The junior geometry seminar meets every Tuesday at 3:45-4:45pm central. What is the junior geometry seminar?
The organizational meeting is on August 23 at 3:45 pm central.
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 titles and abstracts of the talks are also available here. at the official math department seminar calendar.
Title: The (Symplectic) Camel and the eye of a needle
Abstract: Many of you have heard the parable of the camel and the eye of a needle "it is easier for a camel to go through the eye of a needle than for a rich man to enter the kingdom of God". Symplectic geometry (in its role as the one and only geometry given to us by a superior entity) has an analogous result concerning balls symplectically embedded into cylinders: if the radius of the ball is bigger than the radius of the cylinder then there is no way to symplectically embed the former inside the latter. As it happens with math, results that are somewhat easy to state turn out to be very tricky to prove and in fact we will need to rely on some "advance" techniques (by now standard) in symplectic geometry. We will follow Gromov's proof of the theorem which makes use of pseudo-holomorphic curves and some facts about area minimising curves. Everyone is welcome to attend! Prerequisites will be kept very low, I will blackbox something but hopefully the main ideas will be clear.
Title: Equivariant cohomology and (conditional) oriented matroid.
Abstract: According to Taoist wisdom, "Truths, if expressible by language, are not general truths". This has an interesting manifestation in math: theorems, if relying on geometry, are not general theorems. Therefore, I am going to talk about non-geometry. Given a collection of linear hyperplanes in \(\mathbb{R}^d\), one can construct two spaces \(M_1\) and \(M_3\). \(M_1\) is the complement of the union of all hyperplanes, while \(M_3\) is the complement of \(R^{3d}\) of the union of corresponding subspaces of codimension 3. The cohomology of \(M_1\), \(H^*(M_1)\) is a boring ring with an interesting filtration. Gel'fand and Varchenko gave presentations for \(H^*(M_1)\) and later on de Longueville & Schultz gave presentations for \(H^*(M_3)\). It happens to be the case that \(H^*(M_3)\) is exactly the associated graded algebra of \(H^*(M_1)\). Moseley realized that the reason for this connection lies in the fact that both rings are specializations of the torus-equivariant cohomology of \(M_3\). Oriented matroids are combinatorial abstractions of real hyperplane arrangements. This is a vast generalization since almost all matroids don't come from hyperplane arrangements. All the cohomology rings mentioned above only depend on the underlying oriented matroids, so in principle, the above phenomenon should still hold even if the space \(M_1\) and \(M_3\) don't exist. This is proved in full generality in a joint work with Dorpalen-Barry and Proudfoot. Bodies die and vanish, but souls don't. In a similar way, cohomology exists without a space, for geometry dies and vanishes, but combinatorics doesn't.
Title: GIT quotients
Abstract: Let's says you have a group G acting on a variety (or scheme) X and you want to take the quotient. Intuitively you may want to take some version of the "orbit space" but it turns out there is not always an obvious way to put a scheme structure on it. Geometric invariant theory (GIT) gives us a way to solve this problem. The simplest solution is given by the affine GIT quotient, which is given by taking Spec of G-invariant functions on X. If X is an affine variety, points in the affine quotient will turn out to correspond to closed orbits in X. In particular, if G is finite we recover the orbit space as a scheme, although in general the behavior can be pathological. Thus sometimes a more sophisticated answer is needed, which is given by the projective GIT quotient. We shall study both of these and their relationship, with one key example being Kleinian singularities (quotients of A^2 by finite groups). Time permitting we will briefly talk about Hamiltonian reductions and Nakajima quiver varieties. No background will be assumed.
Title: Exotic Covers of Familiar Spaces
Abstract: Relative algebraic geometry provides a framework and common language to com- pare various geometric theores (classical algebraic, complex analytic, Berkovich analytic, rigid analytic, etc). I will give a flash-summary of the idea of algebraic geometry relative to an arbitrary closed symmetric monoidal category C. When we specialize C to Vect_k and Ban_k, we recover usual algebraic and rigid analytic geometry, respectively. This will motivate why the class of homotopy epimorphisms in C are geometrically interesting. We will then show that there is a homotopy epimorphism from Z to a dual Frechet nuclear version of Z_p inside Ind(Ban). The rest of the talk will be devoted to giving more down-to-earth (and neat) pictures of what these ideas are supposed to capture.
