Mathematics

Introduction to Orbifolds

Brannon Basilio (UIUC)

Abstract: We can generalize the notion of a manifold to include singularities; thus we can define a new object called orbifolds. In this talk, we will give an introduction to the notion of orbifolds, including examples, covering orbifold, Euler number of an orbifold, and the classification theorem of 2-orbifolds.

A probabilistic approach to quantizing Yang-Mills theory

Kesav Krishnan (UIUC)

Abstract: I will introduce the problem of rigorously quantizing Yang Mills Theory, and how probability theory can be used to that end. If time permits, I will talk about the discrete gauge-string duality as introduced by Sourav Chatterjee

Vanishing and Realizability

Shane Clark (University of Kentucky)

Abstract: The Reidemeister trace of an endomorphism of a CW complex is a lower bound for the number of fixed points (up to homotopy) of that endomorphism. For an endomorphism $f$, the Reidemeister trace of $f^n$ is a lower bound for the number of fixed points of $f^n$, however it can be a far from an optimal lower bound. One method of addressing this discrepancy constructs an equivariant map, the n^{th} Fuller trace $f$, which carries information about the periodic points of a map $f$. However, we must ask how much information is retained by this equivariant construction? In this talk we show that the n^{th} Fuller trace of $f$ is a complete invariant for describing a minimum set of periodic points for maps of tori.

Bounds on volumes of mapping tori

Heejoung Kim (UIUC)

Abstract: For a surface $S$ and a homeomorphism $f: S\to S$, the mapping torus of $S$ by $f$ is defined by $M_f=(S\times [0,1])/((x,0)\sim (f(x), 1))$. In particular, for a closed surface $S$ of genus at least 2 and a pseudo-Anosov element $f$ of the mapping class group of $S$, $M_f$ is a hyperbolic manifold. Brock provided bounds of the hyperbolic volume of $M_f$ from a hyperbolic structure on $M_f$ by using its Weil-Petersson metric. And then Agol gave a sharp upper bound for the volume in terms of the translation distance on the pants graph $P(S)$ which is associated with pants decomposition on $S$. In this talk, we will discuss mapping class groups and Agol's proof on the upper bound.

Unifying Galois Theories with Categorification

Robert (Joseph) Rennie (UIUC)

Abstract: Since its inception nearly two centuries ago, what we call "Galois Theory" (say in an undergraduate algebra course) has led to many analogous results, and thus attained the status of a sort of metatheorem. In Galois' case, this concept was applied to fields, yielding an equivalence between some lattice of field extensions and a lattice of subgroups of a corresponding "galois group" ... under certain conditions. Later on, the same concept was shown to be present in Topology, with extensions being replaced by their dual notion of covering spaces, and the galois group being replaced by the fundamental group... again, under certain conditions. Even later, Galois' results for fields were generalized to arbitrary rings, introducing new associated data along the way. In this talk, we explore the process of formally unifying all of these "Galois Theories" into one Galois Principle, with the aim of developing an intuition for identifying some of its infinite use-cases in the wilds of Math (e.g. Algebra, Topology, and Logic). Along the way, I aim to discuss explicitly and to motivate categorification to the working mathematician using the results of this talk as concrete examples.

\'Etale K-theory

Akhil Mathew (U Chicago)

Abstract: I will explain some general structural results about algebraic K-theory and its \'etale sheafification, in particular its approximation by Selmer K-theory. This is based on some recent advances in topological cyclic homology. Joint work with Dustin Clausen.

Arnold-Liouville Theorem

Jungsoo (Ben) Park (UIUC)

Abstract: This talk will be an introduction to fundamental concepts of symplectic geometry. Furthermore, we will delve into the proof of Arnold-Liouville theorem: https://en.wikipedia.org/wiki/Liouville–Arnold_theorem.

Approximating higher algebra by derived algebra

William Balderrama

Abstract: Obstruction theories and spectral sequences from the basic computational tools for accessing complex homotopical structure by means of pure algebra. Many of these are constructed via a careful examination of the relevant notion of ``free resolution''; the difficulties in their construction are then in maintaining sufficient control over these resolutions, as well as in the identification of the relevant obstruction groups. I will describe a general conceptual framework for producing these tools, based on a higher categorical variation on the notion of an algebraic theory, which is easily applicable to a wide variety of situations and provides a direct bridge between homotopical structure and algebraic structure.

$C_2$-equivariant homotopy groups of spheres

Mark Behrens (Notre Dame Math)

Abstract: I will explain how $RO(C_2)$-graded $C_2$-equivariant homotopy groups of spheres can be deduced from non-equivariant stable homotopy groups of stunted projective spaces, and the computation of Mahowald invariants.

Poincare duality for singular spaces

Gayana Jayasinghe (UIUC)

Abstract: Poincare duality of manifolds is a classical theorem which can be phrased in terms of the homology and cohomology groups of manifolds. However, when we look at singular spaces, this fails to hold for the usual homology and cohomology groups. In the setting of a certain class of singular spaces know as topological pseudomanifolds, which include orbifolds, algebraic varieties, moduli spaces and many other natural objects, one can extend these groups in order to recover some form of Poincare duality. I'll present how this was achieved by Goresky and MacPherson with their Intersection homology, and by Cheeger using L^2 cohomology and explain how they are related to each other, in similar spirit to the equivalence in the smooth setting. I'll only assume a basic knowledge of homology and cohomology.

CANCELLED

Christina Osborne (OSU Math)

To Be Announced

Ben Antieau (UIC )

Organizational Meeting

Brannon Basilio (UIUC)

Abstract: Organizational meeting to discuss who wants to give talks and when. The link will be sent to gradgeomtopo mailing list, but please email Brannon basilio3 at illinois for edu to join the list.

