The junior geometry seminar meets every Tuesday at 3:45-5pm central. What is the junior geometry seminar?
The organizational meeting is on January 18th at 3:45 pm central.
Zoom links and email reminders will be sent to the mailing list. Click here to be added to the mailing list.
List of past organizers (aka JG ORGANIZER HALL OF FAME )
The participants of the jr. geometry seminar are encouraged to raise awareness about a social justice issue in mathematics that they personally care about.
Click here for more information about the jr. geometry SJA format.
The titles and abstracts of the talks are also available here. at the official math department seminar calendar.
Title: Line Bundles in Representation Theory: The Borel-Weil-Bott Theorem
Abstract: After reviewing the notion of a line bundle, we will give a general algorithm for constructing line bundles on the flag varieties/manifolds, which are certain varieties/manifolds associated reductive groups like GL_n or SL_n. We'll discuss how one can use this algorithm to construct all finite dimensional representations of a given reductive group, known as the Borel-Weil theorem, and discuss its generalization to "higher cohomology groups" in characteristic 0 known as the Borel-Weil-Bott Theorem. Time permitting, we'll discuss Kempf vanishing theorem, a partial generalization of the characteristic p version of the Borel-Weil-Bott theorem.
Title: Lagrangian Cobordism
Abstract: Lagrangian Cobordism is a simple equivalence relation on the lagrangian submanifolds in a symplectic manifold, introduced by Vladimir Arnold in 1980. We will start by introducing some fundamental notions of symplectic geometry, hamiltonian flows and lagrangian submanifolds. Then, we will define lagrangian cobordism and give some examples. Finally, to add some fanciness, we will talk about Nadler and Tanaka's construction of an infinity category of lagrangian cobordisms. In their special setting, they even construct a stable infinity category, so I will attempt to say a few words about what that means.
Title: Geometric Langlands and its daughters
Abstract: In this talk, we concern ourselves with the Geometric Langlands conjecture and its variants. More specifically, we motivate the Geometric Langlands conjecture in its current form by recalling the history leading up to it. We will also consider some new exciting variants of the conjecture and how these are related.
Title: An introduction to the Arnold Conjecture, or how I learned Floer Homology
Abstract: In this talk I will talk about the Arnold conjecture, which is concerned with the minimum number of fixed points that an Hamiltonian symplectomorphism on a (closed) manifold must have. This question was asked in 1972 by V. Arnold in his book - Arnold s problem - and led to the birth of Floer Homology, introduced by A. Floer in the late 80s. Interestingly enough, the history of the Arnold conjecture has some ties with our Math department: B. Siebert and other people extended the proof of the conjecture to any compact symplectic manifold, while A. Blumberg (together with M. Abouzaid) increased the lowerbound on the number of fixed points. Lastly, let us not forget R. Pedrotti who keeps on forgetting the difference between weak and strong Arnold conjecture. But that’s a story for another time maybe.
Title: Blowing up of a cone along its vertex
Abstract: Blowing up is a classical technique in algebraic geometry. It consists of modifying an ambient variety by replacing a chosen subvariety with all the possible "directions of approach to it". The goal of this talk is to introduce (a restricted case of) blow-ups, and study one example in detail: that of an affine cone over a projective variety along its vertex point.
We will work this example out in two ways:
first using mostly classical geometric ideas and language,
then secondly, using the modern technology of algebraic geometry.
In this way, we hope to showcase how the high-flying sophisticated language of contemporary algebraic geometry, though often streamlining and/or obscuring them, nonetheless still encodes fundamentally geometric ideas.
Title: Convolution Algebras and where to find them
Abstract: First we will introduce the formalism of convolution through the example of functions on finite sets. In particular, these are vectors, and convolution will turn out to encode matrices and matrix multiplication. Then we will discuss how to generalize this example, replacing finite sets with other types of geometric objects and functions with other functors valued in vector spaces or categories. One key example will be the Springer resolution, which allows us to construct convolution algebras by taking Homology, K-theory, or Hochschild Homology. Time permitting, we will discuss some examples and properties of categories of sheaves with a monoidal structure given by convolution, such as the Hecke category of constructible sheaves on the flag variety of a lie algebra. No background will be assumed.
Title: Equivariance to Families and the Metaplectic/Spin Anomaly
Abstract: We will start by reviewing classifying spaces and associated bundles. Then we will ask the question: given an algebra A and a space S, when is there a natural bundle over S with fibers modeled on A? Whenever a group G acts on A and we have a principal G-bundle on S, the associated bundle construction provides us with exactly such a bundle. After doing a concrete example of this, we will ask a harder question: given a module M over A, can we find a natural bundle over S with fibers modeled on M? When M doesn't have an action of G this question is subtle, and merely having a principal G-bundle on S will not always be enough structure. In our concrete example, the additional structure needed will be a spin/metaplectic structure on S.
SPRING BREAK
Title: Bundles, bundles, and b -- the Yang-Mills equation
Abstract: I will discuss the Maxwell equations (of electromagnetism) and outline the construction of the Yang-Mills functional, in the hopes of motivating the concepts of bundles, connections and gauge invariance (which I will review). After this, I can comment on some neat applications of this machinery to other areas of math (such as low dimensional topology), or go into Sobolev spaces of sections of fiber bundles and some geometric analysis, depending on people's preferences.
Title: Factorization homology as a homology theory for manifolds
Abstract: We will motivate the theory of factorization homology developed by Ayala and Francis as a source of algebraic invariants fine-tuned to the geometry of manifolds (as distinguished from general topological spaces). We will work out factorization homology of an algebra over the circle as a special case in which the de Rham differential miraculously appears. Time permitting, we will explain in which sense factorization homology provides a complete source of homology theories for manifolds.
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