https://wiki.math.wisc.edu/api.php?action=feedcontributions&user=Eramos&feedformat=atomUW-Math Wiki - User contributions [en]2022-09-24T15:49:41ZUser contributionsMediaWiki 1.35.6https://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Spring_2016&diff=11468Algebraic Geometry Seminar Spring 20162016-02-08T04:30:02Z<p>Eramos: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B113.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Fall_2015 here].<br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2016 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 22<br />
|[http://homepages.math.uic.edu/~tryan8/ Tim Ryan] (UIC)<br />
|Moduli Spaces of Sheaves on \PP^1 \times \PP^1<br />
|Daniel<br />
|-<br />
|January 29<br />
|<strike>[http://www.math.wisc.edu/~yangjay/ Jay Yang] </strike> (Wisconsin)<br />
|<strike>Random Toric Surfaces </strike><br />
|Local<br />
|-<br />
|February 5<br />
|[http://www.math.wisc.edu/~wang/ Botong Wang] (Wisconsin)<br />
|Topological Methods in Algebraic Statistics<br />
|Local<br />
|-<br />
|February 12<br />
|[http://www.math.wisc.edu/~yangjay/ Jay Yang] (Wisconsin)<br />
|Random Toric Surfaces<br />
|Local<br />
|-<br />
|February 19<br />
|[http://www.math.wisc.edu/~derman/ Daniel Erman] (Wisconsin)<br />
|Supernatural Analogues of Beilinson Monads<br />
|Local|<br />
|-<br />
|February 26<br />
|TBD<br />
|<br />
|<br />
|-<br />
|March 4<br />
|[http://www3.nd.edu/~craicu/ Claudiu Raicu] (Notre Dame)<br />
|TBA<br />
|Steven<br />
|-<br />
|March 11<br />
|[http://www.math.wisc.edu/~eramos/ Eric Ramos] (Wisconsin)<br />
|Local Cohomology of FI-modules<br />
|Local<br />
|-<br />
|March 18<br />
|Spring break<br />
|<br />
|<br />
|-<br />
|March 25<br />
|TBD<br />
|<br />
|<br />
|-<br />
|April 1<br />
|TBD<br />
|<br />
|<br />
|-<br />
|April 8<br />
|TBD<br />
|<br />
|<br />
|-<br />
|April 15<br />
|TBD<br />
|<br />
|<br />
|-<br />
|April 22<br />
|TBD<br />
|<br />
|<br />
|-<br />
|April 29<br />
|[http://people.math.osu.edu/anderson.2804/ David Anderson] (Ohio State)<br />
|TBA<br />
|Steven<br />
|-<br />
|May 6<br />
|TBD<br />
|<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Tim Ryan===<br />
Moduli Spaces of Sheaves on \PP^1 \times \PP^1<br />
<br />
In this talk, after reviewing the basic properties of moduli spaces of<br />
sheaves on P^1 x P^1, I will show that they are<br />
$\mathbb{Q}$-factorial Mori Dream Spaces and explain a method for<br />
computing their effective cones.<br />
My method is based on the generalized Beilinson spectral sequence,<br />
Bridgeland stability and moduli spaces of Kronecker modules.<br />
<br />
<br />
===Botong Wang===<br />
Topological Methods in Algebraic Statistics<br />
<br />
In this talk, I will give a survey on the relation between maximum likelihood degree of an algebraic variety and it Euler characteristics. Maximam likelihood degree is an important constant in algebraic statistics, which measures the complexity of maximum likelihood estimation. For a smooth very affine variety, June Huh showed that, up to a sign, its maximum likelihood degree is equal to its Euler characteristics. I will present a generalization of Huh's result to singular varieties, using Kashiwara's index theorem. I will also talk about how to compute the maximum likelihood degree of rank 2 matrices as an application. <br />
<br />
<br />
===Daniel Erman===<br />
Supernatural analogues of Beilinson monads<br />
<br />
First I will discuss Beilinson's resolution of the diagonal and some of the applications of that construction including the notion of a Beilinson. Then I will discuss new work, joint with Steven Sam, where we use supernatural bundles to build GL-equivariant resolutions supported on the diagonal of P^n x P^n, in a way that extends Beilinson's resolution of the diagonal. I will discuss some applications of these new constructions.<br />
<br />
===Eric Ramos===<br />
The Local Cohomology of FI-modules<br />
<br />
Much of the work in homological invariants of FI-modules has been concerned with properties of certain right exact functors. One reason for this is that the category of finitely generated FI-modules over a Noetherian ring very rarely has sufficiently many injectives. In this talk we consider the (left exact) torsion functor on the category of finitely generated FI-modules, and show that its derived functors exist. Properties of these derived functors, which we call the local cohomology functors, can be used in reproving well known theorems relating to the depth, regularity, and stable range of a module. We will also see that various facts from the local cohomology of modules over a polynomial ring have analogs in our context. This is joint work with Liping Li.</div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2015&diff=10643Algebraic Geometry Seminar Fall 20152015-11-06T02:59:29Z<p>Eramos: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B223.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2015 here].<br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2015 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 18<br />
|[http://www.math.harvard.edu/~ebriedl/ Eric Riedl] (UIC)<br />
|Rational Curves on Hypersurfaces<br />
|Jordan<br />
|-<br />
|September 25<br />
|[http://www.mathcs.emory.edu/~dzb/ David Zureick-Brown] (Emory)<br />
|Hilbert schemes of canonically embedded curves of low genus<br />
|Jordan<br />
|-<br />
|October 2<br />
|[https://math.temple.edu/~vald/ Vasily Dolgushev] (Temple)<br />
|A manifestation of the Grothendieck-Teichmueller group in geometry<br />
|Andrei<br />
|-<br />
|October 9<br />
|[https://math.wisc.edu/~maxim/ Laurentiu Maxim] (Madison)<br />
|Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
|local<br />
|-<br />
|October 16<br />
|[http://www.math.wisc.edu/~dewey/ Ed Dewey] (Madison)<br />
|Characteristic Classes of Cameral Covers<br />
|local<br />
|-<br />
|October 23<br />
|[http://people.math.sc.edu/kassj/ Jesse Kass] (South Carolina)<br />
|How to count zeros arithmetically?<br />
|Melanie<br />
|-<br />
|November 6<br />
|[https://www.math.wisc.edu/~eramos/ Eric Ramos] (Wisconsin)<br />
|Homological Invariants of FI-modules<br />
|Daniel<br />
|-<br />
|November 13<br />
|[http://www-personal.umich.edu/~jakelev/ Jake Levinson] (Michigan)<br />
|(Real) Schubert Calculus from Marked Points on P^1<br />
|Daniel<br />
|-<br />
|November 20<br />
|[https://sites.google.com/site/hackenzheng/ Xudong Zheng] (UIC)<br />
|TBA<br />
|Daniel<br />
|-<br />
|December 4<br />
|[http://www.math.wisc.edu/~svs/ Steven Sam] (Wisconsin)<br />
|Ideals of bounded rank symmetric tensors<br />
|Local<br />
|-<br />
|December 11<br />
|[http://www.math.columbia.edu/~danhl/ Daniel Halpern-Leistner] (Columbia)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Spring 2016 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 29<br />
|[http://www.math.wisc.edu/~yangjay/ Jay Yang] (Wisconsin)<br />
|Random Toric Surfaces<br />
|Local<br />
|-<br />
|March 4<br />
|[http://www3.nd.edu/~craicu/ Claudiu Raicu] (Notre Dame)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Eric Riedl===<br />
Rational Curves on Hypersurfaces<br />
<br />
One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the Fano case and try to motivate some of the ideas used to attack this problem.<br />
<br />
===David Zureick-Brown===<br />
Hilbert schemes of canonically embedded curves of low genus<br />
<br />
I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.<br />
<br />
===Vasily Dolgushev===<br />
A manifestation of the Grothendieck-Teichmueller group in geometry<br />
<br />
Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in<br />
1990, the Grothendieck-Teichmueller group GRT. This group has<br />
interesting links to the absolute Galois group of rationals, moduli of<br />
algebraic curves, solutions of the Kashiwara-Vergne problem, and<br />
theory of motives. My talk will be devoted to the manifestation of GRT<br />
