improper analogue of Hadwiger’s Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable with monochromatic components of bounded size. The number of colours is best possible regardless of the size of monochromatic components. This solves an open problem of Edwards, Kang, Kim, Oum and Seymour [SIAM J. Disc. Math. 2015], and concludes a line of research initiated in 2007. Similarly, for fixed $t\geqslant s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is best possible, solving an open problem of van de Heuvel and Wood [J. London Math. Soc. 2018]. We actually prove a single theorem from which both of the above results are immediate corollaries. For an excluded apex minor, the result is strengthened  as follows: for fixed $t \geqslant s \geqslant 3$, and for any fixed apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is $(s+1)$-colourable with monochromatic components of bounded size. The number of colours is again best possible. This is joint work with Vida Dujmović, Louis Esperet and Pat Morin [arXiv:2306.06224].

In critical PDEs, finite-time blow-up may occur while the scaling-critical norm remains bounded, with the singularity arising instead from concentration and loss of compactness. In this talk, we study this phenomenon for the radial focusing energy-critical wave equation in three dimensions. Starting from the radial free wave equation, we discuss incoming and outgoing radiation and the channel of energy method, which quantifies the energy escaping through exterior light cones. We then examine non-radiative dynamics and the rigidity principles arising from the channel of energy. Finally, we discuss how these ideas are related to the analysis of radial Type II blow-up.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

TBA

Most multi-armed bandit and reinforcement learning algorithms typically rely on full optimism, using confidence bonuses large enough to upper-bound unknown values with high probability. This talk presentsquasi-optimism, a new exploration principle that permits a controlled degree of underestimation while retaining provably optimal regret guarantees. We first introduce the principle in tabular reinforcement learning, where a simple inverse-count bonus achieves minimax-optimal regret without empirical-variance estimation, and then discuss its extension to distributional and instance-dependent regret guarantees in bandits and reinforcement learning. Finally, we extend quasi-optimism to linear contextual bandits through a quadratic–linear bonus. Despite not being pointwise optimistic, the new linear bandit algorithm simultaneously achieves minimax and margin-dependent regret guarantees under a single gap-agnostic tuning, while substantially reducing unnecessary exploration in empirical evaluations. Together, these results suggest that full optimism is not necessary for exploration to be both theoretically optimal and practically effective. This talk is based on the following recent works: Oh & Lee,Linear Contextual Bandits with Quasi-Optimism, 2026. Lee & Oh, Unified Framework of Distributional Regret in Multi-Armed Bandits and Reinforcement Learning, COLT 2026.Lee & Oh,Minimax Optimal Reinforcement Learning with Quasi-Optimism, ICLR 2025.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

Singularly perturbed differential equations arise in many problems in fluid mechanics and related areas, where small diffusivity or viscosity can generate sharp boundary, interior, and corner layers. These multiscale structures present significant challenges for both classical numerical methods and standard neural network approaches. We develop neural network methods that use asymptotic analysis to identify and incorporate the dominant structure of singular layers directly into the approximation. By explicitly incorporating the leading singular components into the neural network ansatz, the trainable component is used primarily to approximate the smoother part of the solution. We also consider conservative formulations based on finite volume residuals, which incorporate local flux balance into the learning framework. Numerical examples for a range of singularly perturbed problems demonstrate the accuracy and robustness of these approaches as the perturbation parameter tends to zero.

A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are subdivided into disjoint blocks of variables $X_1,\dots,X_c$, there is a natural multi-grading on $R$, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal $I$ of $R$. However, the full strength of the Bayer-Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals $I$ such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of $R/I$ in terms of almost regular sequences of the multi-graded generic initial ideal of $I$ restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.

TBA

This talk surveys major developments in the stability analysis of nonlinear waves in fluid dynamics. We begin with classical energy methods, which provide robust global $L^2$-bounds and nonlinear stability but often do not capture detailed spatial behavior or sharp decay rates. In many problems on unbounded domains, the absence of a spectral gap also prevents the direct use of standard semigroup theory.Pointwise semigroup methods overcome this difficulty through detailed Green’s function estimates, revealing how perturbations propagate, spread, and decay. Although powerful and sharp, this approach is often technically demanding and requires substantial spectral and resolvent information.We then discuss compactified spectral methods, a more recent and unified framework in which spatial transformations, weighted spaces, and compactification convert the original problem into a spectral or coercivity problem on a bounded interval. This can reveal an effective spectral gap, or a comparable coercive structure, even when none is apparent in the original formulation.These three approaches will be illustrated through examples from fluid mechanics, with the aim of providing a conceptual roadmap for modern stability analysis in nonlinear PDE.

