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Series: Algebra Seminar

Series: Algebra Seminar

Series: Algebra Seminar

Series: Algebra Seminar

In this talk, we introduce rather exotic algebraic structures called
hyperrings and hyperfields. We first review the basic definitions and
examples of hyperrings, and illustrate how hyperfields can
be employed in algebraic geometry to
show that certain topological spaces (underlying topological spaces of
schemes, Berkovich analytification of schemes, and real schemes) are
homeomorphic to sets of rational points of schemes over hyperfields.

Series: Algebra Seminar

According to Plucker's formula, the total inflection of a linear series (L,V) on a complex algebraic curve C is fixed by numerical data, namely the degree of L and the dimension of V. Equipping C and (L,V) with compatible real structures, it is more interesting to ask about the total real inflection of (L,V). The topology of the real inflectionary locus depends in a nontrivial way on the topology of the real locus of C. We study this dependency when C is hyperelliptic and (L,V) is a complete series. We first use a nonarchimedean degeneration to relate the (real) inflection of complete series to the (real) inflection of incomplete series on elliptic curves; we then analyze the real loci of Wronskians along an elliptic curve, and formulate some conjectural quantitative estimates.

Series: Algebra Seminar

The talk reports on joint work with Wayne Raskind and concerns the conjectural definition of a new type of regulator map into a quotient of an algebraic torus by a discrete subgroup, that should fit in "refined" Beilinson type conjectures, exteding special cases considered by Gross and Mazur-Tate.The construction applies to a smooth complete variety over a p-adic field K which has totally degenerate reduction, a technical term roughly saying that cycles acount for the entire etale cohomology of each component of the special fiber. The regulator is constructed out of the l-adic regulators for all primes l simulateously. I will explain the construction, the special case of the Tate elliptic curve where the regulator on cycles is the identity map, and the case of K_2 of Mumford curves, where the regulator turns out to be a map constructed by Pal. Time permitting I will also say something about the relation with syntomic regulators.

Series: Algebra Seminar

Given data and a statistical model, the maximum likelihood estimate is
the point of the statistical model that maximizes the probability of
observing the data. In this talk, I will address three different
approaches to maximum likelihood estimation using algebraic methods.
These three approaches use boundary stratification of the statistical
model, numerical algebraic geometry and the EM fixed point ideal. This
talk is based on joint work with Allman, Cervantes, Evans, Hoşten,
Kosta, Lemke, Rhodes, Robeva, Sturmfels, and Zwiernik.

Series: Algebra Seminar

The nerve complex of an open covering is a well-studied notion. Motivated by the so-called Lyubeznik complex in local algebra, and other sources, a notion of higher nerves of a collection of subspaces can be defined. The definition becomes particularly transparent over a simplicial complex. These higher nerves can be used to compute depth, and the h-vector of the original complex, among other things. If time permits, I will discuss new questions arises from these notions in commutative algebra, in particular a recent example of Varbaro on connectivity of hyperplane sections of a variety. This is joint work with J. Doolittle, K. Duna, B. Goeckner, B. Holmes and J. Lyle.

Series: Algebra Seminar

The structure of non-archimedean curves X and their tame covers f:Y-->X is well understoodand can be adequately described in terms of a (simultaneous) semistable model. In particular, asindicated by the lifting theorem of Amini-Baker-Brugalle-Rabinoff, it encodes all combinatorialand residual algebra-geometric information about f. My talk will be mainly concerned with the morecomplicated case of wild covers, where new discrete invariants appear, with the different function being the most basic one. I will recall its basic properties following my joint work with Cohen and Trushin,and will then pass to the latest results proved jointly with U. Brezner: the different functioncan be refined to an invariant of a residual type, which is a (sort of) meromorphic differential form on the reduction, so that a lifting theorem in the style of ABBR holds for simplest wild covers.

Series: Algebra Seminar

I will explain how to explicitly compute the syntomic regulator for varieties over $p$-adic fields, recently developed by Nekovar and Niziol, in terms of Vologodsky integration. The formulas are the same as in the good reduction case that I found almost 20 years ago. The two key ingrediants are the understanding of Vologodsky integration in terms of Coleman integration developed in my work with Zerbes and techniques for understanding the log-syntomic regulators for curves with semi-stable reduction in terms of the smooth locus.