in the extended moduli of algebraic varieties, which was conjectured<br />
by Maxim Kontsevich in 1999. My talk is partially based on the joint<br />
paper with Chris Rogers and Thomas Willwacher: http://arxiv.org/abs/1211.4230.<br />
<br />
===Laurentiu Maxim===<br />
Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
<br />
I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), <br />
which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical <br />
results in the literature as special cases. Important specializations of these results include generating series for <br />
extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series <br />
for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures <br />
of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula<br />
for equivariant invariants of external products, which includes all of the above-mentioned results as special cases.<br />
This is joint work with Joerg Schuermann.<br />
<br />
===Ed Dewey===<br />
Characteristic Classes of Cameral Covers<br />
<br />
Cameral covers are what you get when you try to diagonalize a family of regular matrices. They form a nice algebraic stack, which means that one can define cohomological invariants of cameral covers by computing the cohomology ring of that stack. With rational coefficients this ring has a presentation in terms of hyperplane arrangements. My talk will be in the style of a "working seminar": I will explain what cameral covers are and try to make you like them, I will tell you what I know about their characteristic classes and the main ideas behind this computation, and then I will tell you where I am stuck.<br />
<br />
===Jess Kass===<br />
How to count zeros arithmetically<br />
<br />
A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local Brouwer degree of a real polynomial function at an isolated zero as the signature of a quadratic form. I will discuss a parallel result in A1-homotopy theory, and time permitting, explain how to study a singularity by applying these results to the gradient of a defining equation. This is joint work with Kirsten Wickelgren.<br />
<br />
===Eric Ramos===<br />
Homological invariants of FI-modules<br />
<br />
An FI-module is a functor from the category FI, of finite sets and <br />
injections, to a module category. These objects were initially studied <br />
by Church, Ellenberg, and Farb in connection with the theory of <br />
representation stability. It was discovered that certain homological <br />
invariants of FI-modules could be used to solve problems in topology, <br />
algebraic geometry, and number theory. In this talk we will study <br />
these invariants using a new theory of depth. With this theory, we <br />
will be able to prove various results, including bounds on the <br />
regularity, as well as the stable ranges of FI-modules.<br />
<br />
===Jake Levinson===<br />
(Real) Schubert Calculus from Marked Points on P^1<br />
<br />
I will describe a family S of Schubert problems on the Grassmannian, defined using flags osculating (tangent to) the rational normal curve at a chosen set of marked points.<br />
<br />
This family is very well-behaved (for example, it is Cohen-Macaulay), particularly when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) showed that the solutions are then "as real as possible", and Speyer (2014) extended the construction to stable curves, showing that the real locus of S is a smooth cover of the moduli space of real stable curves. Moreover, the monodromy of the cover has a remarkable description in terms of Young tableaux and Schützenberger's jeu de taquin.<br />
<br />
I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.</div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2015&diff=10635Algebraic Geometry Seminar Fall 20152015-11-04T20:36:02Z<p>Eramos: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B223.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2015 here].<br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2015 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 18<br />
|[http://www.math.harvard.edu/~ebriedl/ Eric Riedl] (UIC)<br />
|Rational Curves on Hypersurfaces<br />
|Jordan<br />
|-<br />
|September 25<br />
|[http://www.mathcs.emory.edu/~dzb/ David Zureick-Brown] (Emory)<br />
|Hilbert schemes of canonically embedded curves of low genus<br />
|Jordan<br />
|-<br />
|October 2<br />
|[https://math.temple.edu/~vald/ Vasily Dolgushev] (Temple)<br />
|A manifestation of the Grothendieck-Teichmueller group in geometry<br />
|Andrei<br />
|-<br />
|October 9<br />
|[https://math.wisc.edu/~maxim/ Laurentiu Maxim] (Madison)<br />
|Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
|local<br />
|-<br />
|October 16<br />
|[http://www.math.wisc.edu/~dewey/ Ed Dewey] (Madison)<br />
|Characteristic Classes of Cameral Covers<br />
|local<br />
|-<br />
|October 23<br />
|[http://people.math.sc.edu/kassj/ Jesse Kass] (South Carolina)<br />
|How to count zeros arithmetically?<br />
|Melanie<br />
|-<br />
|November 6<br />
|[https://www.math.wisc.edu/~eramos/ Eric Ramos] (Wisconsin)<br />
|Homological Invariants of FI-modules<br />
|Daniel<br />
|-<br />
|November 13<br />
|[http://www-personal.umich.edu/~jakelev/ Jake Levinson] (Michigan)<br />
|(Real) Schubert Calculus from Marked Points on P^1<br />
|Daniel<br />
|-<br />
|November 20<br />
|[https://sites.google.com/site/hackenzheng/ Xudong Zheng] (UIC)<br />
|TBA<br />
|Daniel<br />
|-<br />
|December 4<br />
|[http://www.math.wisc.edu/~svs/ Steven Sam] (Wisconsin)<br />
|Ideals of bounded rank symmetric tensors<br />
|Local<br />
|-<br />
|December 11<br />
|[http://www.math.columbia.edu/~danhl/ Daniel Halpern-Leistner] (Columbia)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Spring 2016 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 29<br />
|[http://www.math.wisc.edu/~yangjay/ Jay Yang] (Wisconsin)<br />
|Random Toric Surfaces<br />
|Local<br />
|-<br />
|March 4<br />
|[http://www3.nd.edu/~craicu/ Claudiu Raicu] (Notre Dame)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Eric Riedl===<br />
Rational Curves on Hypersurfaces<br />
<br />
One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the Fano case and try to motivate some of the ideas used to attack this problem.<br />
<br />
===David Zureick-Brown===<br />
Hilbert schemes of canonically embedded curves of low genus<br />
<br />
I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.<br />
<br />
===Jess Kass===<br />
How to count zeros arithmetically?<br />
<br />
A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local Brouwer degree of a real polynomial function at an isolated zero as the signature of a quadratic form. I will discuss a parallel result in A1-homotopy theory, and time permitting, explain how to study a singularity by applying these results to the gradient of a defining equation. This is joint work with Kirsten Wickelgren.<br />
<br />
===Vasily Dolgushev===<br />
A manifestation of the Grothendieck-Teichmueller group in geometry<br />
<br />
Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in<br />
1990, the Grothendieck-Teichmueller group GRT. This group has<br />
interesting links to the absolute Galois group of rationals, moduli of<br />
algebraic curves, solutions of the Kashiwara-Vergne problem, and<br />
theory of motives. My talk will be devoted to the manifestation of GRT<br />
in the extended moduli of algebraic varieties, which was conjectured<br />
by Maxim Kontsevich in 1999. My talk is partially based on the joint<br />
paper with Chris Rogers and Thomas Willwacher: http://arxiv.org/abs/1211.4230.<br />
<br />
===Laurentiu Maxim===<br />
Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
<br />
I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), <br />
which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical <br />
