STAGS

Welcome to the Student Topology and Geometry Seminar (STAGS) at the University of Minnesota! If you would like to be added to the mailing list or give a talk, please contact Anh Hoang.

Time and Location

Time: 10:10-11:00 am Central Time

Place: Vincent 570

Zoom Link: https://umn.zoom.us/j/96152913121?pwd=bkYyYWo1dTQ2VWg4RENXUG9qdnJ2dz09

Meeting ID: 961 5291 3121

Password: 123stags

This week's talk

Date: 04/25/2023

Speaker: Mathieu De Langis

Title: Towards a Topological Proof of Wright’s Theorem

Abstract: Malle's conjecture concerns the asymptotic behavior of the number of degree n extensions of a number field with Galois group permutation-isomorphic to G. Using the function field analogy, a similar conjecture can be made for finite extensions of F_q(t), where q is a power of a prime. In the case where G is abelian, Wright has proven a version of Malle's conjecture in both contexts, using class field theory. This talk concerns an alternative approach towards proving Malle's conjecture in the function field case, by translating the problem into one of counting points on particular spaces.

Spring 2023

Abstract: Malle's conjecture concerns the asymptotic behavior of the number of degree n extensions of a number field with Galois group permutation-isomorphic to G. Using the function field analogy, a similar conjecture can be made for finite extensions of F_q(t), where q is a power of a prime. In the case where G is abelian, Wright has proven a version of Malle's conjecture in both contexts, using class field theory. This talk concerns an alternative approach towards proving Malle's conjecture in the function field case, by translating the problem into one of counting points on particular spaces.

Abstract: It’s a piece of mathematical folklore that every deformation problem is controlled by a differential graded Lie algebra (dgla) In the late 2000’s a series of authors including Manetti, Pridham, and Lurie went on a mission to make this statement rigorous. But what exactly is a differential graded Lie algebra, why are they objects we should care about, and how is it that they relate to deformation theory? In this talk I hope to persuade you that dgla’s are not scary obtuse objects, but in fact a natural generalization of things we are very familiar with. This talk is based off the following blog post with permission from and thanks to the author: https://francisrlb.wordpress.com/2023/02/25/how-does-a-lie-algebra-encode-a-space-part-1/#more-728.

Abstract: In this talk, I will give an introduction to spectral sequences and discuss some related categorifications. If time permits, I will talk about some power operations in spectral sequences.

Abstract: In 1977, Ravenel and Wilson published their seminal paper "The Hopf ring for complex cobordism", detailing the definition and basic properties of Hopf rings and demonstrating their usefulness in algebraic topology. In this talk, I will give an example of Hopf rings' topological applications, related to computing homology operations and determining orientability of vector bundles. I will then give a sampling of the complicated algebraic structure of the titular example, the Hopf ring for complex cobordism.

Abstract: Most of us have had the wonderful experience of teaching Math 1271 here at the U! We teach our students how to understand the world linearly through answering geometric questions using linear objects. In combinatorics, we too look at geometric questions and aim to understand them using "linear objects." The linear objects we will study in this talk are called Young diagrams: collections of boxes that we stack together in certain ways. These combinatorial objects have deep ties to both representation theory and algebraic geometry. We will use these Young diagrams and tableaux combinatorics to understand an algebraic geometric object called the Grassmannian. Time permitting, we will use this combinatorics to compute the cohomology ring and understand the multiplicative structure of the Grassmannian.

Abstract: The Cohen-Lenstra heuristics are a series of conjectures which developed from the observation that class groups of number fields are cyclic weirdly often. While significant numerical evidence has shown their robustness, to date little progress has been made towards an actual proof of their statistical predictions. Despite this, in the years since their original statement, Cohen-Lenstra-type phenomena have been observed in a growing number of additional contexts, one of these being higher K-theory. In this expository talk, we will explore the results of Jordan, Klagsburn, Poonen, Skinner and Zaytman showing significant evidence of such distributive phenomena in the mod-p K theory of rings of integers of algebraic number fields.

