UIUC Dept. of Mathematics Seminar Calendar

     August 2003           September 2003          October 2003    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
                 1  2       1  2  3  4  5  6             1  2  3  4
  3  4  5  6  7  8  9    7  8  9 10 11 12 13    5  6  7  8  9 10 11
 10 11 12 13 14 15 16   14 15 16 17 18 19 20   12 13 14 15 16 17 18
 17 18 19 20 21 22 23   21 22 23 24 25 26 27   19 20 21 22 23 24 25
 24 25 26 27 28 29 30   28 29 30               26 27 28 29 30 31   
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Tuesday, September 2, 2003

Graduate Student Commutative Algebra/Algebraic Geometry Seminar
2:00 pm in 241 Altgeld Hall, Tuesday, September 2, 2003

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Diana White (UIUC Math)
Gorenstein Dimension
Abstract: In the late 1960's, Auslander and Bridger introduced the Gorenstein dimension of a finitely generated module over a commutative noetherian ring. Gorenstein dimension, like projective dimension, is defined in terms of resolutions by a certain class of modules. While there are many other similarities between these two homological dimensions, there are also significant differences. For example, Gorenstein dimension reflects the property of the underlying ring being Gorenstein, whereas the projective dimension reflects the property of the underlying ring being regular. This talk focuses on the definition and basic properties of Gorenstein dimension, often comparing and contrasting with the projective dimension. It is designed to be a gentle introduction (starting from definitions and examples) and will be accessible to anyone who has had a first course in commutative algebra. It could also be useful background information for 4-6 other talks in the research-level commutative algebra seminar this semester.

Wednesday, September 3, 2003

Tuesday, September 9, 2003

Algebraic Geometry
4:00 pm in 441 Altgeld Hall, Tuesday, September 9, 2003

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Bruce Reznick (UIUC)
Hilbert's 8-Point Construction and Sums of Squares of Polynomials
Abstract: In 1888, Hilbert proved that there exist polynomials p(x,y) which take only non-negative values and cannot be written as a sum of squares of polynomials. This construction led directly to his 17th Problem, yet it was not presented in the literature in detail until the 1960's, and in a singular case which simplified the computations. In this talk, I will attempt to explain that Hilbert's construction is not as hard as this history would suggest. (The only non-trivial prerequisite is knowledge of Bezout's Theorem.) I will also briefly sketch the current state of knowledge about writing polynomials as sums of squares of polynomials.

Wednesday, September 10, 2003

Tuesday, September 16, 2003

Wednesday, September 17, 2003

Gromov-Witten Invariants and Related Topics. A Graduate Student Algebraic Geometry Seminar
4:00 pm in 241 Altgeld Hall, Wednesday, September 17, 2003

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David Murphy (UIUC)
Stable vector bundles
Abstract: As motivation for the Bridgeland paper, we will discuss the notion of (semi-)stability for vector bundles over a smooth connected algebraic curve of genus $g$. Stable vector bundles play the role of simple objects and we have a Jordan-Holder result stating that every algebraic vector bundle has an increasing filtration by vector sub-bundles whose successive quotients, which are uniquely determined by the original bundle, are all stable. See the seminar web site http://www.math.uiuc.edu/~jacox/ag.html for more information on this seminar.

Tuesday, September 23, 2003

Graduate Student Commutative Algebra/Algebraic Geometry Seminar
2:00 pm in 241 Altgeld Hall, Tuesday, September 23, 2003

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Jonathon Cox [email] (UIUC Math)
Why is O(1) called the twisting sheaf?
Abstract: The sheaf O(1) on a projective variety is of central importance in algebraic geometry. In Chapter II, Section 5 of Hartshorne's Algebraic Geometry, the sheaf O(1) is introduced as "the twisting sheaf of Serre." While "twisting" has an explicit algebraic meaning, a name like that should certainly have geometric motivation as well. To see this, we use the language of bundles as well as "translations" between it and the language of sheaves. I will briefly introduce the concepts of vector bundles and their transition functions. References for these topics can be found on my web site. I will assume various results from the above mentioned section of Hartshorne. In particular, I will assume the standard description of global sections of the sheaves O(n). This follows from the above mentioned section of Hartshorne, and is a good exercise to do beforehand if you haven't worked through it before.