results in the literature as special cases. Important specializations of these results include generating series for <br />
extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series <br />
for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures <br />
of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula<br />
for equivariant invariants of external products, which includes all of the above-mentioned results as special cases.<br />
This is joint work with Joerg Schuermann.<br />
<br />
===Ed Dewey===<br />
Characteristic Classes of Cameral Covers<br />
<br />
Cameral covers are what you get when you try to diagonalize a family of regular matrices. They form a nice algebraic stack, which means that one can define cohomological invariants of cameral covers by computing the cohomology ring of that stack. With rational coefficients this ring has a presentation in terms of hyperplane arrangements. My talk will be in the style of a "working seminar": I will explain what cameral covers are and try to make you like them, I will tell you what I know about their characteristic classes and the main ideas behind this computation, and then I will tell you where I am stuck.<br />
<br />
===Eric Ramos===<br />
Homological Invariants of FI-modules.<br />
<br />
An FI-module is a functor from the category FI, of finite sets and <br />
injections, to a module category. These objects were initially studied <br />
by Church, Ellenberg, and Farb in connection with the theory of <br />
representation stability. It was discovered that certain homological <br />
invariants of FI-modules could be used to solve problems in topology, <br />
algebraic geometry, and number theory. In this talk we will study <br />
these invariants using a new theory of depth. With this theory, we <br />
will be able to prove various results, including bounds on the <br />
regularity, as well as the stable ranges of FI-modules.<br />
<br />
===Jake Levinson===<br />
(Real) Schubert Calculus from Marked Points on P^1<br />
<br />
I will describe a family S of Schubert problems on the Grassmannian, defined using flags osculating (tangent to) the rational normal curve at a chosen set of marked points.<br />
<br />
This family is very well-behaved (for example, it is Cohen-Macaulay), particularly when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) showed that the solutions are then "as real as possible", and Speyer (2014) extended the construction to stable curves, showing that the real locus of S is a smooth cover of the moduli space of real stable curves. Moreover, the monodromy of the cover has a remarkable description in terms of Young tableaux and Schützenberger's jeu de taquin.<br />
<br />
I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.</div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2015&diff=10634Algebraic Geometry Seminar Fall 20152015-11-04T20:33:14Z<p>Eramos: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B223.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2015 here].<br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2015 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 18<br />
|[http://www.math.harvard.edu/~ebriedl/ Eric Riedl] (UIC)<br />
|Rational Curves on Hypersurfaces<br />
|Jordan<br />
|-<br />
|September 25<br />
|[http://www.mathcs.emory.edu/~dzb/ David Zureick-Brown] (Emory)<br />
|Hilbert schemes of canonically embedded curves of low genus<br />
|Jordan<br />
|-<br />
|October 2<br />
|[https://math.temple.edu/~vald/ Vasily Dolgushev] (Temple)<br />
|A manifestation of the Grothendieck-Teichmueller group in geometry<br />
|Andrei<br />
|-<br />
|October 9<br />
|[https://math.wisc.edu/~maxim/ Laurentiu Maxim] (Madison)<br />
|Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
|local<br />
|-<br />
|October 16<br />
|[http://www.math.wisc.edu/~dewey/ Ed Dewey] (Madison)<br />
|Characteristic Classes of Cameral Covers<br />
|local<br />
|-<br />
|October 23<br />
|[http://people.math.sc.edu/kassj/ Jesse Kass] (South Carolina)<br />
|How to count zeros arithmetically?<br />
|Melanie<br />
|-<br />
|November 6<br />
|[https://www.math.wisc.edu/~eramos/ Eric Ramos] (Wisconsin)<br />
|Homological Invariants of FI-modules<br />
|Daniel<br />
|-<br />
|November 13<br />
|[http://www-personal.umich.edu/~jakelev/ Jake Levinson] (Michigan)<br />
|(Real) Schubert Calculus from Marked Points on P^1<br />
|Daniel<br />
|-<br />
|November 20<br />
|[https://sites.google.com/site/hackenzheng/ Xudong Zheng] (UIC)<br />
|TBA<br />
|Daniel<br />
|-<br />
|December 4<br />
|[http://www.math.wisc.edu/~svs/ Steven Sam] (Wisconsin)<br />
|Ideals of bounded rank symmetric tensors<br />
|Local<br />
|-<br />
|December 11<br />
|[http://www.math.columbia.edu/~danhl/ Daniel Halpern-Leistner] (Columbia)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Spring 2016 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 29<br />
|[http://www.math.wisc.edu/~yangjay/ Jay Yang] (Wisconsin)<br />
|Random Toric Surfaces<br />
|Local<br />
|-<br />
|March 4<br />
|[http://www3.nd.edu/~craicu/ Claudiu Raicu] (Notre Dame)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Eric Riedl===<br />
Rational Curves on Hypersurfaces<br />
<br />
One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the Fano case and try to motivate some of the ideas used to attack this problem.<br />
<br />
===David Zureick-Brown===<br />
Hilbert schemes of canonically embedded curves of low genus<br />
<br />
I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.<br />
<br />
===Jess Kass===<br />
How to count zeros arithmetically?<br />
<br />
A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local Brouwer degree of a real polynomial function at an isolated zero as the signature of a quadratic form. I will discuss a parallel result in A1-homotopy theory, and time permitting, explain how to study a singularity by applying these results to the gradient of a defining equation. This is joint work with Kirsten Wickelgren.<br />
<br />
===Vasily Dolgushev===<br />
A manifestation of the Grothendieck-Teichmueller group in geometry<br />
<br />
Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in<br />
1990, the Grothendieck-Teichmueller group GRT. This group has<br />
interesting links to the absolute Galois group of rationals, moduli of<br />
algebraic curves, solutions of the Kashiwara-Vergne problem, and<br />
theory of motives. My talk will be devoted to the manifestation of GRT<br />
in the extended moduli of algebraic varieties, which was conjectured<br />
by Maxim Kontsevich in 1999. My talk is partially based on the joint<br />
paper with Chris Rogers and Thomas Willwacher: http://arxiv.org/abs/1211.4230.<br />
<br />
===Laurentiu Maxim===<br />
Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
<br />
I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), <br />
which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical <br />
results in the literature as special cases. Important specializations of these results include generating series for <br />
extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series <br />
for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures <br />
of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula<br />
for equivariant invariants of external products, which includes all of the above-mentioned results as special cases.<br />
This is joint work with Joerg Schuermann.<br />
<br />
===Ed Dewey===<br />
Characteristic Classes of Cameral Covers<br />
<br />
Cameral covers are what you get when you try to diagonalize a family of regular matrices. They form a nice algebraic stack, which means that one can define cohomological invariants of cameral covers by computing the cohomology ring of that stack. With rational coefficients this ring has a presentation in terms of hyperplane arrangements. My talk will be in the style of a "working seminar": I will explain what cameral covers are and try to make you like them, I will tell you what I know about their characteristic classes and the main ideas behind this computation, and then I will tell you where I am stuck.<br />