Title: Triangles and Cubulations
Abstract: In this talk we present Stalling's triangles of groups and discuss the possibility of cubulation.
Title: Why should a(n algebraic) geometer care about categories?
Abstract: We will start by reviewing some basic scheme theory, assuming very little prerequisite knowledge. The goal will be to show why introducing categories, specifically the category of (quasi) coherent sheaves, is useful from a geometric point of view. At the end of the talk we will touch on sheaves of categories and some open questions in the theory of stacks.
Title: What is a gauge, how to fix it and why
Abstract: I'll talk about connections and gauge transformations on principal bundles, and then use Hodge theory as motivation for a certain type of gauge fixing due to Uhlenbeck. There will be some analysis, but I won't assume much prerequisite knowledge.
Title: Supermanifolds: what are they and why are they so super?
Abstract: The Spin-statistics theorem tells us there are two types of particles in the universe, bosons and fermions, and that the classical observables corresponding to these particles should commute and anticommute respectively. Supersymmetry posits that there should be an extra symmetry of the universe where bosons and fermions are exchanged. In this talk we begin by describing the algebraic formalism of superalgebra which allows us to combine both commutative and antifommutative observables into a single object and elegantly describe supersymmetries. We then develop one of the two modern approaches to supermanifolds using the language of sheaves. We discuss Batchelor's theorem, which tells us that every supermanifold may be represented as the sections of a bundle of exterior algebras. If time permits we will discuss the idea of a symplectic supermanifold and the supermanifold Darboux theorem.
Title: Lie 2-Groups, Representations, and Field Theory
Abstract: I'll introduce Lie 2-groups, discuss what it would mean to study their representations, and give subtle hints about what this could mean for symmetries in QFT.
Title: Hyperkahler Geometry
Abstract: Berger's theorem sorts Riemannian geometry into 8 flavours: general holonomy, kahler, quaternionic-kahler, hyperkahler, G2, Spin(7), and symmetric spaces. Hyperkahler geometry is probably 4th most popular, and is a great thing to spend an hour learning about. A Hyperkahler manifold comes with an unsettling amount of extra structure: a 2-sphere worth of kahler structures which all fit together in a way governed by the quaternions. Despite how special these objects seem, there are many natural and interesting hyperkahler manifolds which show up in various places in math. K3 surfaces, coadjoint orbits, Hilbert schemes of points, and character varieties of surfaces are some good examples. We will explain how to construct some of these hyperkahler manifolds, and how to understand them via twistor space.
Title: Matrix multiplication is Fourier theory is Koszul duality, or: How I learned to stop worrying and love integral transforms
Abstract: Integral transforms provide a recipe for mapping between various kinds of linearizations of geometric spaces (such as vector spaces of functions on topological spaces or categories of sheaves on algebraic varieties). After presenting a context-independent sketch of this construction, I will explain how integral transforms provide a conceptual formulation of matrix multiplication, the Fourier transform (and Pontrjagin duality more generally), the equivalence of categories between coherent sheaves on an abelian variety and its dual, and (to a somewhat more mysterious extent) Koszul duality.
Title: Superspace construction of a SUSY QFT
Abstract: Riemannian sigma model is a supersymmetric quantum mechanics which is highly interesting from the mathematical point of view; it has deep connections with Hodge theory, Morse theory, and Fredholm theory. There is an elegant construction of the theory known by physicists, namely superspace construction. I'll discuss how to mathematically understand the construction, using the language of supermanifolds. + Although I ended up using quite a different language, Tudor Dimofte's lecture 10 of BW&TD - Twisted QFT is a good source for this topic. This is where I learned this construction, and it might be interesting to watch.
Title:
Abstract:
Title: Analogies between varifolds and schemes
Abstract: We will discuss some analogies and similarities between, and motivations behind, foundational objects of two different fields with different sets of fundamental techniques: geometric measure theory and algebraic geometry. In particular, we will compare how currents and varifolds on the GMT side, and schemes on the AG side, each in their own setting and their own way enable us to better handle singular spaces, tangential and infinitesimal information, and more.