Given a valuation over a singularity, it is a fundamental problem whether it has finitely generated associated graded rings. This problem has deep connection with the theory of K-stability and moduli, where the finite generation of certain minimizing valuations were shown. For klt singularities, we propose the study of Kollár valuations, which are valuations with finitely generated associated graded rings that induces klt degenerations. We show that the locus of Kollár valuations is path connected. We discuss some open questions and examples. Based on joint work with Chenyang Xu.

One-dimensional persistent homology is arguably the most important and heavily used tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and by utilizing cohomological ideas. In this talk, we introduce a certain 2-dimensional persistence module structure associated with the persistent cohomology ring, where one parameter is the cup-length and the other is the filtration parameter. We show that this new persistence structure, called the persistent cup module, is stable.
In addition, we consider a generalized notion of persistent invariants, which extends the standard rank invariant, Puuska's rank invariant induced by epi-mono-preserving invariants of abelian categories, and the recently-defined persistent cup-length invariant, and we establish their stability. This generalized notion of persistent invariants also enables us to lift the LS-category of topological spaces to a novel stable persistent invariant, called the persistent LS-category invariant.

This is Part II of Benjamin’s talk.

We discuss two problems in stochastic geometry:

1. Sylvester's problem is a classical question that asks for the probability that d+2 random points in R^d are in convex position, that is, that none of them is in the convex hull of the others. We compute this probability when the points are the steps of a random walk. Remarkably, the probability depends only on dimension and is independent of the increment distribution of the walk, provided that it satisfies a mild nondegeneracy condition.

2. An exact formula for the mean volume of the convex hull of d-dimensional Brownian motion at a given time has been known since 2014. What can be said about the inverse volume process? In other words, how much time, on average, is required for the convex hull to attain a given volume? We establish two-sided bounds that capture the correct order of asymptotic growth in the dimension.

We discuss work on the epsilon multiplicity which shows that the epsilon multiplicity exists and is a limit of Amao multiplicities in the most general possible case. These are results from joint work with Cutkosky and more recent work from last month.

Title: Gauge Theory on Hyperkahler Manifolds and 3-Sasakian Links

Abstract: Manifolds with special holonomy — such as Calabi-Yau, hyperkahler, G2, and Spin(7)-manifolds — are challenging objects to study.  In recent years, geometers have proposed various gauge-theoretic PDE (known as “instanton equations”) on such spaces, partly in an effort to define enumerative invariants.  While much work has been done in the Calabi-Yau, G2, and Spin(7) settings, the hyperkahler (HK) case has thus far received less attention.

In this talk, we discuss a natural class of Yang-Mills connections over hyperkahler 4n-manifolds X, known as Sp(n)-instantons, that generalize ASD instantons.  To model their conical singularities, we relate Sp(n)-instantons over HK cones to contact instantons over their links, and establish some dimensional reductions.  We then prove a “Lewis-type theorem” on asymptotically conical (AC) hyperkahler manifolds X to the following effect: If X admits AC Sp(n)-instantons, then any Hermitian Yang-Mills connection over X that decays sufficiently rapidly is necessarily an Sp(n)-instanton.  This is joint work with Emily Windes (Oregon).

Infinity categories, as formulated by André Joyal and developed by Jacob Lurie, have become a useful framework for dealing with higher categorical phenomena in different areas, including representation theory and algebraic geometry. The goal of this talk is to introduce just enough background on simplicial homotopy theory to define infinity categories, including the prerequisites to read the first chapter of Lurie's book, Higher Topos Theory. If time allows, we will discuss basic properties and examples of infinity categories.

To gain their unique biological function, plant cells regulate protein biosynthesis through gene activation and repression along with multiple mRNA mechanisms. The subcellular localization of mRNAs has been reported as a complementary regulatory mechanism of the biology of fungi, yeast, and animal cells. However, studies comprehensively reporting the impact of mRNA localization in plant cells are lacking.

Here, we set to mathematically model the spatial distribution of sub-cellular cytosolic transcripts across multiple cell types and developmental stages. Through the use of high-resolution spatial transcriptomic technology, we first report the comprehensive and differential mapping of millions of plant transcripts between the nuclear and cytoplasmic compartments of various soybean nodule cell types. We then characterize key mathematical features of these transcriptomic spatial distributions using Topological Data Analysis (TDA). TDA offers a comprehensive pattern-quantifying framework that is robust to variations in cell shape, size, and orientation. TDA thus provides us with a common ground to mathematically compare and contrast intrinsic differences in sub-cellular transcript distributions and patterns across cell types and expressed genes.

Our analyses reveal distinct patterns and spatial distributions of plant transcripts between the nucleus and cytoplasm, varying both between and within genes, as well as across different cell types. We believe this differential distribution is an additional, less understood, regulatory mechanism controlling protein translation and localization, cell identity, and cell state and reveals the influence of the sub-compartmentalization of transcripts as another post-transcriptional regulatory mechanism.

We generalize the classical Vieta formulas that express the coefficients of a polynomial in terms of all the roots. In particular, we focus on the case when just some of the roots are known. Our formulas are established by exploiting    some properties of the complete homogeneous symmetric polynomials. 
Further, we provide  new identities for these polynomials when the variables are in a geometric progression and give an application to cyclotomic polynomials. This is joint work with  A. Echezabal and M. Laporta

The study of fragile economic systems is important in identifying systems that are vulnerable to a dramatic collapse. For instance, complex systems like supply chains are at risk of being fragile because they require many parts to work well simultaneously. Even when each individual firm has a small susceptibility to a shock, the global system may still be at great risk. A recent survey by Matthew Elliot and Ben Golub review fragile economic systems from the point of view of networks. In a network, the reliability that the final product (e.g., a car, computer, or lifesaving medication) is made by a firm is determined the probabilities of shocks being in the system. Thus, reliability transitions from being zero to a positive probability depending on the chances of a shock --- characterizing these phase transitions is an important problem in the theory of economic fragility. In our work, we view these phase transitions through the algebraic geometry lens by using resultants. As a result, we bring new tools to econometrics to analyze multi-parameter models, and we fully describe the reliability of many new network models using computational algebraic geometry. Our most significant application is a surprising case study on a mixture of two multi-parameter supply chain models. This is joint work with Jiayi Li (UCLA).