Student Geometry/Topology Seminar

Schedule of Talks

Abstracts

Gallucci's Theorem and Mobius's Theorem each give an interesting result arising from the same configuration of points and lines in three dimensional space. From Mobius's Theorem one can deduce Cox's (d-3)rd Theorem, which gives a specific configuration of 2^(d-1) points and 2^(d-1) planes such that d of the planes pass through each point and d of the points lie on each plane. A special case gives a Euclidean analogue. If there is time, the following observation will lead to an alternative proof of Cox's Theorems: Given seven points in general position, there exists a unique eighth point, such that any quadric containing the first seven points must contain the eighth.

In this talk we'll walk through the proof the so-called Simple Loop ``Conjecture'': given a map between closed connected surfaces which is not pi_1--injective, there is a simple, non-contractible loop mapped to a homotopically trivial loop. I will follow the cute and concise proof by D. Gabai. There will be plenty of pretty pictures of branched covers.

TBA

In algebraic geometry, simple concepts often require intimidating machinery to study. Toric varieties, on the contrary, have an unmotivated definition (informally, a toric variety is a compactification of the algebraic torus) but the tools used in their study are very pleasant. Specifically, toric varieties are described by combinatorial objects called fans, and combinatorial properties of these fans translate into topological properties of the corresponding variety. I will discuss several of these correspondences. The talk will not be completely elementary; I will assume that the audience understands how a domain R gives rise to an algebraic variety Spec(R).

Historically there have been two methods of doing geometry, the analytic and the synthetic. The analytic approach is based on the use of numerical coordinates and the tools of calculus, while the synthetic approach is to deduce theorems of geometry by purely logical means from an initial body of axioms (such as in Euclidean geometry or Desargues's projective geometry). I will describe how, by suitably "enlarging" the category of smooth manifolds, we can dispense with calculus and do Differential Geometry on a purely synthetic (axiomatic) basis.

In this talk, I will try to explain a mathematical expression of topological conformal field theories and how it relates to moduli space of curves.

Enumerative geometry is the study of such natural and elementary problems as determining the number of curves of specified degree passing through a given collection of points in the plane. A beautiful recursion for these numbers was proved by Kontsevich in 1994, using techniques that resulted from a newfound synthesis between enumerative geometry and string theory. This unforeseen connection led more generally to the birth of Gromov-Witten invariants, certain integrals over a moduli space that not only carry enumerative significance, but also arise as the coefficients in the so-called quantum product on cohomology. In this talk, I will define stable maps and explore their moduli space, culminating with a computation that illustrates the idea behind (and the power of) Kontsevich's formula. While the talk will be self-contained, a follow-up talk by Nathan Priddis next week will continue the exposition of basic Gromov-Witten theory.

Last week Emily Clader introduced the moduli space of stable curves and the moduli space of stable maps, and showed us how they lead to the solution to a hard problem in enumerative geometry--the number of degree d curves through 3d-1 general points in the projective plane. It turns out Konstevich's formula for this number can also be derived from the quantum product which arises in Gromov-Witten theory. In this talk, we will introduce Gromov-Witten theory and quantum cohomology, and then show how Konstevich's formula can be derived using these powerful tools. This talk should still be accessible even to those who missed last week's talk.

The path integral formulation in quantum mechanics is a quantization method based on the Lagrangian of a the theory as a starting point rather than the Hamiltonian. I will begin this talk by discussing some of the basics of classical Hamiltonian and Lagrangian mechanics in order to provide the necessary background to introduce the path integral in quantum mechanics.

I will give a very accessible introduction to circle packing, present several nice results in the field, sketch some proof techniques, and finally give some easy to state open questions. This talk will focus on combinatorial circle packing. An example of the kind of question we might ask is, given a planar graph, is there a collection of closed disks in the plane, one for each vertex, so that disks are tangent if they are connected by an edge in the graph? We will not discuss Apollonian circle packings. The ideas and proofs presented will be broadly accessible, but of particular interest to combinatorialists, complex analysts, and anyone who likes (elementary) geometry.

About The Seminar

This seminar is organized by and run by University of Michigan graduate students. Past topics have come from many different areas of geometry and topology, such as:

• Twisted K-theory (algebraic topology)
• Morse theory (differential topology)
• Quasi-isometries of symmetric spaces (differential geometry)
• Growth of groups (geometric group theory)
• Fundamental group of a compact semisimple Lie group (Lie group theory)
• Mathematical physics (differential topology)

For more past topics, see the "past seminar" links on the sidebar. Talks can be a broad survey of a theory or a focused discussion of a single result (or anywhere in between... the space of possibilities is path connected).

For information, or to volunteer to speak on any topic you wish, contact:

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Last Updated:

January 14, 2011

Past seminar pages:

Fall 2010
Winter 2010
Fall 2009
Winter 2009
Fall 2008
Winter 2008
Fall 2007
Winter 2007
Fall 2006
Winter 2006
Fall 2005
Winter 2005
Fall 2004