Abstract: In this talk, I’ll survey a few different examples of homotopical structures in algebraic topology and try to make the case for "thinking homotopically".

Abstract: We will survey a few examples of the homological stability phenomenon.

Past Seminars

Fall 2022

Abstract: One of the most fundamental problems in topology is to compute the homotopy groups of spheres; however, these groups have proved elusive. One standard modification to the problem is to restrict to the so-called stable case, which involves computing only the homotopy groups of a sufficiently high degree. While these computations are still very difficult in practice, there are many more computational tools available in the stable case than in general. In particular, the Adams spectral sequence is a very powerful tool for finding information about these groups. In this talk, I'll define the stable homotopy groups of spheres and give an overview of the use of the Adams spectral sequence. 

Abstract: Hyperplane arrangements are rich objects giving rise to fascinating combinatorics, topology, geometry, and algebra. In particular, a presentation of the fundamental group of the arrangement complement can be expressed in a combinatorial graph. We will describe the construction of this graph and use it to compute the fundamental group of some arrangements. In this introductory-level talk, we will assume current enrollment in MATH 8301: Manifolds and Topology.

Abstract: Operads are a mathematical object that capture and encode abstract notions of operations. The notion of an algebra over an operad allows us to abstract the notion of things such as commutative, associative, and Lie algebras. Moreover, it allows us to have well defined notions of such things in other categories, e.g. commutative multiplication in topological spaces. The study of operads gives us insight into the subtle relations behind all these different types of operations. In this talk I will be giving an example-driven introduction to some of the basic notions behind operads.

Abstract: Reductive groups--as well as their variants the compact Lie groups--are ubiquitous in modern mathematics. Regardless of whether you study algebraic geometry, topology, differential geometry, number theory, or even combinatorics, you will run into reductive groups eventually. However, their definition and construction--as seen in a Lie groups and Lie algebras course--often lacks motivation, becoming tedious to the listener. In this talk, I will give a few situations where reductive groups arise naturally, to give a taste of why the subject is interesting and necessary.

Abstract: We will give an overview of the central ideas in geometric analytic number theory, in particular the application of topological methods in arithmetic statistics. We will discuss the procedure of extrapolating problems in analytic number theory to problems and solutions in arithmetic topology, and some of the tools commonly used in the subject. If time permits, we will demonstrate the geometric Malle’s conjecture and recent results by Ellenberg, Tran, and Westerland.

Spring 2021 (online)

Abstract: I will give an introduction to supergeometry, discuss how it differs from ordinary algebraic geometry, and explain why constructing supermoduli spaces is a highly non-trivial problem.

Abstract: Bokstedt periodicity is one of the fundamental calculations in the theory of topological Hochschild homology. In this talk, I will provide the relevant background to make Bokstedt’s theorem intelligible and discuss two avenues of proof. Time permitting, we will discuss some interesting consequences.

Abstract: For a space X, the space of loops on X shares an intimate connection with the homotopy theory of X. Unfortunately, the specific structure of the loop space of X is often elusive and hard to visualize. For X = G a compact Lie group, understanding the space of loops on G is even more imperative to understanding phenomena such as Bott periodicity. In this talk, we will discuss the construction of the affine Grassmannian, a realization (up to homotopy) of the loop space of G equipped with a concrete cell structure. As time permits we will sketch a way of computing the cohomology of the loop space of G using the affine Grassmannian.

Abstract: We will give a brief motivation for the study of infinity categories from a topological standpoint, then proceed with the construction of quasi-categories and their homotopy categories as given by Lurie and Joyal.