<br />
===Eric Ramos===<br />
Homological Invariants of FI-modules.<br />
<br />
An FI-module is a functor from the category FI, of finite sets and <br />
injections, to a module category. These objects were initially studied <br />
by Church, Ellenberg, and Farb in connection with the theory of <br />
representation stability. It was discovered that certain homological <br />
invariants of FI-modules could be used to solve problems in topology, <br />
algebraic geometry, and number theory. In this talk we will study <br />
these invariants using a new theory of depth. With this theory, we <br />
will be able to prove various results, including bounds on the <br />
regularity, as well as the stable ranges of FI-modules.<br />
<br />
<br />
===Jake Levinson===<br />
(Real) Schubert Calculus from Marked Points on P^1<br />
<br />
I will describe a family S of Schubert problems on the Grassmannian, defined using flags osculating (tangent to) the rational normal curve at a chosen set of marked points.<br />
<br />
This family is very well-behaved (for example, it is Cohen-Macaulay), particularly when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) showed that the solutions are then "as real as possible", and Speyer (2014) extended the construction to stable curves, showing that the real locus of S is a smooth cover of the moduli space of real stable curves. Moreover, the monodromy of the cover has a remarkable description in terms of Young tableaux and Schützenberger's jeu de taquin.<br />
<br />
I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.</div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Algebraic_Geometry_Seminar_Fall_2015&diff=10633Algebraic Geometry Seminar Fall 20152015-11-04T20:32:35Z<p>Eramos: /* Fall 2015 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B223.<br />
<br />
The schedule for the previous semester is [https://www.math.wisc.edu/wiki/index.php/Algebraic_Geometry_Seminar_Spring_2015 here].<br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2015 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|September 18<br />
|[http://www.math.harvard.edu/~ebriedl/ Eric Riedl] (UIC)<br />
|Rational Curves on Hypersurfaces<br />
|Jordan<br />
|-<br />
|September 25<br />
|[http://www.mathcs.emory.edu/~dzb/ David Zureick-Brown] (Emory)<br />
|Hilbert schemes of canonically embedded curves of low genus<br />
|Jordan<br />
|-<br />
|October 2<br />
|[https://math.temple.edu/~vald/ Vasily Dolgushev] (Temple)<br />
|A manifestation of the Grothendieck-Teichmueller group in geometry<br />
|Andrei<br />
|-<br />
|October 9<br />
|[https://math.wisc.edu/~maxim/ Laurentiu Maxim] (Madison)<br />
|Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
|local<br />
|-<br />
|October 16<br />
|[http://www.math.wisc.edu/~dewey/ Ed Dewey] (Madison)<br />
|Characteristic Classes of Cameral Covers<br />
|local<br />
|-<br />
|October 23<br />
|[http://people.math.sc.edu/kassj/ Jesse Kass] (South Carolina)<br />
|How to count zeros arithmetically?<br />
|Melanie<br />
|-<br />
|November 6<br />
|[https://www.math.wisc.edu/~eramos/ Eric Ramos] (Wisconsin)<br />
|Homological Invariants of FI-modules<br />
|Daniel<br />
|-<br />
|November 13<br />
|[http://www-personal.umich.edu/~jakelev/ Jake Levinson] (Michigan)<br />
|(Real) Schubert Calculus from Marked Points on P^1<br />
|Daniel<br />
|-<br />
|November 20<br />
|[https://sites.google.com/site/hackenzheng/ Xudong Zheng] (UIC)<br />
|TBA<br />
|Daniel<br />
|-<br />
|December 4<br />
|[http://www.math.wisc.edu/~svs/ Steven Sam] (Wisconsin)<br />
|Ideals of bounded rank symmetric tensors<br />
|Local<br />
|-<br />
|December 11<br />
|[http://www.math.columbia.edu/~danhl/ Daniel Halpern-Leistner] (Columbia)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Spring 2016 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 29<br />
|[http://www.math.wisc.edu/~yangjay/ Jay Yang] (Wisconsin)<br />
|Random Toric Surfaces<br />
|Local<br />
|-<br />
|March 4<br />
|[http://www3.nd.edu/~craicu/ Claudiu Raicu] (Notre Dame)<br />
|TBA<br />
|Steven<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
<br />
===Eric Riedl===<br />
Rational Curves on Hypersurfaces<br />
<br />
One way to understand the geometry of a variety is to understand its rational curves. Even for some relatively simple varieties, little is known about their spaces of rational curves. Many people have made previous progress on these questions, but there remain many open cases. In joint work with David Yang, we investigate the dimensions of the spaces of rational curves on very general hypersurfaces, and prove that for n > d+1 or d > (3n+1)/2, the spaces of rational curves have the expected dimension, as conjectured (in various cases) by several people, including Coskun, Harris and Starr, and Voisin. In this talk, we focus our attention particularly on the Fano case and try to motivate some of the ideas used to attack this problem.<br />
<br />
===David Zureick-Brown===<br />
Hilbert schemes of canonically embedded curves of low genus<br />
<br />
I'll discuss new work (joint with Aaron Landesman) on smoothability of low genus curves.<br />
<br />
===Jess Kass===<br />
How to count zeros arithmetically?<br />
<br />
A celebrated result of Eisenbud--Kimshaishvili--Levine computes the local Brouwer degree of a real polynomial function at an isolated zero as the signature of a quadratic form. I will discuss a parallel result in A1-homotopy theory, and time permitting, explain how to study a singularity by applying these results to the gradient of a defining equation. This is joint work with Kirsten Wickelgren.<br />
<br />
===Vasily Dolgushev===<br />
A manifestation of the Grothendieck-Teichmueller group in geometry<br />
<br />
Inspired by Grothendieck’s lego-game, Vladimir Drinfeld introduced, in<br />
1990, the Grothendieck-Teichmueller group GRT. This group has<br />
interesting links to the absolute Galois group of rationals, moduli of<br />
algebraic curves, solutions of the Kashiwara-Vergne problem, and<br />
theory of motives. My talk will be devoted to the manifestation of GRT<br />
in the extended moduli of algebraic varieties, which was conjectured<br />
by Maxim Kontsevich in 1999. My talk is partially based on the joint<br />
paper with Chris Rogers and Thomas Willwacher: http://arxiv.org/abs/1211.4230.<br />
<br />
===Laurentiu Maxim===<br />
Equivariant invariants of external and symmetric products of quasi-projective varieties<br />
<br />
I will start by revisiting formulae for the generating series of genera of symmetric products (with suitable coefficients), <br />
which hold for complex quasi-projective varieties with any kind of singularities, and which include many of the classical <br />
results in the literature as special cases. Important specializations of these results include generating series for <br />
extensions of Hodge numbers and Hirzebruch genus to the singular setting and, in particular, generating series <br />
for intersection cohomology Hodge numbers and Goresky-MacPherson intersection cohomology signatures <br />
of symmetric products of complex projective varieties. In the second part of the talk, I will describe a generating series formula<br />
for equivariant invariants of external products, which includes all of the above-mentioned results as special cases.<br />
This is joint work with Joerg Schuermann.<br />
<br />
===Ed Dewey===<br />
Characteristic Classes of Cameral Covers<br />
<br />
Cameral covers are what you get when you try to diagonalize a family of regular matrices. They form a nice algebraic stack, which means that one can define cohomological invariants of cameral covers by computing the cohomology ring of that stack. With rational coefficients this ring has a presentation in terms of hyperplane arrangements. My talk will be in the style of a "working seminar": I will explain what cameral covers are and try to make you like them, I will tell you what I know about their characteristic classes and the main ideas behind this computation, and then I will tell you where I am stuck.<br />
