gradieNTAG

Abstract: In this talk, I will introduce some introductory theorems and facts about number fields and their rings of integers. We will then look at absolute values on number fields. Given a number field \(K\) with a complete discrete non-trivial non-archimedean absolute value and a finite Galois extension \(L/K\) with Galois group \(G\), we will define a filtration \(G \supset G_0 \supset G_1 \supset \dots \) of subgroups of \(G\). The group \(G_0\) is called the inertia group, \(G_1\) the ramification group and the groups \(G_i,\ i>1\) are called the higher ramification groups of \(L\) over \(K\). We will establish some of the basic properties of these groups and what they can tell us about the Galois group \(G\). Time permitting, we will also look at an interesting application of this theory.

Abstract: The complete linear system \(\mathbb{P}H^0(\mathcal{O}_{\mathbb{P}^n}(2))\cong \mathbb{P}^{n(n+3)/2}\) of quadrics in \(\mathbb{P}^n\) admits a natural action by \(\mathrm{PGL}_{n+1}\), and hence so does the Grassmannian \(\mathbb{G}(d, n(n+3)/2)\) parametrizing linear systems of quadrics of dimension \(d\) for \(0\leq d\leq n(n+3)/2\). We can study the orbits of this action—\(d\)-dimensional linear systems of quadrics in \(\mathbb{P}^n\) up to changes of coordinates—using techniques from intersection theory such as Chow rings and Chern classes. The study of the case \(d=1\)--of pencils of quadrics--goes back to the work of Segre from the 1850's. In 1906, Jordan showed that when \(n=2\) and \(d=1\), this action has exactly 8 orbits. We will examine this case in detail, studying the Hasse poset of orbit closures and computing their Chow ring classes in the Chow ring of the Grassmannian. Characters in this beautiful story will include the Cayley cubic surface, the catalecticant of a binary quartic form, and generically nonreduced components in Fano schemes. We will end by mentioning possible extensions of these results and techniques to higher \(n\) and \(d\), which is part of joint work with Mokhtar currently in progress. (Potential collaborators are very welcome!)

Abstract: We show that the partial sums of the long Plücker relations for pairs of weakly separated Plücker coordinates oscillate around 0 on the totally nonnegative part of the Grassmannian. Our result generalizes the classical oscillating inequalities by Gantmacher-Krein (1941) and recent results on totally nonnegative matrix inequalities by Fallat-Vishwakarma (2023). In fact we obtain a characterization of weak separability, by showing that no other pair of Plücker coordinates satisfies this property. Weakly separated sets were initially introduced by Leclerc and Zelevinsky and are closely connected with the cluster algebra of the Grassmannian. Moreover, our work connects several fundamental objects such as weak separability, Temperley-Lieb immanants, and Plücker relations, and provides a very general and natural class of additive determinantal inequalities on the totally nonnegative part of the Grassmannian. We shall begin the talk with the introduction of totally nonnegative matrices and see its important connection with planar networks. Then we shall review some classical results related to the topic to continue further with the more recent discovery. The talk will (mostly) be self-contained.

Abstract: A nice family of simplicial complexes is those that are vertex decomposable as, for instance, their Stanley-Reisner ideals are Cohen-Macaulay. Meanwhile, and seemingly completely unrelated, the theory of Gorenstein liaison can loosely be understood as studying ideals under an equivalence relation of being "linked" via a Gorenstein ideal. In this talk, we will discuss geometric vertex decomposition, a degeneration technique that gives both an ideal-theoretic generalization of vertex decomposability and a more computationally-friendly tool for computing Gorenstein links. This will shine a light on the connection between vertex decomposability and Gorenstein liaison. We will also talk about nice consequences for ideals that are geometrically vertex decomposable and give some examples of families of ideals that posses this property.

Abstract: Du Val singularities appear in the classification of algebraic surfaces and other areas of algebraic geometry. Wahl’s concept of local Euler characteristics of sheaves helps in describing the properties of these singularities. We consider the sheaf of symmetric differentials and compute one ingredient of the local Euler characteristic: the codimension of those symmetric differentials that extend to the resolution of the singularity in the space of those that are regular around it. Singularities of type \(A_n\) can be described with toric varieties. We use Klyachko's theory of toric vector bundles to express this codimension as a lattice point count in a rational polytope. For symmetric differentials of symmetric degree \(m\) at \(A_n\)-singularities we explicitly determine these polytopes and find expressions for the counts in terms of Ehrhart's quasi-polynomials. We also analyse the behaviour of this quantity as a function of \(n\).

Abstract: Curves are central objects of study in Algebraic and Arithmetic Geometry. An important invariant of a curve is its genus. Two fundamental results related to the genus of a curve are Riemann-Roch Theorem and Riemann-Hurwitz Formula. In this talk, we will look at their statements and apply them to classify all curves of genus less than or equal to two.

Abstract: Cluster algebra is a class of commutative algebra that encrypts combinatorial data by construction, founded by Sergey Fomin and Andrei Zelevinsky in 2002. In this talk, I will talk about the definition of cluster algebra of geometric type, show the construction from quivers and mutations, its correspondence with matrix mutations, and give some examples of cluster algebra as coordinate rings. I will also show some combinatorial meaning of the exchange relations. If time allows, I will briefly mention the Laurent phenomenon and the finite type classification.

Abstract: Intersection theory concerns about the invariance of intersection of subvarieties under some equivalence relation, for example, Bézout's theorem states that the number of intersection of two plane curves only depends on their degree, and another example is that there is 27 lines on any smooth cubic surface in \(\mathbb{P}^3\). In this talk, we will be discussing about rational equivalence, Chow ring, and we will use affine stratification to compute the Chow ring of some simple variety like \(\mathbb{P}^n\) and \(\mathbb{P}^m\times \mathbb{P}^n\), and then we will use it to give some results. If time allows, we may talk a bit about Chern classes.

Abstract: We will define the tropical semiring and see how tropical arithmetic turns polynomials into piecewise linear functions. We will later define tropical hypersurface and the tropicalization of an algebraic hypersurface. Armed with that knowledge we will present the connection between tropical and algebraic geometry, given by the fundamental theorem of tropical algebraic geometry, and finally will then define the tropicalization of an algebraic variety and give a sneak into what information about the variety we can recover from its tropicalization.

Abstract: In this talk, I will discuss the definition and motivation behind the mysterious Tate-Shafarevich group, relating to the number of rational points on algebraic curves. Should time allow, I will also share a plan of attack to prove various facts about this group, which does not allow straightforward proofs.

Abstract: In 1849 Arthur Caley and George Salmon proved that there are exactly 27 lines on each cubic smooth surface over an algebraically closed field. In this talk, I will present a modern construction of the arguments to prove this result. We will show that the lines lie on a smooth cubic surface is the vanishing locus of a section of the 3rd symmetric power of the tautological bundle on the Grassmannian of lines in \(\mathbb{P}^3\). Towards this goal, we will briefly introduce the Chern classes as degeneracy loci of the general sections, the Schubert cycles, and the Chow Ring.