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.

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.

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.

This talk explores the intricate interplay between the intrinsic geometry of complex projective varieties and the homological invariants of their defining ideals. We focus on the structure of syzygies—encoding higher order algebraic relations among defining equations—and the Castelnuovo–Mumford regularity, a key invariant governing the complexity of embedded varieties. Finally, we survey classic and recent developments regarding linear syzygies, the status of the Eisenbud–Goto conjecture, and the higher syzygies of secant varieties.