Seminars and Colloquia by Series

A Taste of Extremal Combinatorics in AG

Series
Algebra Seminar
Time
Tuesday, December 7, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Robert WalkerUniversity of Wisconsin, Madison

In this talk, we survey known results and open problems tied to the dual graph of a projective algebraic F-scheme over a field F, a construction that apparently Janos Kollar is familiar with. In particular one can use this construction to answer the following question: if you consider the 27 lines on a cubic surface in P^3, how many lines meet a given line? The dual graph can answer this and more questions in enumerative geometry and intersection theory easily, based on work of Benedetti -- Varbaro and others.

Cayley-Bacharach theorems and measures of irrationality

Series
Algebra Seminar
Time
Tuesday, November 30, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Brooke UlleryEmory University

If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.

Homology representations of compactified configurations on graphs

Series
Algebra Seminar
Time
Tuesday, November 16, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Claudia YunBrown

The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish. Our paper can be found on arXiv with ID 2109.03302.

Clusters and semistable models of hyperelliptic curves

Series
Algebra Seminar
Time
Tuesday, November 9, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jeffrey YeltonEmory University

For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is semistable -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities.  When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the cluster data associated to $f$.  When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated.  I will give an overview of how to construct "nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.

u-regeneration: solving systems of polynomials equation-by-equation

Series
Algebra Seminar
Time
Tuesday, November 2, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jose RodriguezUniversity of Wisconsin, Madison

Solving systems of polynomial equations is at the heart of algebraic geometry. In this talk I will discuss a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems. Our approach uses fewer homotopy continuation paths than the current leading method based on regeneration.  Moreover it is applicable in both projective and multiprojective settings. To motivate the approach I will also give some examples coming from applied algebraic geometry.
This is joint work with Tim Duff and Anton Leykin.

Graded rings with rational twist in prime characteristic

Series
Algebra Seminar
Time
Tuesday, October 26, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Florian EnescuGeorgia State

Prompted by the definition for the Frobenius complexity of a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of graded rings for which this generating function is rational. We will discuss this class of rings.  This work is joint with Yongwei Yao.

Geometric equations for matroid varieties

Series
Algebra Seminar
Time
Tuesday, October 5, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ashley K. WheelerGeorgia Tech

Each point x in Gr(r, n) corresponds to an r × n matrix A_x which gives rise to a matroid M_x on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|M_y = M_x} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the GrassmannCayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich.

Moduli spaces of tropical curves and tropical psi classes

Series
Algebra Seminar
Time
Tuesday, September 28, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Andreas GrossGeorgia Tech

Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts. In my talk, I will report on joint work with Renzo Cavalieri and Hannah Markwig, in which we define tropical psi classes and study relations between them. I will explain how some of the expected identities cannot be recovered from a purely tropical perspective, whereas others can, revealing the tropical nature they have been of in the first place.

Geometric equations for matroid varieties

Series
Algebra Seminar
Time
Tuesday, September 21, 2021 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ashley K. WheelerGeorgia Tech

Each point x in Gr(r,n) corresponds to an r×n matrix A_x which gives rise to a matroid M_x on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets {y∈Gr(r,n)|M_y=M_x} form a stratification of Gr(r,n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich. 

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