<br />
<br />
===Jake Levinson===<br />
(Real) Schubert Calculus from Marked Points on P^1<br />
<br />
I will describe a family S of Schubert problems on the Grassmannian, defined using flags osculating (tangent to) the rational normal curve at a chosen set of marked points.<br />
<br />
This family is very well-behaved (for example, it is Cohen-Macaulay), particularly when the chosen points are all real. For zero-dimensional Schubert problems, work of Mukhin-Tarasov-Varchenko (2007) showed that the solutions are then "as real as possible", and Speyer (2014) extended the construction to stable curves, showing that the real locus of S is a smooth cover of the moduli space of real stable curves. Moreover, the monodromy of the cover has a remarkable description in terms of Young tableaux and Schützenberger's jeu de taquin.<br />
<br />
I will give analogous results on real one-dimensional Schubert problems. In this case, S is a family of curves, whose real points turn out to be smooth, and whose real geometry is described by orbits of tableau promotion and a related operation involving evacuation.</div>Eramoshttps://wiki.math.wisc.edu/index.php?title=742&diff=92697422015-01-29T05:30:54Z<p>Eramos: </p>
<hr />
<div>'''Math 742'''<br />
<br />
Commutative Algebra and Galois Theory<br />
<br />
MWF 11-11:50, Van Vleck B129<br />
<br />
Prof: [http://www.math.wisc.edu/~andreic Andrei Caldararu]. Office hours: Wednesday 2:30-4:00pm, room VV 605.<br />
<br />
Grader: [http://www.math.wisc.edu/~eramos Eric Ramos]. Office hours: Monday 12:00-1:00pm VV 416, or by appointment.<br />
<br />
Late homework may be given directly to the grader, along with either (i) the instructor's permission, or (ii) a polite request for mercy.<br />
<br />
This course, the second semester of the introductory graduate sequence in algebra, will cover the basic aspects of commutative ring theory and Galois theory. The textbook we'll use for the Commutative Algebra portion will be Atiyah-Macdonald "Commutative Algebra". For Galois Theory I plan on using "Fields and Galois Theory" by J.S. Milne which can be found [http://www.jmilne.org/math/CourseNotes/ft.html here].<br />
<br />
For Galois theory you may also look at Emil Artin's notes which are available [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndml/1175197041 here.] <br />
<br />
==SYLLABUS==<br />
<br />
In this space we will record the theorems and definitions we covered each week, which we can use as a list of notions you should be prepared to answer questions about on the Algebra qualifying exam. The material covered on the homework is also an excellent guide to the scope of the course. <br />
<br />
'''WEEKS 0.5-1''':<br />
<br />
Category theory: the notions of category, functor, natural transformation. Full, faithful, essentially surjective functors. Examples. Groupoid. Categories enhanced in (abelian groups, vector spaces, topological spaces). Monomorphism, epimorphism.<br />
<br />
'''WEEKS 1-2''': <br />
<br />
Commutative rings and homomorphisms, integral domains, fields. Ideals, prime and maximal. Existence of maximal ideals. Local rings. Some geometric pictures (Spec and Specm). Nilradical. The nilradical is the intersection of all primes. Relatively prime ideals, Chinese remainder theorem. Extension and contraction, pictures of what happens for the inclusion Z -> Z[i]<br />
<br />
'''WEEK 3''':<br />
<br />
Modules, module homomorphisms. Module Hom, covariance and contravariance. Finitely generated modules, Nakayama's lemma and variants. Short exact sequences. Tensor product and exactness properties.<br />
<br />
'''WEEK 4''':<br />
<br />
An A-B-bimodule M induces a functor Mod-A -> Mod-B given by -- \otimes_A M. The equivalence of categories Mod-k and Mod-M_n(k). Restriction and extension of scalars. Algebras. Tensor product of algebras, the case of C \otimes_R C. Exactness of tensor products, flatness.<br />
<br />
'''WEEK 5'''<br />
<br />
Simple rings. Structure of finite dimensional simple rings over a field. Brauer group, computation of Br(R) = Z/2Z. Semisimple modules and rings. Wedderburn and Artin-Wedderburn theorems. Maschke's theorem on semisimplicity of k[G] for a finite ring. Tensor product of algebras and interpretation as fiber product of affine schemes. Localization (definition and basic properties).<br />
<br />
'''WEEK 6'''<br />
<br />
Rings and modules of fractions. Definitions and universal properties. Examples A_f, A_p. Modules of fractions. The operation S^{-1} is exact, corresponds to tensoring with S^{-1}A, so S^{-1}A is flat. Local properties, examples. Ideals in rings of fractions, in particular primes in A_p are primes in A which are contained in p.<br />
<br />
'''WEEK 7'''<br />
<br />
Integral dependence and valuations. Various characterizations of integral dependence, integral closure forms a ring, which is integrally closed. Relations to rings of fractions. Going up and going down. Valuation rings definition and basic properties. Existence. The integral closure of A is the intersection of all valuation rings which contain A. Nullstellensatz.<br />
<br />
Below you will find a repository of homework problems.<br />
<br />
==HOMEWORK 0 (not to be turned in)==<br />
<br />
1) Show that a fully faithful functor F: C -> D captures the property we called "essentially injective": if F(A) is isomorphic to F(B) for objects A, B of C, then A is isomorphic to B. (If you wanted to think of "functors which are injective on objects", and replaced equality with isomorphism, you'd get this notion of "essentially injective".)<br />
<br />
2) Prove that a natural transformation eta: F => G (where F, G are functors C -> D) such that eta_A is an isomorphism for every A in C, is a natural isomorphism. (A natural transformation is called a natural isomorphism if there exists another natural transformation mu: G => F such that eta o mu = id_G, mu o eta = id_F.) In this case we say that F and G are naturally isomorphic.<br />
<br />
3) Prove that a functor F that is fully faithful and essentially surjective is an equivalence, in the sense that there exists a functor G: D -> C and natural isomorphisms between F o G and id_D, and between G o F and id_C. (You will need to use the axiom of choice.)<br />
<br />
4) Consider the functor ** : Vect -> Vect which takes a vector space V to its double dual V^**. Show that it is isomorphic to the identity when restricted to the subcategory of finite dimensional vector spaces. (We already constructed a map of functors id => ** in class.) If V is not finite dimensional, can you characterize the image of the natural map V -> V^**?<br />
<br />
5) Fill in the blanks: "A category C, enhanced in abelian groups, with only one object, is the same things as a ...........". Same question, with "abelian groups" replaced by "k-vector spaces" for a fixed field k. <br />
<br />
==HOMEWORK 1 (due Feb 6)==<br />
<br />
Atiyah-Macdonald, page 10: 1, 2, 6, 10, 12, 15, 16, 17, 18, 21<br />
<br />
==HOMEWORK 2 (due Feb 14)==<br />
<br />
Atiyah-Macdonald, page 10: 19, 22, 26, 27, 28.<br />
<br />
==HOMEWORK 3 (due Mar 5)==<br />
<br />
Atiyah-Macdonald, page 31: 2, 3, 4, 8, 9, 10, 11 (second part is hard!), 12, 13, 14, 20. Bonus -- if you know about Tor -- do #24.<br />
<br />
==HOMEWORK 4 (due Mar 14)==<br />
<br />
Some non-commutative algebra exercises. You may find some references to read [http://stacks.math.columbia.edu/download/brauer.pdf here]. Or, even better, you can go and read Chapter I of the excellent book "Noncommutative Algebra" by Benson Farb and R. Keith Dennis. (On reserve in the library.) Most of these exercises are from this book.<br />
<br />
