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).

We will finish the proof of Quillen-Suslin. Please note the non-standard meeting time for the seminar this week.

Many problems in computer vision are represented using a parametrized overdetermined system of polynomials which must be solved quickly and efficiently. Classical methods for solving these systems involve specialized solvers based on Groebner basis techniques or utilize randomization in order to create well-constrained systems for numerical techniques. We propose new methods in numerical linear algebra for solving such overdetermined polynomial systems and provide a complete error analysis showing that the numerical approach is stable. Examples will be provided to show the efficacy of the method and how the error in the data affects the error in the solution.

This is the first part of the talk on the Quillen-Suslin theorem, formerly known as Serre's Conjecture. The assertion is that a finitely generated projective module over a polynomial ring over a field is always free. In 1976, Quillen and Suslin independently provided a complete proof of this theorem. Quillen was awarded a fields medal in part because of his proof this conjecture. In this talk we'll discuss a simplification of Suslin's proof given by Vaserstein. 

In the first part, we'll develop the required machinery, starting with the concepts such as stably free modules and unimodular column extension property (UCEP). Based on these, we'll see what stably free modules of a commutative ring having UCEP look like. Also, we'll see the proof of Horrock's theorem and will draw some observations required to prove the main theorem.

When is it possible to project two sets of labeled points lying in a pair of projective planes to the same points on a projective line? Here one answer: such projections exist if and only if the two 2D point sets are themselves images of a common point set in 3D projective space. Furthermore, when the two sets of points are in general position, it is possible to give a complete description of the loci of pairs of projection centers. I will describe the roles of classical invariant theory, Cremona transformations, and geometric computer vision in this description.

Let \(R\) be a Noetherian local ring  of dimension \(d\).   We define an intersection product on schemes \(Y\) which are birational and projective over \(Spec(R)\)  which allows us to interpret multiplicity in \(R\) as an intersection product, generalizing a theorem of Ramanujam and others. We use this to give new geometric formulations and proofs of some classical theorems about multiplicity. In particular, we give a new geometric proof of a celebrated theorem of Rees about degree functions. This is joint work with Jonathan Montaño.