Mathematics

We study minimizers of Cahn-Hilliard energies under strong anchoring (Dirichlet) boundary conditions using a Nehari manifold approach with spectral analysis of the Dirichlet Laplacian.

For the de Gennes energy with quartic potential, we reveal bifurcation phenomena governed by the boundary value and transition layer thickness parameter. When the boundary value equals the phase average, and the parameter exceeds a critical threshold, the minimizer is unique and homogeneous; below this threshold, two symmetric minimizers emerge. Deviating boundary values restore uniqueness with asymmetric minimizers. We derive rigorous bounds for these solutions.

We extend this framework to the Flory-Huggins logarithmic potential, which models polymer blends and presents singular behavior at boundaries. Our analysis, supported by numerical simulations, reveals temperature-mediated bifurcations and demonstrates how the Nehari manifold technique provides a unified treatment of both functionals under strong anchoring conditions.

In this talk, I will describe a strong Briançon-Skoda type result (the integral closure of \(J^{n+k-1}\) is contained in \(J^k\), or a slight enlargement of \(J^k\)) which utilizes the Eagon-Northcott
or Buchsbaum-Eisenbud complex.

This result gives a new proof of Lipman-Sathaye's Briançon-Skoda result for regular rings, it gives the
precise version of Briançon-Skoda for generalizations of pseudo-rational rings (improving Lipman-Teissier's result and implying Aberbach-Huneke in equal characteristic), it also implies the tight closure, plus closure, and epf closure versions of the Briançon-Skoda theorem. It also implies effective Briancon-Skoda results for characteristic free versions of Du Bois singularities, generalizing work of Huneke-Watanabe and Wheeler-Zhang in the F-pure case.  Finally, we show how this result, plus Gabber's weak local unformization, can be used as the missing piece to solve Huneke's conjecture of uniform Briançon-Skoda
and uniform Artin-Rees for (reduced) quasi-excellent rings of finite Krull dimension.

Magic angles are a hot topic in condensed matter physics: when two sheets of graphene are twisted by those angles, the resulting material is superconducting and the so-called energy bands are flat. In 2011, Bistritzer and MacDonald proposed a model that is experimentally very accurate in predicting magic angles. In this talk, I will introduce some recent mathematical progress on the Bistritzer--MacDonald's model, including the mathematical characterization of magic angles and flat bands, and generic existence of Dirac cones. I will also discuss some new mathematical discoveries in twisted multilayer graphene.

Given a finitely generated module \(M\) over a noetherian local ring \(R\), one may assign to it a conical affine variety, called the cohomological support variety of \(M\) over \(R\). This theory was first developed by Luchezar Avramov for local complete intersection rings in 1989, and by the work of many has recently been extended to encompass all commutative noetherian local rings. Geometric properties of this variety encode important homological information about \(M\) as well as \(R\). In this talk I will discuss what cohomological support varieties are, why they are useful, and some recent work on how they behave when restricting along a local homomorphism.

In this talk we will discuss old and not so old inequalities on the
volume of the orthogonal projections (sometimes called local
Loomis-Whitney type estimates).  We will explore connections of those
inequalities to inequalities for mixed volumes as well as inequalities
of the Minkwoski sums of convex bodies.
 

We revisit the problem of estimating the fundamental matrix of a pair of perspective cameras, a cornerstone of geometric computer vision. As is well-known, linear solvers require at least 8 point correspondences, whereas nonlinear minimal solvers require just 7 in the uncalibrated case or 5 in the calibrated case. In this paper, we consider a special case of the 7-point problem where 5 of the points are configured to lie on two lines, which has previously been shown to have a unique solution. As a theoretical contribution, we offer an analysis of how this uniqueness manifests in the standard 7-point algorithm. On a practical level, we provide the first practical linear solver for the minimal problem associated to this special configuration. Additionally, we evaluate a heuristic 5-point fundamental matrix solver based on the construction of virtual midpoints. When combined with early non-minimal fitting, the runtime and accuracy of our solver is competitive with the state-of-the-art on multiple benchmarks. 

