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 $m$.

Department Meeting

Speaker: (Western)

"first Fall meeting"

Time: 15:30 - 16:30

Room: MC 107

agenda by email!

Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

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)

"Higher-Order Networks: Topology and Geometry"

Time: 15:30 - 16:30

Room: MC 108

Networks describe dyadic interactions and discard higher-order ones in complex systems. I will trace a path from weighted networks and their statistical-physics measures and show their limitations, leading to simplicial complexes built from triangles, tetrahedra, and higher simplices that pairwise edges cannot encode. Topological invariants such as the Betti numbers, the Euler characteristic, torsion, and the Ihara–Zeta function are introduced. Connes' spectral triplet then supplies geometric measures, a spectral distance, spectral dimension, and a discrete curvature, that complement these topological invariants. I will carry this out by illustrating the framework on a dataset of musical compositions, where chords (notes played simultaneously) form natural higher-order structures.

Transformation Groups Seminar

Speaker: Suyoung Choi (Ajou University)

"Cohomological rigidity of smooth compact toric varieties"

Time: 14:30 - 15:30

Room: MC 108

Toric varieties are a useful class of spaces where topology, geometry, and combinatorics are closely connected. In this talk, I will discuss how we can classify smooth toric varieties from a topological point of view. I will first explain some basic ideas of topological classification and the role of cohomology. I will then introduce the cohomological rigidity problem and discuss some results for (generalized) Bott manifolds. Finally, I will briefly talk about recent results on the classification of smooth Fano toric varieties.

Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

Geometry and Combinatorics

Speaker: Colin Crowley (Western)

"Cohomological aspects of power ideals"

Time: 15:30 - 16:30

Room: MC 108

Perhaps the most important invariant of an arrangement of hyperplanes is the associated matroid. Many cohomological invariants of the arrangement turn out to depend only on the underlying matroid, and there has been much recent activity in the field of combinatorial Hodge theory which generalizes deep theorems from arrangements to abstract matroids. In this talk, I'll discuss a bridge between these ideas, and an area of algebraic combinatorics called Zonotopal algebra. Via this connection, we can use tools from combinatorial Hodge theory to recover known results about zonotopal algebras as well as prove new ones, including a conjecture of Rhoades, Tewari, and Wilson. We also explore new classes of zonotopal algebras inspired by vanishing results in combinatorial Hodge theory, and vice versa.

Geometry and Physics Seminar

Speaker: Behrad Taghavi (Western)

"Liouville Theory Correlators and Weil–Petersson Geometry of the Moduli Spaces of Hyperbolic Cone Surfaces"

Time: 12:30 - 13:30

Room: MC 108

Two-dimensional conformal field theory has proven a powerful tool in both physics and mathematics, and one of its most striking applications is to the geometry of surfaces. This is clearest in Liouville theory, which can be regarded as a quantum theory of two-dimensional geometry. Semiclassically, a correlator of heavy vertex operators in this theory is governed by the on-shell Liouville action of a hyperbolic metric with conical singularities at the insertion points. In this talk, I will study this on-shell Liouville action as a natural object on a deformation space of hyperbolic conical metrics, and discuss how it encodes the geometry of that space. For compact surfaces of genus g>1 and punctured spheres, this link goes back to Zograf and Takhtajan, who showed that the on-shell Liouville action generates the accessory parameters of the uniformization problem and is a Kähler potential for the Weil–Petersson metric; Park, Takhtajan and Teo later extended this to higher-genus punctured surfaces. I will focus on the extension to case of orbifold Riemann surfaces with genus g>1: after cancelling an anomaly introduced by regularization at the conical points, the modified action becomes a global Kähler potential for the combination of Weil–Petersson and Takhtajan–Zograf metrics appearing in the local index theorem for orbifold Riemann surfaces. Time permitting, I will discuss ongoing work which aims to use CFT methods to obtain an asymptotic expansion of the Weil–Petersson Kähler form for small cone angles, recovering the structure of Mirzakhani's formula for the Weil–Petersson class (with boundary components of purely imaginary length).

Based on joint work with A. Naseh and K. Allameh, Phys. Rev. D 110 (2024) 046018, and work in progress with A. Naseh.

other

Speaker: AI in Mathematics discussion (Western)

"AI in Mathematics discussion"

Time: 15:30 - 16:30

Room: MC 108

The role of LLMs in various aspects of the profession is increasingly of interest. All are welcome to this discussion group.

Graduate Student Research Seminar

Speaker: (Western)

"TBA"

Time: 13:00 - 14:00

Room: MC 204

Geometry and Physics Seminar

Speaker: Masoud Khalkhali (Western)

"From Harmonic Analysis on Trees to p-adic AdS/CFT, Part 1"

Time: 12:30 - 13:30

Room: MC 108

This lecture series approaches \(p\)-adic AdS/CFT through the geometry and harmonic analysis of trees. Our guiding theme is the analogy between hyperbolic spaces, Riemann surfaces, and their non-Archimedean counterparts. Beginning with homogeneous trees and Cartier’s approach to their harmonic analysis, we will study the Bruhat–Tits tree \(\mathcal T_K\) associated with \(\mathrm{PGL}_2(K)\), where \(K\) is a non-Archimedean local field. Its boundary at infinity is canonically identified with \(\mathbb P^1(K)\), and the action on the tree extends the Möbius action on this boundary. This provides a geometric starting point for comparing the Archimedean and non-Archimedean settings. The first part develops the basic dictionary: geodesics and horocycles, adjacency and Laplace operators, spherical functions, and boundary integral representations. Particular emphasis will be placed on the relationship between spectral theory in the interior and function theory on the boundary. We will also examine Schottky uniformization in the classical and non-Archimedean settings, relating Riemann surfaces and hyperbolic handlebodies to Mumford curves and the quotient graphs that encode their skeletal geometry. These comparisons will illustrate both the power and the limitations of the tree analogy.