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Talk 1 Title: Density-Sensitive Bifiltrations in Topological Data Analysis.

Abstract: The standard filtrations of TDA (Rips, Čech, Delaunay) are stable to small perturbations of the input data, but are notoriously unstable to outliers, and can be insensitive to variations in density. One of the main motivations of 2-parameter persistence is to address these limitations. There are several natural ways one can define a density-sensitive bifiltration of point cloud or metric data, offering different tradeoffs between generality, computability, and robustness to outliers. Such bifiltrations have been actively studied in recent years, leading to substantial advances in our understanding of them, as well as new computational tools. In this talk, I will survey this progress.

Talk 2 Title: $\ell^p$-metrics on Multiparameter Persistence Modules and Merge Trees.

Abstract: The standard stability and inference theory for persistent homology is an $\ell^\infty$ theory.  This talk concerns foundational work towards developing an $\ell^p$ extension of this theory.  Bjerkevik and I have introduced a generalization of the $p$-Wasserstein distance on barcodes to multiparameter persistence modules, called the $p$-presentation distance and denoted $d_I^p$.  We have shown that $d_I^{\infty}$ equals the interleaving distance and that on 1- or 2-parameter persistence modules over prime fields, $d_I^p$ is the universal (i.e., largest) metric satisfying a natural stability property.  This universality result extends a stability theorem of Skraba and Turner for the $p$-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance.  Cardona, Curry, Lam and I have subsequently introduced an analogous $p$-presentation distance on merge trees, with a closely analogous theory.
  In spite of these good theoretical properties, fundamental aspects of the $p$-presentation distances are poorly understood.  Most critically, the distances are defined in terms of two infima, and the question of whether these infima are attained has remained open. Recently, Riley Decker and I have proven that for both modules and merge trees, the infima are attained in the case $p=1$. We have also established bounds on the size of the objects attaining these infima. These bounds depend polynomially both on the size of minimal presentations for the input and on a measure of input fineness called the \emph{algebraic spread}.  We conjecture that these results extend to arbitrary $p$ with polynomial bounds that do not depend on the algebraic spread.

Talk 3 Title: Limit Computation Over Posets via Minimal Initial Functors. (Joint work with Tamal Dey)

Abstract: It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation.  We systematically study the algorithmic implications of this idea for diagrams indexed by a poset.  We say an initial functor $F\colon C\to D$ with $C$ small is \emph{minimal} if the sets of objects and morphisms of $C$ each have minimum cardinality among the sources of all initial functors with target $D$.  For $Q$ a finite poset or $Q$ an interval in $\mathbb{N}^d$ (i.e., a convex, connected subposet), we describe all minimal initial functors $F\colon P\to Q$ and in particular, show that $F$ is always a subposet inclusion.  We give efficient algorithms to compute a choice of minimal initial functor. In the case that $Q$ is an interval in $\mathbb{N}^d$, we give asymptotically optimal bounds on $|\hom P|$, the number of relations in $P$, in terms of the number $n$ of minima of $Q$: We show that $|\hom P|=\Theta(n)$ for $d\leq 3$, and $|\hom P|=\Theta(n^2)$ for $d>3$.  We apply these results to give new bounds on the cost of computing $\lim G$ for a functor $G\colon Q\to \mathbf{Vec}$ valued in vector spaces.  For $Q$ connected, we also give new bounds on the cost of computing the generalized rank of $G$  (i.e., the rank of the induced map $\lim G \to \operatorname*{colim} G$), which is of interest in topological data analysis.

Title: The TF equivalences on the real Grothendieck groups

Abstract: Let $\mathcal{A}$ be an abelian length category with finitely many isoclasses of simple objects. Then the dual real Grothendieck group $K_0(\mathcal{A})_\mathbb{R}^*$ is identified with an Euclidean space $\mathbb{R}^n$. The theme of my talk is the TF equivalence on $K_0(\mathcal{A})_\mathbb{R}^*$, which I introduced as a way to visualize important numerical data of $\mathcal{A}$ by using semistable torsion pairs of Baumann-Kamnitzer-Tingley. I will discuss how the TF equivalence is related to stability conditions of King and the wall-chamber structure of Brüstle-Smith-Treffinger. Then I will explain that the TF equivalence recovers the silting fan of 2-term silting complexes in the case $\mathcal{A}$ is the category of finitely generated modules over a finite dimensional algebra.

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Title: On Angle-optimization and Simplification of Degree-1 Homology Representatives

Abstract: In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree 1 and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data. This talk is based on joint work with Yuta Shimada (arXiv:2608.23949). 

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