In this talk, we introduce a two-dimensional semiflexible membrane model whose Hamiltonian is given by $H[\phi]=\sum_x |\nabla \phi_x|^2 +N^\lambda |\Delta \phi_x|^2$, interpolating between the discrete Gaussian free field (DGFF) and the membrane model (MM). We analyze its infinite-volume behavior and identify distinct regimes depending on the parameter $\lambda$. If $\lambda<0$, the covariance behaves similarly to that of the DGFF, while for $\lambda>2$, it resembles a rescaled MM. In the intermediate regime $0\le \lambda\le 2$, we observe a qualitatively different behavior: a nontrivial dependence on the distance between points. We further determine the leading order constant of the covariance. These results provide a unified description of the crossover from gradient-dominated to curvature-dominated behavior in this class of models.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the next two--three talks, I will explain the backgrounds on K_1 and relative K_0 of group rings for finite groups over local/global fields of characteristic 0 and their orders.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the first three talks, I will explain the backgrounds on Selmer groups and flat cohomology.

Diffusion is a macroscopic phenomenon arising from the random movement of particles at the microscopic level. Fick’s law predicts uniform spreading of particles over time, while fractionation is often observed in heterogeneous environments, as in the Soret effect and Darken’s experiment. In this talk, we show that such heterogeneous diffusion can be described by a two-coefficient diffusion equation derived from particle dynamics. In particular, for persistent random walks, fractionation occurs only when both heterogeneity and anisotropy are present. We formally derive the limiting diffusion equation and present a methodology to rigorously establish convergence from a persistent discrete kinetic equation to the macroscopic diffusion equation.

his lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the first two or three talks, I will explain the backgrounds on Selmer groups and flat cohomology.

This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the first two talks, I will explain the backgrounds on Selmer groups and flat cohomology.

The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with sufficiently large degree already contains exponentially many non-isomorphic spanning trees. Indeed, we prove a stronger statement: for every fixed $n$-vertex tree $T$, $$\Pr\bigl[\mathcal{T} \simeq_{\mathrm{iso}} T\bigr] = e^{-\Omega(n)},$$ where $\mathcal{T}$ is a uniformly random spanning tree of a connected $n$-vertex almost regular graph with sufficiently large degree. To prove this, we introduce a graph-theoretic variant of the classical balls--into--bins model, which may be of independent interest.

Many complete Riemannian manifolds of constant curvature, including complete hyperbolic manifolds, can be realized as quotients of convex domains in real projective spaces by discrete linear group actions. Among these discrete groups, an important and broad class is given by linear reflection groups (also called linear Coxeter groups). In dimension 3, many such linear reflection groups arise from hyperbolic reflection groups, using Andreev’s theorem, through small deformations and gluing constructions. In this talk, I will introduce these ideas and concepts, explain the basic picture, and discuss some recent developments related to the projective Andreev theorem.