Department of

Mathematics


Seminar Calendar
for Graduate Student Algebraic Geometry Seminar events the year of Thursday, December 7, 2017.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
    November 2017          December 2017           January 2018    
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Wednesday, October 4, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 4, 2017

Parabolic Higgs bundles

Georgios Kydonakis (Illinois Math)

Abstract: The Narasimhan and Seshadri theorem, one of the seminal first results in the study of the moduli space of vector bundles over a Riemann surface, relates degree zero, stable vector bundles on a compact Riemann surface $X$ with unitary representations of ${{\pi }_{1}}\left( X \right)$. One direction to generalize this theorem is by allowing punctures in the Riemann surface and the correspondence, which now involves parabolic bundles, was carried out by Mehta and Seshadri. The version for fundamental group representations of the punctured Riemann surface into Lie groups other than $G=\text{U}\left( n \right)$ entails introducing the notion of parabolic Higgs bundles. We will describe these holomorphic objects and see examples of those corresponding to Fuchsian representations of the fundamental group of the punctured Riemann surface.

Wednesday, October 18, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 18, 2017

Moment maps in Algebraic and Differential Geometry

Hadrian Quan   [email] (UIUC)

Abstract: In geometry, group actions are both ubiquitous and convenient. In this talk, I’ll survey an interesting circle of ideas relating notions of stability for orbits of an action to the complex geometry of the space being acted on. Time permitting, I’ll mention how some of this story generalizes after passing from finite to infinite dimensional groups.

Wednesday, October 25, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 25, 2017

Zariski tagent space to the moduli space of vector bundles on an algebraic curve

Jin Hyung To   [email] (UIUC)

Abstract: Zariski tagent space to the moduli space of vector bundles on an algebraic curve Abs: We will show how to use deformations to find the Zariski tangent space. The moduli space of vector bundles is the GIT quotient of Hilbert scheme. Using this we find the Zariski tangent space of the moduli space of vector bundles.

Wednesday, November 8, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 8, 2017

Local deformation theory of algebraic schemes

Sungwoo Nam (UIUC)

Abstract: Deformation theory is the study of variations of structure of a given object, which can be used to study the original object. It can also be regarded as local geometry of a moduli space, if it exists. In this talk, I will introduce (local) deformation theory, using the formalism of functor of artin rings and discuss automorphism-deformation-obstruction vector spaces and how these are realized as certain cohomologies. Along the way, I will give examples of application to concrete cases, including deformations of schemes, closed subschemes, nonsingular varieties, quasi-coherent sheaves.

Wednesday, November 15, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 15, 2017

Deformation theory of Galois representations

Ravi Donepudi   [email] (UIUC)

Abstract: The first systematic study of deformation theory in algebraic number theory, specifically its application to the theory of Galois representations, was done by Barry Mazur (1987). The goal of this talk is to motivate why this is a useful and interesting thing to do. We begin with discussing why one should study Galois representations in the first place, let alone deform them. Then, we define appropriate categories that serve as the domains of our deformation functors and discuss aspects of their representability. Finally, we give examples of Galois representations arising “naturally” from arithmetic objects (like elliptic curves and modular forms) and from algebraic geometry (via the étale cohomology of smooth projective varieties). Time permitting, we will discuss some conjectures in the theory of Galois representations and the role deformation theory plays in understanding them better. No scheme theory is assumed.

Wednesday, November 29, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 29, 2017

Introduction to quiver varieties

Ciaran O'Neill (UIUC)

Abstract: I’ll cover all the necessary background to define quiver varieties. This will include giving a reminder of the GIT quotient, defining the twisted GIT quotient and the basics of quivers. I’ll end by stating a theorem guaranteeing when a certain canonical map between quiver varieties is a resolution of singularities.

Wednesday, December 6, 2017

4:00 pm in Altgeld Hall 141,Wednesday, December 6, 2017

A Counterexample to the Weitzenböck Conjecture in Characteristics $p > 2$

Stephen Maguire   [email] (UIUC)

Abstract: A variety $X$ is ruled if there is a bi-rational map $\mathbb{P}^1 × Y \rightarrow X$ for some variety Y . The study of ruled varieties is motivated by the desire to study rational varieties by weakening the condition. If a variety $X$ has a $\mathbb{G}_a$-action, then one sees that it is ruled. Also, simplicial toric varieties often have non-reductive automorphism groups. An example is $\mathbb{P}(1 : 1 : 2)$ whose automorphism group is isomorphic to $GL(2, k)\times \mathbb{G}_a^3$(semi-direct product). As a result one finds that constructing a type of non-reductive GIT is a worthy venture. If one wants to construct this type of non-reductive GIT one first needs to understand actions of $\mathbb{G}_a$, the simplest non-reductive group. The earliest theorem regarding Ga was Weitzenb¨ock’s Theorem, which states that a representation $\mu : \mathbb{G}_a → GL(V_n)$ has a finitely generated ring of invariants $k[X]^{\mathbb{G}_a}$ if the field $k$ is an algebraically close field of characteristic zero. Work then focused on whether the invariant ring of non-linear actions of Ga is finitely generated. The answer is no. At this point people wished to answer the Weitzenbock conjecture, namely whether the result of Weitzenb¨ock’s Theorem still holds if the characteristic of the field is positive. It is this question we wish to answer in the following talks. In this talk we begin the process of producing a representation $\mu : \mathbb{G}_a → GL(V_6)$ over an algebraically closed field k of characteristic $p > 2$ such that the ring of invariants $k[x_1,\ldots,x_6]^{\mathbb{G}_a}$ is not a finitely generated k-algebra. In order to do this, we reduce the problem of finding a counterexample to the Weitzenbock conjecture to a curve counting problem, and then use this reduction to further reduce this problem to a problem about the support of a bi-graded ring.

Wednesday, December 13, 2017

4:00 pm in Altgeld Hall 141,Wednesday, December 13, 2017

Representation and Character Variety of 3 Manifolds

Xinghua Gao   [email] (UIUC)

Abstract: I'll start with defining $SL_2(\mathbb{C})$ representation/character variety of 3 manifold groups. Then I'll talk about how twisted cohomology is used to determine smoothness of these varieties, followed by some applications.