SCHEDULE 2005-2006

• Tuesday October 25th, 4:15pm - 622 Math “How efficiently do 3-manifolds bound 4-manifolds?” by Prof. Dylan Thurston.
• It is known since 1954 that every 3-manifold bounds a 4-manifold. Thus, for instance, every 3-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the 4-manifold. (By "complexity" of a manifold we mean the minimum number of simplices in a triangulation.) Given a 3-manifold M of complexity n, we show how to construct a 4-manifold bounded by M of complexity O(n^2). It is an open question whether this quadratic bound can be replaced by a linear bound. The natural setting for this result is shadow surfaces, a representation of 3- and 4-manifolds that generalizes many other representations of these manifolds; among other things, these shadow surfaces are the natural setting for computing many quantum invariants of links and 3-manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a 3-manifold. Our results can also be phrased in terms of the singularities of smooth maps. In particular, the minimum number of "crossing singularities" of a map from a hyperbolic 3-manifold to the plane is bounded below and above by the hyperbolic volume.

SCHEDULE 2004-2005

• April 19th, “Moduli and canonical metrics in geometry and physics” by Prof. Duong Phong.
• A surface with given topology can admit continuous families of distinct complex structures. These are called “moduli”, a terminology going back to Riemann. We discuss how moduli problems arise in many areas of geometry and physics. We also discuss related problems about modular forms and canonical metrics for moduli of more general geometric structures.
  

• April 5th, “Automorphic $L$-functions” by Prof. Herve Jacquet.
• Click here for abstract.
  

• Friday March 4th, 4 - 5 PM “Brownian Motion, a century after Einstein's Annus Mirabilis,” by Prof. Ioannis Karatzas.
• I will review the definition, physical origins and properties of Brownian Motion, mention some basic results about it, and try to illustrate its significance in applications on a couple of examples.

• Friday February 25th, 4 - 5 PM “Periods, Heights, and L-series” by Prof. Shou-Wu Zhang. 
• I will define the notion in the title, explain its role in number theory and geometry, and survey the related recent progress.

• February 15th, “Floer homology and algebraic geometry” by Prof. Michael Thaddeus.
• The Floer homology of a symplectic diffeomorphism (the original kind defined by Floer himself) is an inspiring but difficult thing to work with.  I will explain how it should be related to various fashionable notions in algebraic geometry, such as quantum cohomology, which are more tractable.  This suggests numerous lines of research in algebraic geometry.  The talk will consist entirely of definitions and conjectures.

• February 1st, “Using a heat-flow type equation to understand the topology of 3-manifolds,” by Prof. John Morgan.
• Closed, orientable surfaces are easy to understand -- they can be embedded in three-dimensional space, so that we can `see' them. Higher dimensional manifolds (even 3-dimensional manifolds) are not so directly accessible, and understanding them has proved extremely difficult. Surprisingly, the hardest to understand are manifolds of dimension 3 and 4. It has long been believed that special types of Riemannian metrics should exist on any 3-manifold, and if they do then they can be used to classify all 3-dimensional manifolds. Proving the existence of such metrics has been a goal for the last 25 years. Recently, following a program laid down by Richard Hamilton, Grisha Perelman has shown how to construct the required nice metrics. The idea is to use a particular evolution or flow equation, named by Hamilton the Ricci flow, to evolve the metric to a nice one. The intuition is that this equation is a non-linear version of the heat equation for metrics. Just as the heat equation smooths out the temperature distribution, so this equation should smooth out the metric.

• January 25th, “Some Sieves,” by Prof. Patrick Gallagher.
• An introduction to some of the upper bound sieves in analytic number theory: Selberg's Sieve, the Large Sieve, and the Larger Sieve. The emphasis will be on listing some of their applications, or giving a few of the general inequalities used in the proofs of these sieve upper bounds, or both.

• Monday, December 6th, “Heegaard diagrams and holomorphic disks,” by Prof. Peter Ozsvath, room 621 Math, 6-7 PM.
  
• Heegaard Floer homology is an invariant for three-dimensional manifolds which is defined using techniques adapted from symplectic geometry. I will describe some applications of this theory to topological questions.

• November 23rd, “Kummer's criterion, modular forms and Galois representations,” by Prof. Eric Urban.
• Kummer's criterion decides when a prime is regular. This notion has been introduce by Kummer in 1857 to prove many cases of Fermat's Last Theorem. The purpose of this lecture is to introduce the various tools that come into the proof of a refinement of this criterion and to its generalizations in modern number theory.

• November 16th, “Topics in Number Theory,” by Prof. Dorian Goldfeld.
• We will describe the Riemann Hypothesis and many startling equivalent formulations.
  

• October 12th, “Recognizing the Unknot,” by Prof. Joan Birman.
• Recognizing the unknot is one of the most intuitively clear, yet surprisingly non-trivial problems in algorithmic topology. In this talk we will review the state of the art, and explain our reasons for feeling that a polynomial solution exists and will be found soon.

• October 5th, “Combinatorial Commutative Algebra,” by Prof. Dave Bayer.
• Ezra Miller and Bernd Sturmfels have recently completed a textbook which is available online,
  
    Combinatorial Commutative Algebra
    Springer Graduate Texts in Math, to appear in late 2004
    http://www.math.umn.edu/~ezra/cca.html
  
  following in the tradition of Mel Hochster, Richard Stanley,  et. al. I have been active in this area; various chapters are based on my work with Bernd Sturmfels and coauthors.
  
  Monomial and binomial ideals can be used to various combinatorial problems, such as integer programming. The homological algebra of these ideals can be studied using techiques from elementary combinatorial topology; the free resolution of the twisted cubic curve, for example, can be pictured as a triangulated torus. I will give an introduction to these topics.