The theory of F-singularities uses the Frobenius morphism to measure how "singular" a ring is, that is, how far it is from being a regular ring. We describe two ways to measure singularities using frobenius, respectively called the quasi-F-split height and the F-pure threshold. We describe a relationship between these two invariants, which is latent in the literature, under very specific assumptions. The proof of this statement uses sophisticated geometric machinery (including deformation theory and crystalline cohomology). We then describe a more general proof, joint with Jagathese, that uses only algebra.

The basic question of enumerative geometry can be simply stated as:
How many geometric objects of a given type satisfy given geometric conditions?

For instance, one may ask for the number of lines through 2 points in the plane, or the number of conics through 5 points in the plane. 

The purpose of this talk is to give an introduction to counting problems in algebraic geometry. 

The Voronoi box based two point flux finite volume method provides a path to discretization approaches for systems of partial differential equations which conform to natural constraints of solutions and basic principles of thermodynamics.  As a case in point, the talk introduces an adaptation of the well-known Scharfetter-Gummel upwind scheme from semiconductor physics to generalized Nernst-Planck-Poisson systems taking into accout finite ion size and solvation effects ^[1].

The method has been implemented in the Julia programming language  using the package VoronoiFVM.jl ^[2].  It provides solution methods for coupled nonlinear reaction-convection-diffusion problems in one-, two- and three-dimensional spatial domains.  A key ingredient of this package is the utilization if automatic differentiation to tackle complex nonlinearities in realistic physical models, see e.g. ^[3,4,5].

We will discuss a number of work-in-progress examples demonstrating the utility of this approach in the context of electrolyte simulations including
- Model problem based simulation of double layer effects on electrochemical reactions
- Calculation of electroosmotic flows by coupling with pressure robust finite element methods (Julia based re-implementation of the approach in ^[6])
- Automatic generation of reaction terms from chemical equations using Catalyst.jl <sup>[7]


[1] B. Gaudeul and J. Fuhrmann, "Entropy and convergence analysis for two finite volume schemes for a Nernst-Planck-Poisson system with ion volume constraints", Numerische Mathematik, vol. 151, no. 1, pp. 99–149, 2022
[2] J. Fuhrmann and contributors, https://urldefense.com/v3/__https://github.com/WIAS-PDELib/VoronoiFVM.jl__;!!Mih3wA!DMZTW98Az4xv1B69eOyiSWXUGZQn0qdiyc21NPoVpmuEkF-9hHD07uK0NO4d1mPc84rpGuX0fuTTmJ0Bp9cROjRyOD0C-vQ46g$
[3] Ch. Keller, J. Fuhrmann, and M. Landstorfer, "A model framework for ion channels with selectivity filters based on continuum non-equilibrium thermodynamics", Entropy 2025, 27(9), 981
[4] V. Miloš, P. Vágner, D. Budáč, M. Carda, M. Paidar, J. Fuhrmann, and K. Bouzek, "Generalized Poisson-Nernst-Planck-based physical model of an O2 | LSM | YSZ electrode", Journal of the Electrochemical Society,  no. 169, p. 044505, 2022
[5] D. Brust, K. Hopf, A. Cheilytko, M. Wullenkord, and Ch. Sattler, "Transport of heat and mass for reactive gas mixtures in porous media: Modeling and application", Chemical Engineering Journal 516(15) 2025, 162027
[6] J. Fuhrmann, C. Guhlke, A. Linke, C. Merdon, and R. Müller, “Induced charge electroosmotic flow with finite ion size and solvation effects,” Electrochimica Acta, vol. 317, pp. 778–785, 2019
[7] Loman, T. E., Ma, Y., Ilin, V., Gowda, S., Korsbo, N., Yewale, N., Rackauckas, Ch & Isaacson, S. A. (2023). Catalyst: Fast and flexible modeling of reaction networks. PLOS Computational Biology, 19(10), e1011530.</sup>

The contributions of this dissertation advance both the methodology and theory of modern statistical inference. On the one hand, they establish a distributional theory and resampling framework for spectral analysis in high-dimensional time series. On the other, they provide new probability and moment inequalities for quadratic forms under weak moment conditions. The combined results offer versatile tools for analyzing high-dimensional and heavy-tailed data, thereby addressing fundamental challenges in contemporary statistics.

With the objective of characterizing the stationary behavior of the scaling limit for shortest remaining processing time (SRPT) queues with a heavy-tailed processing time distribution, as obtained in Banerjee, Budhiraja, and Puha (BBP, 2022), we study reflecting coupled Brownian motions (RCBM) $(W_t(a), a, t \geq 0)$. These RCBM arise by regulating coupled Brownian motions (CBM)
$(\chi_t(a), a,t \geq 0)$ to remain nonnegative. Here, for $t\geq 0$, $\chi_t(0)=0$ and
$\chi_t(a):=w(a)+\sigma B_t-\mu(a)t$ for $a>0$, $w(\cdot)$ is a suitable initial condition, $\sigma$ is a positive constant, $B$ is a standard Brownian motion, and $\mu(\cdot)$ is an unbounded, positive, strictly decreasing drift function. In the context of the BBP (2022) scaling limit, the drift function is determined by the model parameters, and, for each $a\geq 0$, $W_{\cdot}(a)$ represents the scaling limit of the amount of work in the system of size $a$ or less. Thus, for the BBP (2022) scaling limit, the time $t$ values of the RCBM describe the random distribution of the size of the remaining work in the system at time $t$. Our principal results characterize the stationary distribution of the RCBM in terms of a maximum process $M_*(\cdot)$ associated with CBM starting from zero. We obtain an explicit representation for the finite-dimensional distributions of $M_*(\cdot)$ and a simple formula for its covariance. We further show that the RCBM converge in distribution to $M_*(\cdot)$ as time $t$ approaches infinity. From this, we deduce the stationary behavior of the BBP (2022) scaling limit, including obtaining an integral expression for the stationary queue length in terms of the associated maximum process. While its distribution appears somewhat complex, we compute the mean and variance explicitly, and we connect with the work of Lin, Wierman, and Zwart (2011) to offer an illustration of Little’s Law.   This is joint work with Marvin Pena (CSUSM) and Sixian Jin (CSUSM).