Primitive/Primitive knots are defined to be simple closed curves lying on a Heegaard surface of genus 2 in S3 with certain conditions. These knots admit lens space surgeries. Primitive/Seifert knots are defined in a similar way and they admit Seifert fibered space surgeries. Berge constructed 12 types of Primitive / Primitive knots. He also showed that the 12 types fall into three groups, which are called a Knot is Solid Tori, a Spine Of A Fiber, and a Special Seifert-Fibered Configuration. I will give the definition of these groups and explain to which the 12 types belong. I will show that there is an analog of this grouping for Primitive / Seifert knots and then I will explain how much larger the set of Primitive/Seifert knots is than Primitive/Primitive knots.

This talk is an introduction to symplectic manifolds and Hamiltonian dynamics. I will start by reviewing the notion of symplectic manifold, give some examples, and enumerate some of their basic properties. Then I will explain the classical structure theorems of Kostant, Atiyah, Guillemin-Sternberg and Delzant on Hamiltonian torus actions, and describe some new results on general symplectic torus actions. A Hamiltonian torus action may be viewed as a particular case of a completely integrable Hamiltonian system. In the second part of the talk I will introduce completely integrable Hamiltonian systems, focusing on the so called "semitoric systems". Semitoric systems lie somewhere in between the more rigid toric systems and the usually complicated completely integrable systems. Finally, I will briefly discuss the spectral theory of quantum semitoric systems, and the so called "Semitoric Spectral Conjecture". Parts of this talk are based on joint work with with Johannes J. Duistermaat and San Vu Ngoc.

Morse theory is a powerful and widespread tool in both geometry and topology, which gives rise to CW-structures and handle decompositions for smooth manifolds. But how unique are Morse functions and the handle decompositions which arise from them? The space of Morse functions is an open and dense subset of all functions, but is highly disconnected. Cerf theory teaches us that by adding functions with cubic singularities we can build a larger space of functions which is connected, and thereby learn how to pass between Morse functions (and hence handle decompositions) on manifolds. Adding still higher singularities yields a more highly connected space, giving us information about higher dimensional families of Morse functions, however this process is doomed to failure. In this talk I will discuss work by Soren Galatius proving that that K. Igusa's space of generalize framed Morse functions on any manifold is contractible. In addition to providing a robust family version of Morse Theory, this result greatly simplifies the Hopkins-Lurie proof of the Cobordism Hypothesis.

Resultants are a fundamental computational and theoretical tool in algebraic geometry and commutative algebra. We'll introduce the definitions and basic properties of the classical resultant, and look at some examples.

Note the new room for the seminar this semester.

A dual equivalence for an arbitrary collection of combinatorial objects endowed with a descent set is a relation for which equivalence classes group together terms according to the Schur expansion of the corresponding generating function. After outlining the definition of dual equivalence, we'll present three main applications: the Schur expansion of Macdonald polynomials, Schur positivity of k-Schur functions (joint with S. Billey), and a combinatorial rule for the Littlewood-Richardson coefficients of the Grassmannian in the special case of a Schubert polynomial times a Schur function (joint with N. Bergeron and F. Sottile). [The speaker is an Assistant Professor candidate.]

See http://www.ma.utexas.edu/seminar/Rosenthal-abs1-27-11.pdf for details.

In this talk, I will state some conjectures and examples concerning the central derivatives of automorphic L-series in terms of heights of special cycles on Shimura varieties.

The geometry of U(1) gerbes is intimately related to the concept of a Dirac structure. I will explain what this means, and give several examples of this idea. I will focus on the Dirac structure on a Lie group, which gives a beautiful explanation for the fusion of Q-Hamiltonian spaces invented by Alekseev, Malkin and Meinrenken. Finally, I will explain how this structure on the group may be obtained from the Hilbert manifold of connections on the circle, solving a puzzle which has vexed us for quite a number of years.

We will explore several problems involving counting.