Algebraic Geometry
4:00 pm in 441 Altgeld Hall, Tuesday, September 23, 2003

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Mihnea Popa (Harvard)
Asymptotic intersection theory and restricted volumes
Abstract: I will describe work (partly in progress) on defining asymptotic intersection numbers of big (or pseudoeffective) line bundles on smooth projective varieties. Intuition is provided by intersecting with the positive part of a Zariski decomposition, in case it exists. The technical tool is a notion of volume for restrictions of linear series. One shows that the asymptotic intersection numbers and the restricted volumes define continuous functions of the big cone of the ambient variety, and obtain an interesting decomposition of this cone given by their zero-locus. This is joint work with Ein, Lazarsfeld, Mustata and Nakamaye.

Wednesday, September 24, 2003

Gromov-Witten Invariants and Related Topics. A Graduate Student Algebraic Geometry Seminar
4:00 pm in 241 Altgeld Hall, Wednesday, September 24, 2003

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David Gepner (UIUC)
Stability Conditions on Triangulated Categories
Abstract: We will cover this paper by Tom Bridgeland. The motivation for defining stablility in triangulated categories comes from string theory. The definition here is in a special case a mathematical version of the \Pi-stability for branes defined by Douglas. The notion of t-structure on a triangulated category is generalized to that of a slicing. The Harder-Narasimhan property is also generalized to this category. The idea of a stability condition combines the notions of slicing and slope functions, which results in nice deformation properties. The central result is that the set Stab(T) of stability conditions on a triangulated category has a natural topology which in fact gives it the structure of a manifold. See the seminar web site http://www.math.uiuc.edu/~jacox/ag.html for more information on this seminar.

Tuesday, September 30, 2003

Graduate Student Commutative Algebra/Algebraic Geometry Seminar
2:00 pm in 241 Altgeld Hall, Tuesday, September 30, 2003

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Sean Sather-Wagtaff [email] (UIUC Math)
Homological dimensions and the Frobenius endomorphism
Abstract: For a local (commutative Noetherian) ring R, it is known that homological properties of the residue field k reflect ring theoretic properties of R. For instance, a theorem of Auslander, Buchsbaum, and Serre states that R is regular if and only if k has finite projective dimension over R. The Gorenstein dimension of Auslander and Bridger plays a similar role for the Gorenstein property as R is Gorenstein if and only if k has finite Gorenstein dimension.

When R contains a field of positive characteristic p, the Frobenius endomorphism is the ring endomorphism f: R -> R given by f(r) = r^p. A theorem of Kunz et. al. states that R is regular if and only if f has finite projective dimension, indicating that the homological properties of the residue field are like those of the Frobenius endomorphism. The analogous result for Gorenstein dimension was proved recently by the speaker and Iyengar: the ring R is Gorenstein if and only if f has finite Gorenstein dimension.

This lecture will serve as a gentle introduction to the results above and will include a review of notions not covered in Math 401--403. It should be considered as an introduction for the talks to be given in the Commutative Ring Theory Seminar, October 9 and 16.

Algebraic Geometry
4:00 pm in 441 Altgeld Hall, Tuesday, September 30, 2003

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Tom Nevins (U Michigan)
Solitons and many-body systems via algebraic surfaces
Abstract: The 1970s saw the discovery of a puzzling phenomenon in the theory of soliton equations, that the motion of poles of certain meromorphic solutions of the KP hierarchy and other soliton systems is governed by the dynamics of some integrable particle systems. I will describe joint work with D. Ben-Zvi that explains this phenomenon geometrically by means of a Fourier-Mukai transform for moduli spaces of vector bundles on certain noncommutative algebraic surfaces. In particular, we obtain extensions of results of Krichever, G. Wilson and others.

Wednesday, October 1, 2003

Tuesday, October 7, 2003

Wednesday, October 8, 2003

Gromov-Witten Invariants and Related Topics. A Graduate Student Algebraic Geometry Seminar
4:00 pm in 241 Altgeld Hall, Wednesday, October 8, 2003

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Jonathan Cox (UIUC)
Intersection theory on stacks and on their moduli spaces
Abstract: From an algebraic geometry viewpoint, Gromov-Witten invariants are degrees of intersections on moduli stacks of stable curves or stable maps. Thus for a rigorous treatment of the Gromov-Witten theory, we need to have a good understanding of how intersection theory works on stacks. We will summarize the first three sections of this paper by Vistoli, which develops intersection theory in the spirit of Fulton for Deligne-Mumford stacks. Time restrictions will again largely limit us to stating important concepts and results. See the seminar web site http://www.math.uiuc.edu/~jacox/ag.html for more information on this seminar.