Pure ring maps arise naturally in algebraic geometry and commutative algebra. For example, inclusions of rings of invariants by linearly reductive groups, split maps, and faithfully flat maps are all pure. It is useful to know when a map is pure because pure maps satisfy effective descent properties and many classes of singularities are preserved under the operation of taking pure subrings. For example, a theorem of Boutot (for rings of finite type over a field) and myself (in general) says that for Noetherian \(\mathbf Q\)-algebras, pure subrings of rings with (at worst) rational singularities have (at worst) rational singularities. Such results are often called Boutot-type theorems.

In this talk, I will discuss a new class of pure ring maps which arise geometrically from families of varieties of the same dimension. In fact, this yields a characterization of the splinter property: A Noetherian ring \(R\) is a splinter (i.e., all module-finite extensions \(R \to S\) split) if and only if every locally equidimensional surjective morphism \(Spec(S) \to Spec(R)\) is pure. Since not all pure maps are locally equidimensional and Boutot-type theorems fail for some classes of singularities like \(F\)-rationality, this raises the question: Are there "weak" Boutot-type theorems for pure ring maps that are also equidimensional? I will discuss my affirmative solution to this question for \(F\)-rationality.

The quantum many-body problem lies at the heart of modern physics and chemistry, yet its complexity continues to challenge both theory and computation. In this talk, I will provide a brief introduction to the quantum many-body problem and outline several mathematical questions that may help advance the field. Particular emphasis will be placed on coupled cluster–based approaches, embedding methods, and emerging quantum computational strategies. Throughout the presentation, I will highlight how mathematical analysis and algorithmic development can contribute to improving accuracy, scalability, and conceptual understanding in computational many-body theory.

Guidance mechanisms enable controllable generation from diffusion models at inference time. Classifier guidance steers sampling using gradients from a noise-aware classifier, offering principled control but requiring a separately trained network. Classifier-free guidance eliminates the external classifier by interpolating conditional and unconditional predictions, yet demands paired training. Training-free methods such as universal guidance repurpose off-the-shelf networks, but rely on per-step gradient optimization that is expensive and often unstable.

In this talk, I present a general recipe for efficiently steering unconditional diffusion models without gradient guidance during inference. Our approach rests on two structural observations. First, noise alignment: even at early, highly corrupted stages of the reverse process, coarse semantic steering is possible using a lightweight, offline-computed guidance signal—no per-step or per-sample gradients required. Second, transferable concept vectors: a concept direction in activation space, once learned, transfers across both timesteps and samples. A single fixed steering vector learned near low noise levels remains effective when injected at intermediate noise levels for every generation trajectory, providing refined conditional control at negligible cost. These directions are identified via Recursive Feature Machines (RFM), a backpropagation-free feature learning method. Experiments on CIFAR-10, ImageNet, and CelebA demonstrate improved accuracy and generation quality over gradient-based guidance, with significant inference speedups.

Two convex bodies $K$ and $L$ in $\mathbb{R}^n$ are affine-equivalent if there exists an affine transformation $T$ such that $T(K)=L$. To measure how close two convex bodies are when they are not exactly affinely equivalent, one introduces the Banach--Mazur distance. Roughly speaking, this is the smallest factor $R \ge 1$ such that, after an appropriate affine transformation, one body is contained in the other, and the other is contained in its dilation by $R$ (with respect to some center $\xi$).

A consequence of Fritz John’s theorem is that for any symmetric convex body, one can approximate $K$ by a polytope $P$ with $O(n)$ facets and Banach--Mazur distance $O(\sqrt{n})$, which is sharp for the Euclidean ball. In contrast, for general convex bodies, the same theorem implies that even with $O(n^2)$ facets, one can only guarantee distance $O(n)$. Thus, there is a gap of order $\sqrt{n}$ for Coarse Polytope Approximation (coarse means the allowance of facets/vertices can only be polynomial in $n$.)

While this problem has been known for over two decades, it remains open whether this $\sqrt{n}$ gap is essential (up to polylogarithmic factors).

In this talk, we will. show that the $O(n)$ bound is essential if one requires the scaling center to be a classical center, such as the barycenter. In other words, either the $\sqrt{n}$ gap is inherent for general convex bodies, or one must go beyond such classical choices of centers. We also include a concrete open problem that appears approachable with current technique is also included.

(This is joint work with Mark Rudelson.)