Algebra Seminar

Vasily Dolgushev, Temple University

In 1990, V. Drinfeld introduced the Grothendieck-Teichmueller group $GT$. This group receives a homomorphism from the absolute Galois group $G_Q$ of rational numbers, and this homomorphism is injective due to Belyi's theorem. In his 1990 ICM talk, Y. Ihara posed a very hard question about the surjectivity of this homomorphism from $G_Q$ to $GT$. In my talk, I will introduce the groupoid of $GT$-shadows and show how this groupoid is related to the group $GT$. I will formulate a version of Ihara's question for $GT$-shadows and describe a family of objects of the groupoid for which this question has a positive answer. My talk is loosely based on the joint paper with I. Bortnovskyi, B. Holikov and V. Pashkovskyi.

Yelena Mandelshtam, IAS Princeton

The Kadomtsev-Petviashvili (KP) equation is a partial differential equation whose study yields fascinating connections between integrable systems, algebraic geometry, and combinatorics. In this talk I will describe some of the various approaches to connecting KP solutions to algebraic objects such as algebraic curves (due to Krichever) and the positive Grassmannian (due to Sato and later Kodama-Williams). I will then discuss recent and ongoing work to build bridges between these approaches.

Group and Galois cohomology is an important tool that gets used in many areas of mathematics.  We will have a learning seminar on this topic this semester.  During this meeting we will explain a bit of the motivation for choosing this topic and assign topics for future talks.  If you are interested in giving a talk this semester, please attend this meeting!

Vasily Dolgushev, Temple University

GT-shadows are morphisms of a groupoid GTSh who objects are finite index $B_3$-invariant subgroups of the free group on two generators. They may be thought of as approximations of elements of the Grothendieck-Teichmueller group GT. After a brief reminder of the groupoid GTSh, I will introduce the dihedral poset as a concrete subposet of the poset of objects of GTSh. For every element of the dihedral poset, we will describe its connected component in the groupoid GTSh. We will use connected components of certain objects of the dihedral poset to produce the first examples of finite non-abelian quotients of GT. My talk is based on the joint paper with Ivan Bortnovskyi, Borys Holikov and Vadym Pashkovskyi.

Vasily A. Dolgushev, Temple University

Grothendieck's child's drawings (a.k.a. dessins d'enfant) connected topology to number theory in a fascinating way. In my first talk, I will present several equivalent definitions of a child's drawing. You will see permutation pairs, bipartite ribbon graphs and finite index subgroups of the free group on two generators. If time permits, I will start talking about Belyi pairs. The absolute Galois group of rational numbers acts on child's drawings and Belyi pairs allow us to introduce this action.   

Vasily Dolgushev, Temple University

I will introduce the action of the groupoid of GT-shadows on child's drawings and discuss how this is related to the action of the Grothendieck-Teichmueller group and the action of the absolute Galois group (of rational numbers) on child's drawings. My talk is loosely based on this paper.  If time permits, I will mention an open question motivated by the paper by J. Ellenberg.

Violet Nguyen, Temple University

This talk is the first in a series of seminar talks discussing Galois Cohomology. We will introduce profinite spaces, profinite groups, and discrete modules over profinite groups, which will be necessary in order to define their cohomology groups. We will end with the statement of Pontryagin duality, which naturally associates to each compact abelian group (and hence each profinite group) a discrete abelian group.

Holly Miller, Temple University

This talk is the second in a series of seminar talks discussing Galois Cohomology. The cohomology groups of a profinite group (with coefficients in a module over said group) will be introduced twice, first via inhomogeneous cochains and then through homogeneous cochains. The first method will be used to give interpretations for the cohomology groups of low dimension. The second will be used to show that these groups are limits of the cohomology groups of finite quotients of the profinite group in question. Lastly, the Tate cohomology -- which extends the usual cohomology -- will be discussed.

Sean O'Donnell, Temple University

This talk will introduce the long exact cohomology sequence and some basic theory surrounding it. We will begin by defining the cochain and cohomology functors, and then introduce the long exact sequence and discuss its naturality. The structure of this exact sequence will motivate us to investigate acyclic, cohomologically trivial, and induced modules, which we will then use to introduce the technique of dimension shifting. During this, we will also discuss the application of these structures to the cohomology of Galois groups acting on the additive groups of their relevant fields.

Chathumini Kondasinghe, Temple University

This talk aims to explore how long exact sequences of group cohomology provide insights into field theory and elliptic curves. We begin with a proof of Hilbert's theorem 90, and then use the long exact sequence of Galois cohomology to establish Kummer theory for fields. In the second half, we will talk about the idea that goes into the proof of the weak Mordell-Weil theorem, a finiteness result for elliptic curves.

Aniruddha Sudarshan, Temple University

We go over results of Section 1.4 from the book "Cohomology of Number Fields" by Neukirch, Schmidt and Wingberg. We start with the basic definitions of cup-product and their functorial properties. If we have time, we end with a result on cohomology of cyclic groups.

Dianbin Bao, Temple University

We will go over the results in Section 1.5 and 1.6 in "Cohomology of Number Fields". We will focus on properties of the cohomology groups $H^n(G,A)$ if we change the group $G$, in particular the properties of restriction and corestriction maps and the $p$-primary part of $\widehat{H}^n(G, A)$.