Treewidth is a graph parameter commonly used to quantify how “close” a graph is to a tree. Although it is a cornerstone of structural graph theory and algorithm design, it is nearly useless for algorithmic purposes in many dense graph classes. In this talk, we discuss the tree-independence number, a more versatile graph parameter that replaces the standard width measure with the stability number. We will present recent results aimed at characterizing the graph classes in which this parameter enables sub-exponential time algorithms for problems that are, in general, NP-hard.

Vertex algebras first appeared in physics as the algebraic structures of 2D CFT (conformal field theory) and were mathematically defined by Borcherds in 1986 during the resolution of the Moonshine conjecture. Since then, the theory has been developed from both mathematical and mathematical physical perspectives. In this talk, we introduce the structural characteristics and complexity of vertex algebras, as well as the interesting phenomena and research topics arising from them.

Geometric group theory, as the name suggests, uses invariants of a geometric nature, e.g., Cayley graphs and growth, to study finitely generated groups. The study of nilpotency, especially the works of Gromov and Bass–Guivarc'h, revealed the importance of LCS ranks, numerical invariants computed from the lower central series. Unfortunately, LCS ranks are notoriously hard to compute explicitly, so they are usually replaced by more tractable approximations: the Chen ranks. Works by Papadima and Suciu showed that Chen ranks connect directly to algebro-geometric objects: Koszul modules and resonance loci. I will explain how Koszul modules and their resonance varieties can be used to prove Green's conjecture on the syzygies of canonical curves, via vanishing theorems for vector bundles.

We introduce the concept of the saturation of a (bi)graph: the union closure after inductively adding its virtual elements, which are weighted ε-good (respectively ε-excellent sets) as in the Stable Regularity Lemma. In the Littlestone class and stable graph case, we show that if the saturation has bounded Littlestone dimension, then it is the smallest ε-saturated object containing the initial one. We show that for certain values of ε, the saturations of Littlestone classes are Littlestone, although not necessarily of the same dimension. For ε large enough, we find examples to show that VC and Littlestone dimensions may grow arbitrarily. For certain ε, we bound Littlestone dimension of the saturation by a finite value depending on VC dimension, by using techniques including the Fundamental Theorem of Statistical Learning and the Littlestone Minimax Theorem. We will focus on the class (or bigraph) case and time permitting, we will discuss the stable graph case. Joint work with Maryanthe Malliaris and Shay Moran.

We present some basics of L-functions usually (to be) covered in the undergraduate and graduate entry level number theory classes.

The stability theory of entropy solutions to the compressible Euler system on a half-line with prescribed boundary conditions, such as inflow, outflow, and impermeable boundary conditions, remains largely open. It is worth emphasizing that establishing the stability of even a single reflected shock at the boundary has been a highly challenging problem. In this talk, I will briefly review the classical stability theory for small-BV entropy solutions in the whole-space setting and then turn to the half-line problem, where I will present recent progress on the stability of entropy solutions in the presence of a boundary.

Let $P$ be a set of points in $PG(n+d,q)$, and let $L$ be a set of $n$-flats. Here, $n$-flat is a short name for $n$-dimensional projective subspaces. A classical bound of Haemers, rediscovered in an influential paper of Vinh, gives an upper bound on the difference between the number of incidences between $P$ and $L$ and the expected number of incidences for random sets of points and flats with the same cardinalities as $P$ and $L$. Haemers’ bound is tight as a function of $|P|$ times $|L|$. Recent work of Kong and Tamo improves the bound under the assumption that $|L|$ is not too large. I will discuss recent work, joint with Tao Zhang, that improves the bound of Kong and Tamo. The proof depends on an independently interesting upper bound on the number of pairs $(l_1,l_2)$ of flats in $L$ such that $\dim(l_1 \cap l_2)=j$, for $0 \leq j \leq n$.