Fall 2020 (online)

Abstract: The concept of a braided vector space V plays an important role in the study of Hopf algebras, notably in defining the Nichols algebra which is central to the "lifting method" program to classify pointed Hopf algebras. The tensor powers of V form a family of braid group representations. When equipped with the quantum shuffle product, they also form a non-commutative, non-cocommutative braided Hopf algebra called a quantum shuffle algebra. Ellenberg, Tran and Westerland identified the homology of the braid groups (Artin groups of type A) with these coefficients as the cohomology of this algebra, using the Fox-Neuwirth cellular stratification of configuration spaces of the complex plane. In this talk, we will extend their techniques to study configuration spaces of the punctured complex plane and prove a similar result for the homology of the Artin groups of type B.

Abstract: In this talk, I will introduce the notion of a (ring) spectrum, which is the central object of study in stable homotopy theory. The collection of commutative ring spectra forms a vast enlargement of the category of commutative rings, and includes exotic objects like complex cobordism and complex K-theory. The goal of the talk is to motivate these constructions for people outside of homotopy theory. If time permits, I will talk about the notion of "stability" and "brave new algebra."

Abstract: "A prism is a 'de-perfection' of a perfectoid ring, and prismatic cohomology specializes to most known integral p-adic cohomology theories."--Bhargav Bhatt. In these two talks, we will give a quick introduction to the central ideas of prisms and prismatic cohomology based on Bhargav Bhatt's lecture notes and his joint paper with Peter Scholze. We will introduce the definitions, main comparison theorems, and ideas of the proofs.

Abstract: We will give an overview of the central ideas in geometric analytic number theory, in particular arithmetic statistics. We will discuss the procedure of extrapolating problems in analytic number theory to problems and solutions in arithmetic topology, and some of the tools commonly used in the subject. If time permits, we will further demonstrate the geometric Malle’s conjecture and recent results by Ellenberg, Tran, and Westerland.

Abstract: From the dihedral group of order 2p, there is a simple construction which gives us the quadratic cover of a Nichols algebra. When p=3, this algebra is isomorphic to the third Fomin-Kirillov algebra which is a well-studied object, but for p>3, very little is known about this algebra or its cohomology. In this talk, we'll construct these algeras, motivate the study of their cohomology, and use their Koszul complexes to find patterns in their cohomology and to introduce a conjecture relating the dimensions of these algebras to one of the most famous sequences in all of mathematics.

Abstract: This week we will provide a more careful discussion of THH and TC, as promised in the previous lecture. In particular, we will define these invariants and sample some difficult computations. Time permitting, we will discuss the Dundas-Goodwillie-McCarthy theorem in greater detail.

Abstract: In this talk, we will introduce the stars of trace methods, state some theorems, and discuss a theoretical schematic for computing algebraic K-theory. If time allows, we will more carefully introduce topological Hochschild homology and topological cyclic homology.

Abstract: In this talk we will motivate and introduce the most basic definition of algebraic K-theory, the zeroth K-group of a ring. In service of doing so, we will introduce group completion of abelian monoids and give a number of results about projective modules over a ring. As time permits, we will extend this to the sister theory of K-theory of an exact category, hopefully finishing with applications of the resolution theorem.

Fall 2019

Abstract: In this talk, I will define algebraic de Rham cohomology for schemes over a field of characteristic zero. Briefly, for a finite type scheme Y that admits an embedding into a smooth scheme X over a field k, the algebraic de Rham cohomology of Y is determined by computing the hypercohomology of particular complexes of sheaves on X (or on the formal completion of Y in X). I will begin the talk by defining formal completion of a complex as well as hypercohomology and useful associated spectral sequences. Then, I will discuss some recent results of Switala and Bridgeland in which Y is the spectrum of a complete local ring and when Y is an affine variety respectively. 

Abstract: In this talk, I'll introduce the theory of commutative differential graded (dg) schemes defined by Ciocan-Fontanine and Kapranov.  Using classical theorems from geometry due to Hochschild, Kostant, Rosenberg, and Serre as motivation, we'll discuss the general ideas of derived geometry and dg schemes.  I'll then cover some basic definitions and constructions that are useful to the theory.  From there I hope to convince you that dg schemes are a useful framework to think about generalizations of HKR isomorphism type theorems using intersection theoretic constructions.

