This talk presents recent progress in differentially private hypothesis testing, focusing on the interplay between privacy, validity, and statistical efficiency. I will discuss a framework for private permutation testing that preserves finite-sample validity and extends naturally to kernel-based procedures. These ideas yield private testing methods with strong theoretical guarantees, including optimality properties in several regimes. I will then turn to minimax results for two-sample testing under central differential privacy, which reveal a rich structure in the privacy–power trade-off. The overall message is that rigorous privacy protection can be incorporated into modern hypothesis testing without sacrificing principled statistical guarantees.

(This is a reading seminar given by the PhD Student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.

For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory.

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight (Sym^4, det^-1) with at most pole of order 1, and that this construction is functorial with respect to degeneration, namely the K-theory elevator for the cycle corresponds to the Siegel operator for the modular form.

Using an operator-theoretic approach, we provide a unified framework for Optimal Transport (OT) between Gaussian measures on separable Hilbert spaces. This formulation allows us to fully characterize the Monge and Kantorovich problems without imposing any regularity or non-degeneracy conditions on the covariance operators. We then develop the dynamic picture, explicitly characterizing 2-Wasserstein geodesics and particle dynamics in this general setting. Extending these results to Entropic OT, we show that the optimal entropic coupling operates as a precise spectral shrinkage of the correlation operator. Time permitting, I will discuss the algorithmic advantages of this spectral perspective and present complementary viewpoints connecting these transport problems.

Functional principal component analysis (FPCA) is a fundamental tool for exploring variation in samples of random curves or surfaces. We propose a new approach to FPCA for functional data observed irregularly and sparsely over their domains, based on smoothing directly at the level of the eigenfunctions. Our formulation leads to an efficient optimization-based procedure whose computational and storage costs are comparable to those of standard multivariate PCA for regularly observed data. The method is flexible with respect to domain geometry and model class, accommodates structural constraints and penalties, and facilitates uncertainty quantification via resampling and asymptotic theory.

Wall's stabilization principle suggests that exotic phenomena in dimension four in the orientable category disappear after taking connected sums with sufficiently many S2xS2. Since most known exotic pairs of closed 4-manifolds become diffeomorphic after one stabilization, a natural question was: is a single S2xS2 enough? Recently, Jianfeng Lin constructed an exotic diffeomorphism on a closed 4-manifold-a diffeomorphism topologically isotopic to the identity but not smoothly isotopic-that survives one stabilization. In this talk, we provide a relative exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations. This gives the first exotic phenomenon in the orientable category that survives two stabilizations. The obstruction to stabilization comes from equivariant Seiberg–Witten theory, together with a version of lattice homology. I will also survey some background and recent developments in equivariant gauge theory. This is joint work with Sungkyung Kang and JungHwan Park.