Introduction to Seifert surfaces and Knot Invariants

Brannon Basilio (UIUC)

Abstract: Seifert surfaces are used to study invariants of knots. In this talk, we introduce the notion of Seifert surfaces and how they give rise to the Alexander invariant and ultimately the Alexander polynomial through the universal abelian covering space.

How do field theories detect the torsion in topological modular forms?

Dan Berwick-Evans   [email] (UIUC)

Abstract: Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric, Euclidean) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from for the supersymmetric sigma model with target determines a cocycle representative of the generator of pi_3(TMF) = Z/24. (Contact Jeremiah Heller for Zoom info: jbheller@illinois.edu)

Floer theory and low-dimensional topology

Nachiketa Adhikari (UIUC)

Abstract: This talk will be an introduction to Floer theory and the key ideas behind it. If time permits, I will also sketch its connections to low-dimensional topology and, in particular, knots. Please email Brannon at basilio3 AT illinois DOT edu for the Zoom details.

Framed Cobordism and the Homotopy Groups of Spheres

Brian Shin

Abstract: The computation of the homotopy groups of spheres has dominated the minds of homotopy theorists since the 1930s. To this day, much is still not known about these groups. In this expository talk, I'd like to discuss one of the earliest approaches to understanding these groups: the geometry of framed cobordisms. Please contact basilio3 (at) illinois (dot) edu for Zoom details.

Stability theorems in geometry

Sambit Senapati

Abstract: We take a look at stability results arising in different geometric contexts, including one for group actions and one for foliations (Reeb-Thurston). We follow this up with a theorem concerning the stability of symplectic leaves in Poisson manifolds. Is there any common framework connecting all these geometric structures? Is there a meta-stability result that encompasses all of these? Come find out on Friday. Note- This will be an introductory talk; in particular I won't be assuming familiarity with Poisson geometry for the most part.

The Picard group of the stable module category for quaternion groups

Richard Wong (University of Texas at Austin)

Abstract: One problem of interest in modular representation theory of finite groups is in computing the group of endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by Carlson-Thévenaz using the theory of support varieties. However, I provide new, homotopical proofs of their results for the quaternion group of order 8, and for generalized quaternion groups, using the descent ideas and techniques of Mathew and Mathew-Stojanoska. Notably, these computations provide conceptual insight into the classical work of Carlson-Thévenaz. (Please email vesna@illinois.edu for Zoom link.)

Index theory through loop spaces

Gayana Jayasinghe

Abstract: I'll present some ideas in Atiyahs paper, "Circular symmetry and stationary phase approximation", where he goes through Wittens argument for a proof of the Index theorem by applying the Duistermaat-Heckman formula on loop spaces.

Decomposition of topological Azumaya algebras

Niny Arcila Maya (University of British Columbia)

Abstract: Let $\mathcal{A}$ be a topological Azumaya algebra of degree $mn$ over a CW complex $X$. We give conditions for the positive integers $m$ and $n$, and the space $X$ so that $\mathcal{A}$ can be decomposed as the tensor product of topological Azumaya algebras of degrees $m$ and $n$. Then we prove that if $m$<$n$ and the dimension of $X$ is higher than $2m+1$, $\mathcal{A}$ has no such decomposition. Please email vesna@illinois.edu for Zoom link.

R-motivic homotopy theory and the Mahowald invariant

Eva Belmont (Northwestern)

Abstract: Abstract: The Mahowald invariant is a highly nontrivial map (with indeterminacy) from the homotopy groups of spheres to itself, with deep connections to chromatic homotopy theory. In this talk I will discuss a variant of the Mahowald invariant that can be computed using knowledge of the R-motivic stable homotopy groups of spheres, and discuss its comparison to the classical Mahowald invariant. This is joint work with Dan Isaksen.

Please email Jeremiah Heller for Zoom info (jbheller@illinois.edu)

Minimal Surfaces and Its Applications

Xiaolong Hans Han

Abstract: In this talk, we will talk about some basic definitions of minimal surfaces, its stability and connection to Heegaard splitting. We also explain transversal vs. non-transversal intersections of minimal surfaces and how to do surgery. We then use results of existence and intersections of minimal surfaces to provide some applications in 3-manifolds. Depending on time, we will outline applications of minimal surfaces to prove non-sharpness of a recent inequality by Brock and Dunfield.

Vector bundles on projective spaces

Morgan Opie (Harvard Univfersity)

Abstract: Given the ubiquity of vector bundles, it is perhaps surprising that there are so many open questions about them -- even on projective spaces. In this talk, I will outline some results about vector bundles on projective spaces, including my ongoing work on complex rank 3 topological vector bundles on CP^5. In particular, I will describe a classification of such bundles which involves a surprising connection to topological modular forms; a concrete, rank-preserving additive structure which allows for the construction of new rank 3 bundles on CP^5 from simple ones; and future directions related to this project.

Please email vesna@illinois.edu for Zoom info.

The Geometry of Modular Forms

Robert Dicks

Abstract: Modular forms are usually introduced as holomorphic functions on the upper-half plane satisfying certain transformation laws and growth conditions. It then would seem obvious that their "natural home" is in the realm of those topics most fruitfully studied using tools from complex analysis with only occasional interactions with other fields. The aim of this talk is to highlight how one can view modular forms from a geometric viewpoint. In particular, the speaker will start with motivation from the classical theory of modular forms, take a detour into the basics of scheme theory (his notion of what "geometry" is), and then discuss how modular forms can be viewed as line bundles on certain schemes and whatever comes from that.