Abstract: I will describe some basics of super Riemman surfaces (SUSY curves), their moduli space, and their relevance to supersymmetric string theory. We describe a supersymmetric generalization of the argument of Kontsevich in 1987 that there exists a flat holomorphic connection on line bundles of the Mumford isomorphism. His result follows form the relationship of between the representations of the Virasoro algebra and the infinitesimal geometry of the moduli spaces of curves. In a compatible way, the moduli space of triples; a Riemann surface, a marked point, and a formal parameter; maps injectively into the Sato Grassmannian. We discuss possible applications of the super Sato Grassmannian to integration over the moduli space of SUSY curves.

Abstract: In the Student Commutative Algebra Seminar/Meetings (aka SCAM) we've been learning about Boij--Soederberg Theory, which describes the Betti diagrams of graded modules (over the standard graded polynomial ring) up to multiplication by a rational number. There is a duality between the cone of betti diagrams of graded modules and the cone of cohomology tables of vector bundles over projective space. In this talk, I'll start by describing the Boij--Soderberg conjectures (now theorems) and hopefully end with describing the supporting hyperplanes of the Boij--Soederberg cone. Understanding the equations of these hyperplanes is necessary in order to understand the duality with the cohomology tables of vector bundles.

Abstract: A key result in the celebrated paper of Bhatt, Morrow, and Scholze is their proof that the cotangent complex satisfies faithfully flat descent for commutative rings. However, it is possible to extend their argument to work for faithfully flat maps of simplicial commutative rings by exploiting some results in derived algebra. I will introduce language required to state this result, and time permitting, I will sketch the argument and state a conjecture which I believe has a positive answer. 

Abstract: The Zariski topology for a scheme can often fail to be well-behaved, for instance when computing cohomology of a constant sheaf. However, there are ways of endowing a scheme with a (Grothendieck) topology which resolve some of these problems. In this talk, I will define various topologies on the category Sch/S, discuss some properties of a strict subset of these, and sketch a proof that the cohomology of quasicoherent sheaves is independent of the choice of topology (as long as your choice is reasonable). 

Abstract: Actions of reductive groups G on varieties X induce corresponding actions of sheaves (and their generalizations) on G on those of X.  When X is a spherical variety for G, and we replace sheaves with perverse sheaves with a weight for the action of B, the relationship is especially intricate.  By passing to Grothendieck groups, we get that the category of perverse sheaves with a B weight on V is, in a sense, a module over the corresponding category on G. 

Abstracts: 

Mike Loper - In order to study varieties, we can study the sheaves that lie over them. When the varieties are toric, this is equivalent to studying modules over a polynomial ring. In this talk, I will review Fitting ideals, which are invariants of a module. I will discuss what Fitting ideals tell us about the corresponding sheaf when the variety  is projective space or more generally, a smooth projective toric variety. 

Greg Michel - It has been conjectured that the number of degree n number fields with bounded discriminant is asymptotically linear in the discriminant bound. Moreover some limited data about the second order asymptotics has been computed for small n. In this talk, we’ll relate recent work of Ellenberg, Tran, and Westerland on Hurwitz spaces and Quantum Shuffle Algebras to this topic, and we’ll explore the relation between Nichols Algebras and this asymptotic point count. Finally, we’ll explain some preliminary results describing Nichols Algebras when the underlying group is a dihedral group.

Spring 2019

Abstract: After introducing the theory of FI-modules in 2012, the collaborative unit consisting of Thomas Church, Jordan Ellenberg and Benson Farb applied their framework to asymptotically stable counting problems in a certain classes of FI-varieties over finite fields in their 2013 paper Representation stability in cohomology and asymptotics for families of varieties over finite fields. The paper serves as a proof-of-concept, unifying a number of previously-known combinatorial results. The key to their method is the Grothendieck-Lefschetz fixed-point theorem with twisted statistics, which relates the rational cohomology of an algebraic variety over the complex numbers with the trace of the Frobenius map applied to the étale cohomology with coefficients in an $\ell$-adic sheaf of that variety over a finite field. In this talk, we shall introduce the Grothendieck-Lefschetz formula and its associated machinery as well as FI-modules and representation stability, then use these ideas to give an exposition of some results of Church, Ellenberg and Farb as they relate to configuration spaces and the braid group.  