1) Let A, A' be k-algebras, with subalgebras B, B', respectively. If the centralizers of B, B' are C, C' (in A, A', resp.), then the centralizer of B\otimes_k B' in A\otimes_k A' is C\otimes C'.<br />
<br />
2) Find all the two-sided ideals of M_n(R), where R is a ring. Conclude that if D is a division algebra, M_n(D) is simple. (Hint: all two-sided ideals are of the form M_n(I) for some two-sided ideal I of R.)<br />
<br />
3) Let N be a submodule of the R-module M. If N and M/N are semisimple, does it follow that M is semisimple?<br />
<br />
4) Let M be a module such that every submodule is a direct summand. Show that M is semisimple as follows:<br />
<br />
(a) Show that every submodule of M inherits the property that every submodule is a direct summand.<br />
<br />
(b) Show that M contains a simple submodule: choose any finitely generated non-zero submodule M' of M. Let M" be a maximal submodule of M' such that M" is not equal to M' (why does it exist?). Then M'/M" is simple.<br />
<br />
(c) Let M_1 be the submodule of M generated by all simple submodules. Show that M_1 = M.<br />
<br />
5) Let A be a simple k-algebra with center k such that [A:k] = p^2 for a prime number p. Prove that A is either a division algebra or A is M_p(k).<br />
<br />
==HOMEWORK 5 (due Mar 28)==<br />
<br />
Atiyah-Macdonald page 31: 5, 6, 10; page 43: 1, 3, 5, 12, 13, 14<br />
<br />
==HOMEWORK 6 (due Apr 4)==<br />
<br />
Atiyah-Macdonald page 43: 15, 18, 19, 20, 21, 22, 25; page 67: 3, 9<br />
<br />
==HOMEWORK 7 (due Apr 11)==<br />
<br />
Atiyah-Macdonald page 67: 1, 2, 17, 28, 30, 31.<br />
<br />
Also prove that the integral closure of R = k[x,y]/(y^2-x^3) is isomorphic to k[t], as follows. Consider the map R -> k[t] given by x|->t^2, y|->t^3. Show that it is injective, so we can consider R as a subring of k[t]. Moreover, they have the same field of fractions. Now on one hand, k[t] is integrally closed, so the closure of R must be included in it. On the other hand, t is integral over R, so k[t] is contained in the integral closure of R.<br />
<br />
==HOMEWORK 8 (due Apr 18)==<br />
<br />
Atiyah-Macdonald page 84: 2, 4, 5, 7, 14, 26, 27<br />
<br />
==HOMEWORK 9 (due Apr 25)==<br />
<br />
Milne page 24: 1.1--1.4; page 31: 2.1--2.6<br />
<br />
==HOMEWORK 10 (due May 9)==<br />
<br />
1. ''Standard Facts about Finite Fields''<br />
<br />
Let p be a prime. Observe that if f(x) is an irreducible polynomial of degree d over F_p (the field with p elements), then (F_p)[x]/(f(x)) is a field with p^d elements, which we call F_(p^d).<br />
<br />
a. Show that x^(p^d) - x splits into a product of distinct linear factors over F_(p^d) by showing that every element of F_(p^d) is a root of this polynomial.<br />
<br />
b. Show that the splitting field of x^(p^d) - x is F_(p^d). Conclude that the field with p^d elements is unique up to isomorphism (i.e., calling it "the field" is justified).<br />
<br />
c. Let sigma be the Frobenius automorphism a->a^p. Prove that the Galois group of F_(p^d)/F_p is cyclic of degree d and is generated by sigma. Conclude that the subfields of F_(p^d) are the fields F_(p^k) where k divides d.<br />
<br />
d. Prove that x^(p^d) - x factors over F_p as the product of all the monic irreducible polynomials over F_p whose degree divides d. Use this to find the number of irreducible cubic polynomials over F_7.<br />
<br />
[Note: as all of these are "standard facts" you can likely look all of them up. Do this only after you have tried to prove them from scratch.]<br />
<br />
<br />
2. Let k be a field, f be a polynomial of degree n in k[x], and K be the splitting field of f over k.<br />
<br />
a. Show that [K:k] divides n!.<br />
<br />
b. Show that in order to have [K:k] = n!, it is necessary but not sufficient for f to be irreducible.<br />
<br />
<br />
3. Find the Galois group of x^7 - 2 explicitly as a permutation group on the roots.<br />
<br />
<br />
4. a. Show that the splitting field K of x^8-2 is Q(2^(1/8), zeta_8) where zeta_8 is a primitive eighth root of unity.<br />
<br />
b. Despite the facts that [Q(2^(1/8)) : Q] = 8 and [Q(zeta_8) : Q] = 4, prove that [K:Q] is actually 16, not 32. (Optional: also explain how you will avoid making similar mistakes in the future, if you have made them in the past.)<br />
<br />
c. Find generators for Gal(K/Q) and write explicitly their permutation action on the roots of x^8-2.<br />
<br />
<br />
5. Let E=k(alpha) where alpha is algebraic over k.<br />
<br />
a. If [E:k] is odd, prove that k(alpha^2) = k(alpha).<br />
<br />
b. Show more generally that k(alpha^2) = k(alpha) if and only if the minimal polynomial for alpha has an odd-degree term. (In other words, if it has a term b*x^c where b is nonzero and c is odd.)<br />
<br />
c. Is it true in general that if m and n are relatively prime and alpha is algebraic of degree m over k, then k(alpha) = k(alpha^n) ?<br />
<br />
<br />
6. Find the splitting fields and Galois groups of the following polynomials (if they exist):<br />
<br />
a. x^3 - 3 over Q.<br />
<br />
b. x^3 - x + 1 over Q.<br />
<br />
c. x^3 - 3 over Q(sqrt 3).<br />
<br />
d. x^3 - 3 over Q(sqrt(-3)).<br />
<br />
e. x^4 - 2 over Q.<br />
<br />
f. x^4 - 7 over Q.<br />
<br />
==HOMEWORK 11 (due May 9)==<br />
<br />
<br />
1. a. Find the splitting field K of x^4 - 4x^2 - 1 over Q.<br />
<br />
b. Show that Gal(K/Q) is (isomorphic to) the dihedral group of order 8.<br />
<br />
c. Find the 8 nontrivial subfields of K and say which 4 of them are Galois over Q.<br />
<br />
<br />
2. Prove or disprove: Every field of degree 4 over Q has a subfield of degree 2 over Q.<br />
<br />
<br />
3. Let K/F be an algebraic extension. We say that an element alpha in K is "abelian" if F[alpha] is a Galois extension of F and the Galois group Gal(F[alpha]/F) is abelian. Prove that the set of abelian elements of K is a field.<br />
<br />
<br />
4. a. Let F < K < L be a tower of field extensions with [L:F] finite, and let alpha be an element of L. If p(x) is the minimal polynomial of alpha over F, prove that K tensor_F F(alpha) is isomorphic to K[x]/p(x) as a K-algebra.<br />
<br />
b. Let K1 and K2 be two algebraic extensions of a field K contained in a field L of characteristic zero. Prove that the K-algebra K1 tensor_K K2 has no nonzero nilpotent elements.</div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2017&diff=6907Graduate Algebraic Geometry Seminar Fall 20172014-04-20T18:23:52Z<p>Eramos: /* April 30 */</p>
<hr />
<div>'''Wednesdays 4pm, Room - Van Vleck B219'''<br />
<br />
The purpose of this seminar is to have a talk on each week by a graduate student to help orient ourselves for the Algebraic Geometry Seminar talk on the following Friday. These talks should be aimed at beginning graduate students, and could try to explain some of the background, terminology, and ideas for the grown-up AG talk that week, or can be about whatever you have been thinking about recently.<br />
<br />
If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@lists.math.wisc.edu<br />
<br />
The list registration page is here: [https://lists.math.wisc.edu/listinfo/gags https://lists.math.wisc.edu/listinfo/gags]<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:clement@math.wisc.edu Nathan], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Spring 2014 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Ed Dewey<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Hitchin's System ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5 <br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| TBA ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12 <br />
| bgcolor="#C6D46E"| Nathan Clement<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Grothendieck's Theorem on V.B. on P^1 ]] <br />
|-<br />
| bgcolor="#E0E0E0"| February 19 <br />