Xiaoyu (Coco) Huang, Temple University

We will present several duality theorems concerning cohomology groups in the context of finite and profinite groups. As an application, we will discuss the reciprocity isomorphism and the abstract formulation of class field theory. The reference is Section 3.1 in "Cohomology of Number Fields" by Neukirch, Schmidt and Wingberg. Some results from Section 1.8 will be reviewed.

Organizational Meeting

This is the organizational meeting of the Algebra Seminar for the fall semester. Please come if you are interested.

Vasily Dolgushev, Temple University

Let K be an algebraically closed field and k be a subfield of K. I will give a brief review of varieties and their morphisms over K. I will talk about k-structures on varieties and introduce morphisms of varieties defined over the smaller field k. If time permits, I will also talk about the functor of points. I will follow Borel's presentation of this material from Chapter "AG" in his book "Linear Algebraic Groups".  

Sean O'Donnell, Temple University

Continuing from last week's talk, we will start by covering some of the necessary pre-requisite algebraic geometry, primarily focusing on k-structures when the field k is not algebraically closed. We will define algebraic groups and their morphisms, sketch proofs for some basic results and review a couple of particularly important examples. We will then discuss actions of algebraic groups on varieties and go over a result analogous to Cayley's Theorem in group theory.

Violet Nguyen, Temple University

In this talk, we continue with Section 2 of Armand Borel's "Linear Algebraic Groups." We start by defining the group closure of a subset of a $k$-group $G$ and listing many properties of this operation. Using a result of R. Baer, we then extend the notion of solvability and nilpotence to $k$-groups.

Holly Miller, Temple University

The focus of this talk is Section 1.3 of Borel's book "Linear Algebraic Groups". Restricted Lie algebras are defined, and we give a few examples thereof. The tangent space of an algebraic group is constructed, and we show that it receives the structure of a restricted Lie algebra. After providing a few examples of the Lie algebra of an algebraic group, we conclude with a discussion on the adjoint representation.

Vasily Dolgushev, Temple University

This is a continuation of the series of talks on linear algebraic groups. I will recall the Jordan decomposition for endomorphisms of a finite dimensional vector space as well as its multiplicative version for the general linear group. Then I will extend the Jordan decomposition and its related notions to an arbitrary affine algebraic group. I will talk about consequences of the theorem about the Jordan decomposition for affine algebraic groups. If times permits, I will also talk about semi-invariants.

Vasily Dolgushev, Temple University

This is a continuation of the series of meetings about linear algebraic groups. After tying up loose ends related to the Jordan decomposition, I will introduce characters, weights and semi-invariants. We will show that, for every normal subgroup $N$ of an affine algebraic group $G$, there exists a morphism $\alpha$ of algebraic groups from $G$ to $GL(V)$ such that $N$ is the kernel of $\alpha$ and the Lie algebra of $N$ is the kernel of the differential of $\alpha$.  

Aditya Sarma Phukon, Temple University

This is a continuation of the series of meetings on algebraic groups. For an algebraic subgroup $H$ of a larger group $G$, we will define what a reasonable quotient $G/H$ should be. We shall justify when these quotients, called Homogeneous spaces, have a variety and, even better, an affine $k$-group structure. Towards this, we shall use representation machinery introduced before by Vasily and apply a neat trick of placing $G$ in a projective space and looking at particular orbits. We will end on a few remarks on cross-sections and, if time permits, on a closely related type of quotient called categorical quotient. 

Pedro Lemos, Temple University

This is a continuation of the series of meetings on algebraic groups. This week I will discuss tori and diagonalizable algebraic groups. I will cover some of their properties, provide examples and describe a structure theorem for diagonalizable groups. I also hope to have some time left to briefly go over the topic of root systems.

Ben Neifeld, Temple University

This is a continuation of the series of meetings on algebraic groups. We will study the conjugation action of closed subgroups $H$ of an affine algebraic group $G$ on $G$, as well as the associated action on the Lie algebra of $G$. For elements of $G$ that are semi-simple and normalize $H$, we will show that their conjugacy classes are closed. We will then discuss a connectedness result for the centralizers of unipotent subgroups and applications to diagonalizable groups.

Thomas Goller, Temple University

This is a continuation of the series of meetings on algebraic groups. We'll begin by looking at some examples of projective varieties to set the stage for a discussion of complete varieties, a sketch of the classification of one-dimensional connected affine groups, and a proof of a fixed point theorem for connected solvable groups acting on complete varieties. Finally, we'll discuss a structure theorem for connected solvable groups, both by looking at a key example of upper-triangular matrices and by working through parts of the proof.

Stephen Liu, Temple University

This is a continuation of the sequence of meetings on algebraic groups. Generalizing the example of upper triangular matrices in $GL_n$, we will introduce Borel subgroups. We will prove that the quotient of a connected affine group by a Borel subgroup is a projective variety. We will also define parabolic subgroups and Cartan subgroups, if time permits, and conclude with some results on the structure of parabolic subgroups.

Xiaoyu Huang, Temple University

This is a continuation of the series of meetings about linear algebraic groups. We will discuss the definition and basic properties of Cartan subgroups. Along the way, we will further explore the importance of Borel subgroups in the proofs of these results. After quickly introducing the definition of Weyl groups, we will define reductive groups and explore the first layer of its properties. The main references are Borel’s book "Linear Algebraic Groups" and Florian Herzig’s lecture notes of the same title.