Thursday, October 9, 2003

Commutative Ring Theory Seminar
3:00 pm in 243 Altgeld Hall, Thursday, October 9, 2003

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Sean Sather-Wagstaff [email] (UIUC)
Gorenstein dimension over local homomorphisms I
Abstract: I will present joint work with S. Iyengar in which we develop a theory of Gorenstein dimension over local homomorphisms which encompasses Auslander and Bridger's classical theory for finitely generated modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if some power of its Frobenius endomorphism has finite Gorenstein dimension.

An introduction to these ideas geared toward graduate students will be given in the Graduate Student Commutative Algebra/Algebraic Geometry Seminar on 30 Sepember.

Tuesday, October 14, 2003

Graduate Student Commutative Algebra/Algebraic Geometry Seminar
2:00 pm in 241 Altgeld Hall, Tuesday, October 14, 2003

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David Murphy [email] (UIUC Math)
A glimpse at invariant theory
Abstract: Like a phoenix, invariant theory has seemed to die and then rise again from its ashes several times to return to the forefront of mathematics. Each reincarnation has benefited from and contributed to commutative algebra and algebraic geometry, and thus this topic and its history should be of significant interest to us today. In this talk, we will trace some of this development and the key results obtained along the way (e.g., Hilbert's 14th Problem and Mumford's Conjecture) from both an algebraic and a geometric perspective.

Wednesday, October 15, 2003

Gromov-Witten Invariants and Related Topics. A Graduate Student Algebraic Geometry Seminar
4:00 pm in 241 Altgeld Hall, Wednesday, October 15, 2003

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Jonathan Cox (UIUC)
Cycle groups for Artin stacks
Abstract: In this paper, Andrew Kresch defines Chow groups of Artin stacks. He also shows how intersection theory with integer coefficients can be carried out on Deligne-Mumford stacks. So we will be expanding in two directions on the previous talks about intersection theory with rational coefficients on Deligne-Mumford stacks, as well as looking ahead toward intrinsic normal cones, which are Artin stacks. However, intersection theory on Artin stacks is much more subtle, and requires different techniques. See the seminar web site http://www.math.uiuc.edu/~jacox/ag.html for more information on this seminar.

Tuesday, October 21, 2003

Graduate Student Commutative Algebra/Algebraic Geometry Seminar
2:00 pm in 241 Altgeld Hall, Tuesday, October 21, 2003

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Jinjia Li [email] (UIUC Math)
Generalized intersection multipicities and the Frobenius endomorphism
Abstract: In his book "Local Algebra", Serre defined the intersection multiplicity of two modules over a regular local ring and made a conjecture about it. Part of that conjecture is still open. Generalized versions of that conjecture are to drop the regularity of the ring and assume one (or both) of the modules have finite projective dimension. I will introduce these conjectures and discuss the progress which has been made on them. I will also discuss some other related problems concerning the Frobenius endomorphism.

Wednesday, October 22, 2003

Gromov-Witten Invariants and Related Topics. A Graduate Student Algebraic Geometry Seminar
4:00 pm in 241 Altgeld Hall, Wednesday, October 22, 2003

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Josh Mullet (UIUC)
The intrinsic normal cone
Abstract: We will discuss this groundbreaking paper by Behrend and Fantechi. The intrinsic normal cone and the obstruction theory introduced here allow an algebraic construction of the virtual fundamental class of any moduli space of stable maps to a variety. As a result, Gromov-Witten invariants can be constructed for any genus and any target variety. (This last construction will be the subject of a future talk by Yong Fu.) See the seminar web site http://www.math.uiuc.edu/~jacox/ag.html for more information on this seminar.

Tuesday, October 28, 2003

Wednesday, October 29, 2003

Gromov-Witten Invariants and Related Topics. A Graduate Student Algebraic Geometry Seminar
4:00 pm in 241 Altgeld Hall, Wednesday, October 29, 2003

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Josh Mullet (UIUC)
The intrinsic normal cone (continued)
Abstract: We will discuss this groundbreaking paper by Behrend and Fantechi. The intrinsic normal cone and the obstruction theory introduced here allow an algebraic construction of the virtual fundamental class of any moduli space of stable maps to a variety. As a result, Gromov-Witten invariants can be constructed for any genus and any target variety. (This last construction will be the subject of a future talk by Yong Fu.) See the seminar web site http://www.math.uiuc.edu/~jacox/ag.html for more information on this seminar.