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1. Classification of Frobenius Forms in Five Variables

   https://doi.org/10.1007/978-3-030-91986-3_15  

   Kadyrsizova, Zhibek; Kenkel, Jennifer; Page, Janet; Singh, Jyoti; Smith, Karen E.; Vraciu, Adela; Witt, Emily E. (January 2021, Women in Commutative Algebra: Proceedings of the 2019 WICA Workshop)

   Miller, Claudia; Striuli, Janet; Witt, Emily E. (Ed.)

   Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a “triangle”. 

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2. Lower bounds on the F-pure threshold and extremal singularities

   https://doi.org/10.1090/btran/106  

   Kadyrsizova, Zhibek; Kenkel, Jennifer; Page, Janet; Singh, Jyoti; Smith, Karen; Vraciu, Adela; Witt, Emily (October 2022, Transactions of the American Mathematical Society, Series B)

   We prove that if f f is a reduced homogeneous polynomial of degree d d , then its F F -pure threshold at the unique homogeneous maximal ideal is at least 1 d − 1 \frac {1}{d-1} . We show, furthermore, that its F F -pure threshold equals 1 d − 1 \frac {1}{d-1} if and only if f ∈ m [ q ] f\in \mathfrak m^{[q]} and d = q + 1 d=q+1 , where q q is a power of p p . Up to linear changes of coordinates (over a fixed algebraically closed field), we classify such “extremal singularities”, and show that there is at most one with isolated singularity. Finally, we indicate several ways in which the projective hypersurfaces defined by such forms are “extremal”, for example, in terms of the configurations of lines they can contain. 

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3. The moduli spaces of sheaves on surfaces, pathologies and brill-noether problems

   https://doi.org/10.1007/978-3-319-94881-2_4  

   Cosun, I.; Huizenga, J. (November 2018, Abel symposia)

   Brill-Noether Theorems play a central role in the birational geometry of moduli spaces of sheaves on surfaces. This paper surveys recent work on the Brill-Noether problem for rational surfaces. In order to highlight some of the difficulties for more general surfaces, we show that moduli spaces of rank 2 sheaves on very general hypersurfaces of degree d in P3 can have arbitrarily many irreducible components as d tends to infinity. 

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4. Shadow surface states in topological Kondo insulators

   https://doi.org/10.1088/1367-2630/ac4124  

   Ghazaryan, Areg; Nica, Emilian M.; Erten, Onur; Ghaemi, Pouyan (December 2021, New Journal of Physics)

   Abstract The surface states of 3D topological insulators in general have negligible quantum oscillations (QOs) when the chemical potential is tuned to the Dirac points. In contrast, we find that topological Kondo insulators (TKIs) can support surface states with an arbitrarily large Fermi surface (FS) when the chemical potential is pinned to the Dirac point. We illustrate that these FSs give rise to finite-frequency QOs, which can become comparable to the extremal area of the unhybridized bulk bands. We show that this occurs when the crystal symmetry is lowered from cubic to tetragonal in a minimal two-orbital model. We label such surface modes as ‘shadow surface states’. Moreover, we show that the sufficient next-nearest neighbor out-of-plane hybridization leading to shadow surface states can be self-consistently stabilized for tetragonal TKIs. Consequently, shadow surface states provide an important example of high-frequency QOs beyond the context of cubic TKIs. 

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5. Geometric approaches to matrix normalization and graph balancing

   https://doi.org/10.1017/fms.2025.10105  

   Needham, Tom; Shonkwiler, Clayton (September 2025, Forum of Mathematics, Sigma)

   Normal matrices, or matrices which commute with their adjoints, are of fundamental importance in pure and applied mathematics. In this paper, we study a natural functional on the space of square complex matrices whose global minimizers are normal matrices. We show that this functional, which we refer to as the non-normal energy, has incredibly well-behaved gradient descent dynamics: despite it being nonconvex, we show that the only critical points of the non-normal energy are the normal matrices, and that its gradient descent trajectories fix matrix spectra and preserve the subset of real matrices. We also show that, even when restricted to the subset of unit Frobenius norm matrices, the gradient flow of the non-normal energy retains many of these useful properties. This is applied to prove that low-dimensional homotopy groups of spaces of unit norm normal matrices vanish; for example, we show that the space of $$d \times d$$ complex unit norm normal matrices is simply connected for all $$d \geq 2$$. Finally, we consider the related problem of balancing a weighted directed graph – that is, readjusting its edge weights so that the weighted in-degree and out-degree are the same at each node. We adapt the non-normal energy to define another natural functional whose global minima are balanced graphs and show that gradient descent of this functional always converges to a balanced graph, while preserving graph spectra and realness of the weights. Our results were inspired by concepts from symplectic geometry and Geometric Invariant Theory, but we mostly avoid invoking this machinery and our proofs are generally self-contained. 

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