# Seminars & Events for 2012-2013

##### The Yang-Mills Equations over Klein Surfaces

(This talk is aimed at graduate students and postdocs.) In "The Yang-Mills equations over Riemann surfaces," Atiyah and Bott studied the Yang-Mills functional over a Riemann surface from the point of view of Morse theory, and derived results on the topology of the moduli space of algebraic bundles over a complex algebraic curve.

##### Long time existence of minimizing movement solutions of Calabi flow

In 1982 Calabi proposed studying gradient flow of the L^2 norm of the scalar curvature (now called Calabi flow) as a tool for finding canonical metrics within a given Kahler class. The main motivating conjecture behind this flow (due to Calabi-Chen) asserts the smooth long time existence of this flow with arbitrary initial data.

##### A resolution of the Yang-Mills-Dirichlet Problem in super-critical dimensions'

In the early 2000 a series of geometric works, by Donaldson-Thomas and Tian in particular, have stimulated the analysis of Yang-Mills fields in dimension larger than the conformal one. We shall recall the main ingredients, developed mostly in the 80's, for the study of Yang-Mills Fields in critical and sub-critical dimensions. PLEASE CLICK ON SUMMARY TITLE FOR COMPLETE ABSTRACT.

##### Open-closed Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds

We study open-closed orbifold Gromov-Witten invariants of toric Calabi-Yau 3-orbifolds with respect to Lagrangian branes of Aganagic-Vafa type. We prove an open mirror theorem which expresses

generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curves. This is a joint work with Bohan Fang and Hsian-Hua Tseng.

##### An integrability theorem for harmonic maps of interest in General Relativity

Einstein's Equations have been extensively studied in the context of integrability for several decades, drawing on results from inverse scattering to algebraic curves. In this talk, we will give a generalized notion of integrability for axially symmetric harmonic maps into symmetric spaces and prove that under some mild restrictions, all such maps are integrable.

##### Controlled Active Vision/Image Processing with Applications to Medical Image Computing

In this talk, we will describe some theory and practice of controlled active vision. The applications range from visual tracking (e.g., laser tracking in turbulence, flying in formation of UAVs, etc.), nanoparticle flow control, and sedation control in the intensive care unit. Our emphasis will be on the medical side, especially image guided therapy and surgery. PLEASE CLICK ON SEMINAR TITLE FOR FULL ABSTRACT.

##### (1) Quantum Beauty; (2) Beauty in Mathematics

**1)** ** **Does the world embody beautiful ideas? That is a question that people have thought about for a long time. Pythagoras and Plato intuited that the world should embody beautiful ideas; Newton and Maxwell demonstrated how the world could embody beautiful ideas, in specific impressive cases.

##### Atoms, Nuclei, and 3d Triangulations

Based on the work of Durhuus-Jonsson and Benedetti-Ziegler, we revisit the question of the number of triangulations of the 3-ball. We introduce a notion of nucleus (a triangulation of the 3-ball without internal nodes, and with each internal face having at most 1 external edge). We show that every triangulation can be built from trees of nuclei.

##### The holomorphic height pairing

In joint work with Mirel Caibar we show that the Beilinson-Bloch, Gillet-Soulé height pairing between algebraic (n-1)-cycles on a (2n-1)-dimensional complex projective manifold X is the imaginary part of a natural (multivalued) complex quantity that varies holomorphically on components of the Hilbert scheme of X.

##### Categorification at a prime root of unity

Quantization parameter q becomes a grading shift after categorification. When q specializes to a root of unity, categorification becomes more subtle. We'll discuss an approach to categorification at a prime root of unity p via p-differentials in characteristic p.

##### Honeycombs

We will discuss Horn's conjecture and a visually appealing way to understand it.

##### The Strange World of Quantum Computing

This talk will give an introductory overview of quantum computing in an intuitive and conceptual fashion. No prior knowledge of quantum mechanics will be assumed. This talk is intended to be a preamble to my next talk on quantum knots.

##### Local Global Principles for Galois Cohomology

We consider Galois cohomology groups over function fields $F$ of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for $n=3$, we show that local-global principles hold for $H^n(F, {\mathbb Z}/m{\mathbb Z} (n-1))$ for all $n>1$. In the case $n=1$, a local-global principle need not hold.

##### Quantum Knots

In this talk, we show how to reconstruct knot theory in such a way that it is intimately related to quantum physics. In particular, we give a blueprint for creating a quantum system that has the dynamic behavior of a closed knotted piece of rope moving in 3-space. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Arnold Conjecture for Clifford Symplectic Pencils

A symplectic pencil is a linear family of symplectic forms, i.e., a linear space of two-forms, each of which, except of course the zero form, is symplectic. Symplectic pencils arise, for instance, from representations of Clifford algebras and can be thought of as an analogue of the symplectic structure in one interpretation of the least action principle with multi-dimensional time. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Derived categories and variation of GIT quotients

Semi-orthogonal decompositions of derived categories of coherent sheaves on varieties provide an efficient and flexible means of capturing deep geometric ties to between varieties. I will describe semi-orthogonal decompositions relating the derived categories of GIT quotients obtained via different linearizations of the action. |

##### Counting Rational Points on Cubic Surfaces

##### Abelian varieties with maximal Galois action on their torsion points

Associated to an abelian variety A/K is a Galois representation which describes the action of the absolute Galois group of K on the torsion points of A. In this talk, we shall describe how large the image of this representation can be (in terms of a number field K and the dimension of A). We achieve this by considering abelian varieties in families and then using a special variant of Hilber

##### Automorphic Levi-Sobolev Spaces, Boundary-Value Problems, and Self-Adjoint Operators

Application of Plancherel's theorem to integral kernels approximating compact period functionalsyields estimates on (global) automorphic Levi-Sobolev norms of the functionals. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Local existence and uniqueness of Prandtl equations

The Prandtl equations, which describe the boundary layer behavior of a viscous incompressible fluid near the physical wall, play an important role in the zero-viscosity limit of Navier-Stokes equations. In this talk we will discuss the local-in-time existence and uniqueness for the Prandtl equations in weighted Sobolev spaces under the Oleinik's monotonicity assumption.