The junior geometry seminar meets every Tuesday at 3:45-5pm central. What is the junior geometry seminar?
The organizational meeting is on 08/31 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: The symplectic proof of the Rectangular Peg Problem by Greene and Lobb
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 in May 2020. 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. The first half of this talk will be very accessible to everyone, the second half will use some elementary notion from symplectic geometry.
Title: Non-connective Algebraic Geometry
Abstract: Homotopical generalizations of algebraic geometry (be it derived or spectral) begins by replacing ordinary commutative rings in the foundations of algebraic geometry with some homotopical version of commutative rings (commutative differential graded algebras, simplicial commutative rings, E_infty-rings, etc.). Tradition dictates that we nonetheless require these homotopical rings to be connective, i.e. (depending on the choice of foundations) to have no non-zero negative homotopy groups or no non-zero positive-degree homotopy groups. Breaking this taboo and admitting non-connective homotopical rings as affine objects leads to non-connective algebraic geometry. It is well-known to admit some unusual, less-geometric, and all-around pathological behavior. In this talk, we will explore some of those, as well as some examples that suggest why we might nonetheless sometimes want to put ourselves in such a setting.
Title: Categorification of the Radon transform
Abstract: The Brylinski-Radon transform provides a relationship between the categories of D-modules on dual projective spaces. In this talk I'll review its statement, and explain an analogue in the context of noncommutative algebraic geometry, which relates the 2-categories of categorical sheaves on dual projective spaces. Time permitting, I'll explain how this categorical Radon transform features in recent advances in the study of derived categories of projective varieties.
Title: Condensed Mathematics
Abstract: Is there a common playing field for complex, real, and rigid analytic geometry? This is one of the driving questions behind Scholze and Clausen's "condensed mathematics". In this talk we will give a basic overview of how one can redo point-set topology to accommodate certain homological shortcomings of the category of topological groups, why this viewpoint seems "natural", and what interesting objects pop out of the general theory.
Title: Weil's Rosetta stone from the viewpoint of covering space theory
Abstract: This talk will aim to motivate a far-reaching analogy between arithmetic curves associated to number fields, algebraic curves defined over a finite field, and Riemann surfaces/algebraic curves over the complex numbers known as Weil's Rosetta stone. We will mostly focus on the theory of branched coverings and its incarnations in all three contexts, leading to various versions of the Riemann-Hurwitz theorem and a geometric perspective on some basic results in algebraic number theory. Time permitting, we will hint at how many areas of modern mathematical activity can be traced back to this paradigm, including arithmetic and algebraic Riemann-Roch theorems, the classical and geometric Langlands programs, and the Riemann-Weil conjecture as a finite field analogue of the Riemann hypothesis.
Title: Categorification of quantization
Abstract: We will start by reviewing what it means to "quantize" a symplectic vector space. Then we will consider how to categorify this procedure. In particular, we will look at categorified D-modules and some results about the Morita theory of higher algebras.
Title: Logarithmic construction of the moduli space of admissible covers
Abstract: In this talk, we shall give a brief survey on the modular compactification of the classical Hurwitz moduli space using log admissible covers following Mochizuki. The classical Hurwitz space parametrizes d-sheeted, simple branched coverings of P^1 with b branched points. Harris and Mumford first introduced the notion of admissible covers as a tool for compactifying the classical Hurwitz space. The use of logarithmic structures is necessary for obtaining a full fledged modular interpretation of the space of admissible covers. In this talk we shall illustrate the key ideas in the more standard case of moduli space of logarithmic stable curves.
Title: The affine grassmannian and its applications to algebraic geometry
Abstract: The affine grassmannian plays a remarkable role in algebraic geometry and representation theory and is probably best known for its feature in
the Geometric Satake correspondence. In this talk, however, we will concern ourselves with some lesser known applications of the geometry of the affine
grassmannians to algebraic geometry. Namely, we will study its relation to the moduli stack of principal G-bundles and from that highlight two consequences:
- A rederivation of the Atiyah-Bott formula
- A proof of the Tamagawa number formula
We highlight that while the study of the affine grassmannian takes place in the world of higher categories, the two applications do not (in fact, the
latter is simply an equality of two positive real numbers).
Title: Holomorphic-Flat Connections
Abstract: Vector bundles with holomorphic structure, and vector bundles with flat connection have been studied in great detail, but, until recently, people haven't been so tempted by bundles which are holomorphic in some directions and flat in others. In this talk, we will see how such "holomorphic-flat" connections arise in nature, and some things I've found while studying them in their own right. For the historical perspective, I will show how a solution to the KdV wave equation gives rise to a family of flat SL(2,C) connections on R^2 parametrized by the complex plane. Then, we will look at families of flat connections parametrized by the formal punctured disk, and see that these can give rise to lagrangians in the cotangent bundle.
Title: Fun with Quivers
Abstract: First we will explain the basics of quivers (directed graphs) and their representations. Next we will discuss the relationship between the representation theory of lie algebras and the representation theory of their Dynkin diagrams, which are quivers. The representations of quivers can further be given a convolution product, yielding an algebra called the Hall algebra. We will explain this construction, as well as the theorem that the Hall algebra is isomorphic to the positive part of the quantum group of the corresponding lie algebra, a modification of the universal enveloping algebra. If time permits, we will further explain how Nakajima used quiver varieties to construct all the representations of the quantum group geometrically. Except for possibly for the last 10-15 minutes, no knowledge will be assumed from the audience and everything we work with will be defined.
Title: Three and a half ways to curve counting
Abstract: Curve counting theories try to answer questions like: How many curves are there in a smooth manifold X that pass through some general submanifolds
and tangent to some submanifold? Since the space parametrizing the smooth curves in X is non-compact, different compactifications lead to different
enumeration of curves. In this talk, I will explain why the naive curve counting theory (1/2 of a method) is not satisfying. I will introduce three principal
ways of counting curves and the conjectural relations between different theories. If time allows, I will introduce the common approaches to computations and
how the partial results in proving the conjectures go.
The talk is based on 7/13 of the paper "13/2 ways of counting curves" by Pandharipande and Thomas.