| bgcolor="#C6D46E"| Yihe Dong<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Group Cohomology ]] <br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Serkan Sakar<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Lie Algebra Homology/Cohomology ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 5<br />
| bgcolor="#C6D46E"| Eric Ramos<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 5| Classification of Injective Modules Over Noetherian Rings ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 12<br />
| bgcolor="#C6D46E"| Marci Hablicsek <br />
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#March 12| Non-commutative resolutions and McKay-correspondence ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 19<br />
| bgcolor="#C6D46E"| Spring Break<br />
| bgcolor="#BCE2FE"| No Seminar <br />
|-<br />
| bgcolor="#E0E0E0"| March 26<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 26| Prep Talk for Kevin Tucker ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 2<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 2| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 9<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 9| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 16<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 16| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 23<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 23| Prep Talk for Charles Doran ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 30<br />
| bgcolor="#C6D46E"| Eric Ramos<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 30| GL-Equivariant Modules Over Polynomial Rings In Infinitely Many Variables ]] <br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ed Dewey'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Hitchin's System<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Nathan Clement'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Grothendieck's Theorem on Vector Bundles on P^1<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will begin by briefly discussing line bundles and vector bundles, primarily in the context of smooth curves. I will introduce Serre Duality and Riemann-Roch in this context. My target application is to give a proof of Grothendieck's Theorem on the decomposition of vector bundles on P^1. I intend this talk to be accessible to anyone who has taken one semester of Algebraic Geometry!<br />
|}<br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Serkan Sakar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lie Algebra Homology/Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== March 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Eric Ramos'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Classification of Injective Modules Over Noetherian Rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== March 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Marci Hablicsek'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-commutative resolutions and McKay-correspondence.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Classical method for resolving singularities is to somehow associate an ideal to the singular space. Non-commutative resolutions associate instead a non-commutative algebra to the space. In the talk, through explicit examples, I'll illustrate how to get quivers from singular spaces and how to find resolutions through the quivers.<br />
|} <br />
</center><br />
<br />
== March 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
|} <br />
</center><br />
== April 2 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
|} <br />
</center><br />
== April 9 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== April 16 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Andrew Bridy'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The Artin-Mazur zeta function of a rational map in positive characteristic".<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: TBA<br />
|} <br />
</center><br />
== April 23 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== April 30 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Eric Ramos'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: GL-Equivariant Modules Over Polynomial Rings In Infinitely Many Variables<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center></div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2017&diff=6906Graduate Algebraic Geometry Seminar Fall 20172014-04-20T18:23:28Z<p>Eramos: /* Spring 2014 */</p>
<hr />
<div>'''Wednesdays 4pm, Room - Van Vleck B219'''<br />
<br />
The purpose of this seminar is to have a talk on each week by a graduate student to help orient ourselves for the Algebraic Geometry Seminar talk on the following Friday. These talks should be aimed at beginning graduate students, and could try to explain some of the background, terminology, and ideas for the grown-up AG talk that week, or can be about whatever you have been thinking about recently.<br />
<br />
If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@lists.math.wisc.edu<br />
<br />
The list registration page is here: [https://lists.math.wisc.edu/listinfo/gags https://lists.math.wisc.edu/listinfo/gags]<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:clement@math.wisc.edu Nathan], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Spring 2014 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Ed Dewey<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Hitchin's System ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5 <br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| TBA ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12 <br />
| bgcolor="#C6D46E"| Nathan Clement<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Grothendieck's Theorem on V.B. on P^1 ]] <br />
|-<br />
| bgcolor="#E0E0E0"| February 19 <br />
| bgcolor="#C6D46E"| Yihe Dong<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Group Cohomology ]] <br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Serkan Sakar<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Lie Algebra Homology/Cohomology ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 5<br />
| bgcolor="#C6D46E"| Eric Ramos<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 5| Classification of Injective Modules Over Noetherian Rings ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 12<br />
| bgcolor="#C6D46E"| Marci Hablicsek <br />
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#March 12| Non-commutative resolutions and McKay-correspondence ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 19<br />
| bgcolor="#C6D46E"| Spring Break<br />
| bgcolor="#BCE2FE"| No Seminar <br />
|-<br />
| bgcolor="#E0E0E0"| March 26<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 26| Prep Talk for Kevin Tucker ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 2<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 2| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 9<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 9| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 16<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 16| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 23<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 23| Prep Talk for Charles Doran ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 30<br />
| bgcolor="#C6D46E"| Eric Ramos<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 30| GL-Equivariant Modules Over Polynomial Rings In Infinitely Many Variables ]] <br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ed Dewey'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Hitchin's System<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Nathan Clement'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Grothendieck's Theorem on Vector Bundles on P^1<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will begin by briefly discussing line bundles and vector bundles, primarily in the context of smooth curves. I will introduce Serre Duality and Riemann-Roch in this context. My target application is to give a proof of Grothendieck's Theorem on the decomposition of vector bundles on P^1. I intend this talk to be accessible to anyone who has taken one semester of Algebraic Geometry!<br />