Abstract: Topological Complexity (TC) has its roots in the robot motion planning problem. If X is a topological space representing every position a robot can be in, TC(X) is the minimal number of motion planning rules required to instruct that robot to move from one position into another position. There is nothing stopping us, however, from determining the TC of spaces that do not pertain to robots. In this talk, I will discuss the TC of a space that represents images that can appear on an n×m-pixel computer screen. This is analogous to determining the minimal number of image processing rules required to continuously morph one digital image into another digital image. In particular, we will be looking at images that have been compressed via partitioning into regions outlined by Jordan curves. I will formally define TC and present some basics in finite topology. Next, I will prove that the TC of such a space is defined, followed by some specific sizes of computer screen for which I have computed the TC.   

Abstract: You're probably familiar with tangles: your tied shoelaces, your knotted headphones, etc. But how do we study these tangles using topology, where we are allowed to untie these things? We would have to fuse the ends of our laces or headphones together to preserve the knotted mess. In this talk we will discuss knots in a mathematical sense, and talk about how to tell when 2 knots are (k)not the same. In particular, we will define the Jones polynomial and Khovanov homology. 

Abstract: I'll be talking about singularities of varieties and one way of measuring them using D-modules while trying to avoid technicalities by sticking to examples of varieties that I can draw on a blackboard. Along the way, we'll see multiplier ideals, log canonical threshold, jumping numbers, and what (if anything) those things tell us about singularities. If there's more time, I'll also talk about Hodge ideals and how these objects can be computed. 

Abstract: In this talk, we will take a very informal and somewhat pictorial look at quotients in geometry.  I'll start by giving some topological intuition on why we need quotients to be a stack, what that means, and then try to give some approachable, but exotic and hopefully interesting examples.

Abstract: Free resolutions over the standard graded polynomial ring can be used to study subvarieties in projective space. Unfortunately, sometimes checking if a given chain complex is actually a free resolution can be difficult by computing homology. Luckily, there is a criterion for exactness that depends only on the rank of the maps and the depth of ideals coming from the maps in the complex. In 2017, Berkesch, Erman, and Smith introduced virtual resolutions to study subvarieties of more general smooth toric varieties. Virtual resolutions allow homology so long as it is killed by a power of a certain ideal. One wonders if there is a criterion for virtuality that is similar to the criterion for exactness. Come find out this week in STAGS! 

Abstract: Many important problems in mathematics concern classification. Moduli spaces are geometric solutions to geometric classification problems. In this talk, I will explain what this means, the problems encountered when trying to find such solutions and different ways to handle these problems. To illustrate, I'll discuss the construction of the moduli of stable curves, focusing on stable reduction. I'll end with mentioning various reasons people care about moduli, hopefully giving you an exposition of the different flavours these problems can take on.

Abstract:  I will define Hochschild homology for algebras, state necessary results, and prove the HKR isomorphism theorem. In addition, I will explicate geometric interpretations of HKR related to derived algebraic geometry. Time permitting, I will mention some vistas toward algebraic K-theory and topological Hochschild homology.  

Abstract:  I will talk about how theorems in arithmetic/combinatorics (more specifically, about counting solutions to polynomial equations over finite fields) can turn into theorems or conjectures in topology.

Fall 2018

Abstract: String scattering amplitudes are fundamental quantities of interest in string theory and are expressed as certain integrals over the moduli spaces of super Riemann surfaces. In this talk I will discuss the moduli spaces involved and introduce the super Mumford isomorphism, a tool which gives rise to a measure whose integral computes the desired scattering amplitudes. 

