FALL ORIENTATON LUNCH

Speaker: MATH'S GRADUATE FALL ORIENTATION LUNCH (Western)

"GRAD CLUB"

Time: 12:00 - 13:00

Room: GRAD CLUB

Ph.D. Candidacy Exam Lecture

Speaker: Alex Zwart (Western)

"Variation of Geometric Invariant Theory and Derived Categories for Toric Varieties."

Time: 13:30 - 14:30

Room: MC 108

A smooth projective toric variety $X$ corresponding to the fan $\Sigma$ with rays $\Sigma(1)$ induces an action of the Picard torus $G$ on the affine space $\mathbb{A}^{\Sigma(1)}$. A character $\chi$ of $G$, i.e. an element of $\mathrm{Pic}(X)$, determines a $G-$linearized line bundle $L_\chi$ which can be used to form a GIT quotient. The characters that give non-trivial GIT quotients are precisely the ones that live in the effective cone in $\mathrm{Pic}(X)_{\mathbb{R}}$. We can set an equivalence relation on this cone by specifying characters to be equivalent if they give the same GIT quotient. This decomposes the effective cone into a fan where the relative interiors of the cones are the equivalence classes, we call this fan the GIT fan and denote it by $\Sigma_{GIT}(X)$ or $\Sigma_{GIT}$ when $X$ is understood. A chamber is the relative interior of a maximal cone in this fan and a wall is the relative interior of a codimension 1 cone. Two chambers are adjacent if they share a wall and we study how the derived category is affected by ``crossing a wall". Specifically, crossing a wall induces a semi-orthogonal decomposition for the toric variety corresponding to one of the chambers. We study this semi-orthogonal decomposition for the generalized del Pezzo varieties following [BDM20]. In [BDM20] they use this semi-orthogonal decomposition to show they have a full exceptional collection of line bundles for this class of varieties. At the end of this paper, we present the beginning results towards a generalization of this work to the case of the Klyachko varieties of [VK85]. This work is motivated by [Pal25] where the author produces a strong $S_m\times S_n$-stable, but not full, exceptional collection of line bundles for the Klyachko varieties in the case $n$ is congruent to 1 modulo $m$.

Ph.D. Candidacy Exam Lecture

Speaker: Mieke Fink (Western)

"The polytope of all matroids"

Time: 15:30 - 16:30

Room: MC 108

In this talk, I will introduce the valuative group of matroids, which allows to represent any matroid on the groundset $n$ of rank $r$ as a point in $\mathbb R ^{n \choose r}$. Many well-known matroid invariants behave as linear functions on the underlying vectorspace. Ferroni and A. Fink initiated the study of the convex hull of all such points corresponding to matroids, which they call the 'polytope of all matroids'. One reason to study the polytope is to understand extremal behavior of matroid invariants. I will explain the construction and present first results on their polyhedral structure, assuming no prior knowledge on matroids.

Ph.D. Candidacy Exam Lecture

Speaker: Roger (Qinghan) Yang (Western)

"Galois Category: Galois theory of Topology and Normal Curves"

Time: 11:30 - 12:30

Room: MC 108

The purpose of this report is to explain why several categories which at first do not look like classical Galois theory nevertheless carry the same kind of Galois structure. We first study finite sets and finite continuous G-sets, focusing on the categorical properties which they have in common. For the field theory, the extension theorem motivates the fibre functor A ↦−→ Homk(A, ks), and leads to the anti-equivalence between finite Etale k-algebras and finite continuous Gk-sets. These examples then motivate the formal definition of a Galois category. We also discuss finite topological coverings as a Galois category and the topological Galois correspondence. The main example from algebraic geometric is obtained from normal curves. After reviewing the commutative algebra needed for normalization, ramification, residue-field extensions, and etaleness, we fix a nonempty open subset U of an integral proper normal curve X with function field K. We construct the compositum KU of the finite separable extensions of K whose corresponding normalizations are etale over U, and define π1(U) := Gal(KU/K). Using the anti-equivalence between normal curves and their function fields, together with the field-theoretic Galois correspondence, we show that the category of finite covers of X which are etale over U is equivalent to the category of finite continuous π1(U)-sets. This gives the curve category its Galois-category structure. Finally, we study what a finite quotient q : π1(U) ↠ G means geometrically. Its kernel determines a finite Galois extension of K, a corresponding cover of X, and a finite-level Galois subcategory equivalent to the category of finite G-sets. Finally, it concludes with monodromy and the Kummer cover x ↦→ xn, which gives a concrete realization of the cyclic quotient ˆ︁ Z/nˆ︁ Z ∼= Z/nZ.

Colloquium

Speaker: Sara Najem (American University of Beirut)

"TBA"

Time: 15:30 - 16:30

Room: MC 108

Transformation Groups Seminar

Speaker: Suyoung Choi (Ajou University)

"TBA"

Time: 14:30 - 15:30

Room: MC 108