|}<br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Serkan Sakar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lie Algebra Homology/Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== March 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Eric Ramos'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Classification of Injective Modules Over Noetherian Rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== March 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Marci Hablicsek'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-commutative resolutions and McKay-correspondence.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Classical method for resolving singularities is to somehow associate an ideal to the singular space. Non-commutative resolutions associate instead a non-commutative algebra to the space. In the talk, through explicit examples, I'll illustrate how to get quivers from singular spaces and how to find resolutions through the quivers.<br />
|} <br />
</center><br />
<br />
== March 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
|} <br />
</center><br />
== April 2 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
|} <br />
</center><br />
== April 9 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== April 16 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Andrew Bridy'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The Artin-Mazur zeta function of a rational map in positive characteristic".<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: TBA<br />
|} <br />
</center><br />
== April 23 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== April 30 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center></div>Eramoshttps://wiki.math.wisc.edu/index.php?title=Graduate_Algebraic_Geometry_Seminar_Fall_2017&diff=6715Graduate Algebraic Geometry Seminar Fall 20172014-02-26T01:37:52Z<p>Eramos: </p>
<hr />
<div>'''Wednesdays 4pm, Room - Van Vleck B219'''<br />
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The purpose of this seminar is to have a talk on each week by a graduate student to help orient ourselves for the Algebraic Geometry Seminar talk on the following Friday. These talks should be aimed at beginning graduate students, and could try to explain some of the background, terminology, and ideas for the grown-up AG talk that week, or can be about whatever you have been thinking about recently.<br />
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If you want to get emails regarding time, place, and talk topics (which are often assigned quite last minute) add yourself to the gags mailing list: gags@lists.math.wisc.edu<br />
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The list registration page is here: [https://lists.math.wisc.edu/listinfo/gags https://lists.math.wisc.edu/listinfo/gags]<br />
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== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:clement@math.wisc.edu Nathan], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
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== Spring 2014 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Ed Dewey<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Hitchin's System ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5 <br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| TBA ]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12 <br />
| bgcolor="#C6D46E"| Nathan Clement<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Grothendieck's Theorem on V.B. on P^1 ]] <br />
|-<br />
| bgcolor="#E0E0E0"| February 19 <br />
| bgcolor="#C6D46E"| Yihe Dong<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Group Cohomology ]] <br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Serkan Sakar<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Lie Algebra Homology/Cohomology ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 5<br />
| bgcolor="#C6D46E"| Eric Ramos<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 5| Classification of Injective Modules Over Noetherian Rings ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 12<br />
| bgcolor="#C6D46E"| Marci Hablicsek <br />
| bgcolor="#BCE2FE"| [[Graduate Algebraic Geometry Seminar#March 12| Non-commutative resolutions and McKay-correspondence ]] <br />
|-<br />
| bgcolor="#E0E0E0"| March 19<br />
| bgcolor="#C6D46E"| Spring Break<br />
| bgcolor="#BCE2FE"| No Seminar <br />
|-<br />
| bgcolor="#E0E0E0"| March 26<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 26| Prep Talk for Kevin Tucker ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 2<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 2| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 9<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 9| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 16<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 16| TBA ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 23<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 23| Prep Talk for Charles Doran ]] <br />
|-<br />
| bgcolor="#E0E0E0"| April 30<br />
| bgcolor="#C6D46E"| TBA<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 30| TBA ]] <br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ed Dewey'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Hitchin's System<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Nathan Clement'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Grothendieck's Theorem on Vector Bundles on P^1<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: I will begin by briefly discussing line bundles and vector bundles, primarily in the context of smooth curves. I will introduce Serre Duality and Riemann-Roch in this context. My target application is to give a proof of Grothendieck's Theorem on the decomposition of vector bundles on P^1. I intend this talk to be accessible to anyone who has taken one semester of Algebraic Geometry!<br />
|}<br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Serkan Sakar'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lie Algebra Homology/Cohomology<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== March 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Eric Ramos'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Classification of Injective Modules Over Noetherian Rings<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== March 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Marci Hablicsek'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Non-commutative resolutions and McKay-correspondence.<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: Classical method for resolving singularities is to somehow associate an ideal to the singular space. Non-commutative resolutions associate instead a non-commutative algebra to the space. In the talk, through explicit examples, I'll illustrate how to get quivers from singular spaces and how to find resolutions through the quivers.<br />
|} <br />
</center><br />
<br />
== March 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
|} <br />
</center><br />
== April 2 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: <br />
|} <br />
</center><br />
== April 9 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== April 16 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Andrew Bridy'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: The Artin-Mazur zeta function of a rational map in positive characteristic".<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract: TBA<br />
|} <br />
</center><br />
== April 23 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center><br />
== April 30 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''TBA'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title:<br />
|-<br />
| bgcolor="#BCD2EE" | <br />
Abstract:<br />
|} <br />
</center></div>Eramos