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. 

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.
 

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.

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.

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.

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.

Abstract: Let us think of a convex body in R^n (convex, compact set, with non-empty interior) as an opaque object, and let us place point light sources around it, wherever and however far from the body we want, to illuminate its entire surface. What is the minimum number of such light sources that we would have to use? The Hadwiger-Boltyanski illumination conjecture from 1960 states that we need at most as many light sources as for the n-dimensional hypercube, and more generally, as for n-dimensional parallelotopes. For the latter their illumination number is exactly 2^n, and they are conjectured to be the only equality cases.

The conjecture is still open in dimension 3 and above, and has only been fully settled for certain classes of convex bodies (e.g. zonoids, bodies of constant width, etc.). In this talk I will briefly discuss some of its history, and then focus on joint works with Wen Rui Sun that settle the conjecture for all 1-symmetric convex bodies (by complementing a method previously developed by K. Tikhomirov for this class of bodies), and that also deal with certain cases of 1-unconditional bodies, and possible extensions of those.

The Bernstein-Sato Polynomial is a classical \(D\)-module invariant that has been used to measure the singularities of a hypersurface \(f\). In this talk we will review some of the classical theory before considering the Bernstein-Sato polynomial over a numerical semigroup ring \(R\). We will see that in this case the Bernstein-Sato polynomial detects not just of an element in \(R\) but of the ambient numerical semigroup ring itself. 

We analyze the space of steady rotating solutions to the two-dimensional incompressible Euler equations nearby vortex patch solutions satisfying a nondegeneracy condition. We address the question of desingularization and prove that such vortex patch states are the limit of rotating Euler solutions that are smooth to infinite order, have compact vorticity support, and respect dihedral symmetry. Our nondegeneracy condition is proved to be satisfied by Kirchhoff ellipses and along the local bifurcation curves emanating from the Rankine vortex. The construction, that is based on a local stream function formulation in a tubular neighborhood of the patch boundary, is a synthesis of analysis on thin domains, nonlinear a priori estimates, and Newton's method. Our techniques additionally allow us to construct nearby exotic families of singular rotating vortex patch-like solutions. This is joint work with Razvan-Octavian Radu.

This talk is about joint work with I. Swanson.

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A longstanding question in algebraic geometry is the classification of  reduced and 
irreducible local complete one--dimensional domains $R$ over an algebraically closed 
field $k$. It is known that such a ring is completely determined once it is known up 
to a "sufficiently high" power of its maximal ideal, where this sufficiently 
high power depends on the singularity degree $\delta$ of the ring.

In this talk we show that two curve singularities $(R, \mathfrak m)$ and $(R', \mathfrak m')$ 
are already isomorphic if there exists an isomorphishm 
$\varphi: R/ \mathfrak m^{j+1} \longrightarrow R'/ {\mathfrak m'}^{j+1}$ of 
$k$--algebras for some $j \geq 2 \delta +1$, and that the isomorphism may be chosen to agree with $\varphi 
\pmod{\mathfrak m^{j-2 \delta+1}}$. This strengthens a result of Hironaka, who obtained the
bound $3 \delta + 1$.