The organizational meeting is on Tuesday September 1.
Zoom links and email reminders will be sent to the mailing list.
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In Fall 2020, 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.
Abstract: Gromov-Witten invariants are the answers to questions like: How many curves of degree \(d\) are there in \(\mathbb{P}^2\) that pass through \(3d-1\) general points? Though the computation of them are in general hard, the property that they are deformation invariants provides a way that leads to simplification. In particular, it is desirable to have a Gromov-Witten theory on a degenerate target space. Logarithmic Gromov-Witten theory provides a beautiful answer to the question. In this talk, I will first give an introduction to the ordinary GW-invariants, and then talk about log GW-theory and how we can use it to decompose the ordinary GW-invariants into log GW-invariants over certain tropical types.
Abstract: The aim of this talk will be to give a non-intimidating introduction to the Cobordism Hypothesis. First proposed by Baez and Dolan and later proved by Hopkins and Lurie, the Cobordism Hypothesis provides a remarkable classification of fully extended topological field theories. We will introduce the main concepts that underpin this theorem, such as duality in symmetric monoidal categories. We will sketch a proof of the theorem in dimension 1, before talking about its higher dimensional versions.
Abstract: One problem of interest in modular representation theory 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 \(\mathrm{StMod}(kG)\). This group was computed by work of Carlson-Thevenaz, using group cohomology and the theory of support varieties of modules.
Recently, work of Mathew and Mathew-Stojanoska introduced a new, homotopical approach to this computation through faithful Galois descent. While Galois descent often holds in \(\mathrm{StMod}(kG)\), it is too much to expect faithful Galois descent. Nevertheless, I show that one can extend their methods to the non-faithful setting in the case that \(G\) is a quaternion or generalized quaternion group. This is a practice research talk.
Abstract: Flat connections, also known as local systems, are of importance in various areas of math. They are important in representation theory, and mathematical physics, to name only a few. A light generalization are D-modules, which come equipped with an action by differential operators. Many of us in this Seminar, and our advisors even more so, deal with D-modules in characteristic 0 on a nigh-daily basis. In this talk, we will take a journey through the theory of D-modules in positive characteristic, focusing on some particularly interesting phenomena that occur and set it apart from the more-familiar theory in characteristic zero (Frobenius descent, crystals). If time permits, we will end on a version of the Riemann-Hilbert correspondence in that context, due to Emmerton-Kisin and Bhatt-Lurie.
Abstract: As the title suggests, I'll talk about the Rectangular Peg Problem and how symplectic geometry was used to find a positive solution to a stronger version of it by Greene and Lobb last May. This seemingly easy problem asks us to find 4 points on any given smooth Jordan curve in C that correspond to the vertices of a rectangle. Greene and Lobb proved that you can find 4 points that are the vertices of a rectangle that it's similar to any given rectangle. Surprisingly, the square peg problem over continuous Jordan curve (formulated in 1911) is still open.
Abstract: We shall concern ourselves with category O, which is the abelican category consisting of certain nice representations of a complex semisimple Lie algebra. While we cover the basics, our main goal is to formulate and describe the progress made towards the proof of the Kazhdan-Lusztig conjecture. In particular, we highlight the work made by Verma, Jantzen and Beilinson-Bernstein.
Abstract: As it turns out, Grassmannians (and various related friends) are important for machine learning. We will see that Schubert varieties play a key role in a method for distinguishing subspaces of data.
Abstract: In 1958, Serre proposed a definition for the intersection multiplicity of two algebraic subvarieties of complex projective space at an isolated point \(p\), in the form of an alternating sum of the dimensions of the Tor groups of the corresponding rings of functions (localized at \(p\)). Following notes in progress by Daniel Dugger, we will try to motivate this formula, guided by the intuition coming from classical intersection products of smooth submanifolds. In the process, we will take a detour through the worlds of algebraic and topological K-theory, highlighting ways in which information can be transferred between these frameworks to arrive at Serre's intersection formula.
Abstract: We will talk about how to give statistical distributions a Riemannian manifold structure, complete with metrics, connections and curvature. Just like General Relativity is a global version of Special Relativity, Information Geometry can be thought of as a "global" version of statistics. We will discuss some basic structures, including conjugate connections, statistical manifolds, divergences, and curvature. I will give some example of such things for the statistical manifold of normal distributions (spoiler: it's the Poincare Disk). Finally, I will also briefly touch on a recent Deep Mind paper that uses information geometry in machine learning.
Abstract: Let G be a semisimple group, for example, \(G = SL_n\). One pervasive theme in representation theory is recovering information about representations of Lie(G) from a maximal torus T in G (for example, T may be identified with the diagonal matrices of \(SL_n\) ) with its natural action by the Weyl group \(W:= N_G(T)/T\). In this talk, we will explore historical incarnations of this theme -- specifically, finite dimensional \(\mathrm{Lie}(G)\) representations and the study of the BGG category \(O_0\) -- and then discuss a recent theorem which identifies a "varying central character" version of category O with sheaves on a space determined by the action of W on T.
Abstract: In this talk I will discuss categorification of Fourier theory. The original setting for Fourier theory is in analysis, where the Fourier transform gives an equivalence of the Hilbert space of L^2 functions on the real number. We will do a version of discrete Fourier transform for finite abelian groups, and get an isomorphism between the vector space of functions of a finite abelian group A and its Pontryagin dual. Then we will categorify this picture, which recovers character theory for finite groups. Then we categorify once more, getting an equivalence of 2-categories. If time allow, I will discuss how to view this in the framework of TFTs.
Abstract: The moduli space of principal G-bundles is a Kähler variety and can be constructed in two different ways: Via Kähler reduction, which sheds light on its symplectic structure, and as an algebraic (GIT) quotient which explains its algebraic structure. I will explain the two different approaches and if time permits give some indication of how they can be compared.
Abstract: First we cover some Lie theory, explaining what Cartan and Borel subalgebras are as well as the Jordan decomposition. We will use this to construct two central objects of geometric representation theory: the Springer resolution and the Steinberg variety. Next we will try to motivate the theorem that the top homology of the Steinberg variety is the group algebra of the Weyl group. Finally we will use this to geometrically construct all the representations of the Weyl group as top homology groups of Springer fibers.
Abstract: We will discuss the Rigidity Conjecture of Goette and Igusa, which states that, after stabilizing and then rationalizing, there are no exotic smoothings of manifold bundles with closed even dimensional fibers. After motivating this work through classical results such as Smale's h-cobordism theorem, and later work such as that of Dwyer, Weiss, and Williams, we will discuss fiberwise Poincare-Hopf theory and the duality theorems that lead to a proof of the aforementioned conjecture.