Abstract: Coherent sheaves over projective space locally look like finitely generated graded modules over the polynomial ring of a field. Minimal free resolutions can be used to study these modules. Unfortunately, minimal free resolutions do not reflect the geometry as well when we move to a product of projective spaces. Virtual resolutions seem to be the answer to  this problem. In this talk, I will explain how virtual resolutions are closer tied through the geometry and hopefully convince you that these are the objects we should be working with in multiprojective space.

I will explore these concepts through case studies of a couple of examples so very little background is necessary.

Abstract: The goal of this talk is to explore the Maximal Rank Conjecture, which answers a natural question about how two basic ways of describing an algebraic curve in projective space, namely as a map $C\to\mathbb{P}^r$ (Parametric) or as a homogeneous ideal in a graded ring (Cartesian), relate to each other. We begin by talking about the Brill-Noether Theorem, which describes the space of maps from a general curve to projective space and was proven by Griffiths and Harris, among others, using the technique of degeneration to a reducible curve. Time permitting, I'll discuss how this technique fits in a recent proof of the Maximal Rank Conjecture by Eric Larson.

Abstract:  Equivariant intersection theory came about in the last 50 years as a way to study the cohomological structure of invariant subvarieties of a group action on a variety.  This talk focus on the generalization of the Riemann-Roch theorem, one of the well known results of classical intersection theory, to the equivariant case.  Along the way, I'll introduce the necessary background and take light detours into some of the more interesting related topics, such as quotient stacks and equivariant vector bundles. 

Abstract: Operads are tools used to describe operations on a mathematical object, as well as the compatibilities between them. They were first formally defined in the early 1970s by J. Peter May, who was studying iterated loop spaces. These spaces have a natural operation (or multiplication) on them given by loop composition, which fails to be strictly associative/commutative; the degree to which this happens can be described using the language of operads. In this talk, I will describe what an operad is, with the specific example of iterated loop spaces in mind. I will also discuss a major result in this area, also due to May — the Recognition Principle. 

Abstract: Let FI be the category of finite sets and injections. This category naturally "acts" on the configuration space(s) of any topological space. When the space is nice enough, the action has remarkable finiteness properties that severely restricts the cohomology groups as the number of points in the configuration increases. After introducing this "stabilization phenomenon", I will discuss two settings in which the action of FI extends to a larger category. Equipped with the extended structure, the stable ranges become (almost) sharp.

A basic familiarity with the words "cohomology", "colimit", "homotopy" and "symmetric group" is essentially all the background needed. 

Abstract: The main part of this talk will be an overview of the space of stable maps and Gromov-Witten invariants. Gromov-Witten invariants are birational and symplectic invariants that have applications in enumerative geometry (counting algebraic objects with certain fixed conditions) as well as String Theory, specifically Mirror Symmetry. They also tells us quite a bit about the cohomological structure of moduli of curves. When the target is a GIT quotient, these GW invariants appear as an extremal case of a family of invariants obtained by imposing a different kind of stability conditions, parametrized by the non-negative real line. Here, I am referring to quasimap theory, which I will talk about briefly towards the end of the talk.

My goal is to keep it accessible enough for people currently taking the AG class to get a sense of what the subject is like so I will not delve into details, but I am happy to give references when I can and chat afterwards.

Abstract: I will provide some motivation for why the cotangent complex is a useful gadget, sketch its construction using model categories and simplicial methods, enumerate some of its properties, and time allowing, state an open problem.  

Abstract: In this talk, I will give an introduction to the field of supergeometry, focusing on the elementary definitions and some basic examples. Following the introduction, I will discuss SUSY curves and their supermoduli spaces.  

Abstract:  In this talk, I'll introduce an interesting family of algebras called the Fomin-Kirillov algebras. We'll explore some basic properties and talk about how better understanding these algebras could lead to some fascinating results in Algebraic Topology.

Useful links