Department of

Mathematics


Seminar Calendar
for Graduate Student Algebraic Geometry Seminar events the year of Tuesday, November 14, 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.
     October 2017          November 2017          December 2017    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
  1  2  3  4  5  6  7             1  2  3  4                   1  2
  8  9 10 11 12 13 14    5  6  7  8  9 10 11    3  4  5  6  7  8  9
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 22 23 24 25 26 27 28   19 20 21 22 23 24 25   17 18 19 20 21 22 23
 29 30 31               26 27 28 29 30         24 25 26 27 28 29 30
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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.