# Mathematics Colloquium Fall 2018

For further information please call the Mathematics Department at 459-2969

**Tuesday, October 9th, 2018**

**Pedro Morales, University of California Santa Cruz**

**Summation of divergent series**

**Here I explore the notion of convergence of series and how it can be generalize to capture divergent series. This goes back to Euler and Leibniz and their work for series such as the harmonic series and Grandi's series. Another famous divergent series is the sum of positive integers, which has some applications in Quantum Field Theory and String Theory.**

**Tuesday, October 16th, 2018**

**David Stork**

**When computers look at art:****New rigorous approaches to the study of paintings and drawings****Mathematics and new rigorous computer algorithms have been used to shed light on a number of recent controversies in the study of art. For example, illumination estimation and shape-from-shading methods developed for robot vision and digital photograph forensics can reveal the accuracy and the working methods of masters such as Jan van Eyck and Caravaggio. Computer box-counting methods for estimating fractal dimension have been used in authentication studies of paintings attributed to Jackson Pollock. Computer wavelet analysis has been used for attribution of the contributors in Perugino's Holy Family and works of Vincent van Gogh. Computer methods can dewarp the images depicted in convex mirrors depicted in famous paintings such as Jan van Eyck's Arnolfini portrait to reveal new views into artists' studios and shed light on their working methods. New principled, rigorous methods for estimating perspective transformations outperform traditional and ad hoc methods and yield new insights into the working methods of Renaissance masters. Sophisticated computer graphics recreations of tableaus allow us to explore "what if" scenarios, and reveal the lighting and working methods of masters such as Caravaggio.**

**How do these computer methods work? What can computers reveal about images that even the best-trained connoisseurs, art historians and artist cannot? How much more powerful and revealing will these methods become? In short, how is the "hard humanities" field of mathematics and computer image analysis of art changing our understanding of paintings and drawings?**

**This profusely illustrate lecture for scholars interested in computer vision, pattern recognition and image analysis will include works by Jackson Pollock, Vincent van Gogh, Jan van Eyck, Hans Memling, Lorenzo Lotto, and several others. You may never see paintings the same way again.**

**Joint work with Antonio Criminisi, Andrey DelPozo, David Donoho, Marco Duarte, Micah Kimo Johnson, Dave Kale, Ashutosh Kulkarni, M. Dirk Robinson, Silvio Savarese, Morteza Shahram, Ron Spronk, Christopher W. Tyler, Yasuo Furuichi and Gabor Nagy**

**Tuesday, October 23rd, 2018**

**TBA**

**Tuesday, October 30th, 2018**

**Ovidiu Muntean, University of Connecticut**

**Analysis of the singularities of the Ricci flow in dimension four****Ricci flow is a geometric partial differential equation that has been very successful in understanding the structure of manifolds. It was the main tool in proving several big questions about three dimensional manifolds, like the Poincare conjecture and more recently the Smale and Anderson-Cheeger-Colding-Tian conjectures. These results rely on the classification of three dimensional singularities of the Ricci flow. ****In this talk, I will present some recent progress made in understanding the singularities of Ricci flow in dimension four. This is joint work with Jiaping Wang.**

**Tuesday, November 6th, 2018**

**Weiyong He, University of Oregon**

*Chen-Cheng’s breakthrough on scalar curvature type equations on compact Kahler manifolds and its extensions*

Recently Xiuxiong Chen and Jingrui Cheng have made a breakthrough on the existence of constant scalar curvature metrics on compact Kahler metrics, in view of Calabi-Donaldson program and Yau-Tian-Donaldson conjecture.

The essential new input is a highly nontrivial a priori estimates for scalar curvature type equation, which is a fully nonlinear fourth order elliptic PDE. We will discuss the exciting developments regarding the existence of constant scalar curvature metrics and their extensions.

**Tuesday, November 13th, 2018**

**Richard Taylor, Stanford University**

*Galois Groups and Locally Symmetric Spaces*

Langlands proposed an extraordinary correspondence between representations of Galois groups and automorphic forms, which has deep, and completely unexpected, implications for the study of both objects. The simplest special case is Gauss' law of quadratic reciprocity. In the so called `regular, self-dual' case much progress has been made in the roughly 45 years since Langlands made these conjectures. In this talk I will discuss recent progress in the regular, but non-self-dual case. In this case the automorphic forms in question can be realized as cohomology classes for arithmetic locally symmetric spaces, i.e quotients of symmetric spaces by discrete groups. Thus instead of the Langlands correspondence being a relationship between algebra and analysis, it can be thought of as a relationship between algebra and topology. This realization of the Langlands correspondence is in many ways more concrete. It also admits to generalizations not envisioned by Langlands, for instance relating mod p Galois representations with mod p cohomology classes.

In this talk I will describe the expected Langlands correspondence in the setting of locally symmetric spaces. I will try both to present the general picture and to give numerical examples. I will also describe recent theorems alluded to above. I will not attempt to describe the proofs.

**Tuesday, November 20th, 2018**

**Colin Guillarmou, Paris-Sud**

*The marked length spectrum of Anosov flows *

We review recent results on the following rigidity problem: do the length of closed geodesics for compact manifolds with negative curvature (or more generally with Anosov geodesic flow) determine the metric up to isometry?

**Tuesday, November 27th, 2018**

**Michael Hortmann, University of Bremen**

**Bitcoin, started in 2009, was the first "Cryptocurrency". There were predecessors, based on Chaum's blind signatures, but Bitcoin solved the "problem of double spending" for a decentralized system: there is no central bank that can inflate or deflate "money", for example. Mathematical tools in Bitcoin: Public Key Cryptography, in particular Elliptic Curve Digital Signatures; and Cryptographic hash functions, in particular SHA-256 and RIPEMD-160. In our talk we discuss these tools, as well as general questions concerning Bitcoin and other cryptocurrencies.**

**Tuesday, December 4th, 2018**

**Qiang Guang, University of California Santa Barbara**

*Compactness for minimal surfaces with free boundary*

Abstract: Given a compact manifold M with boundary, a hypersurface in M is called a free boundary minimal hypersurface ("FBMH") if it is minimal (i.e., mean curvature vanishes) and meets the boundary of M orthogonally. Such hypersurfaces arise naturally as critical points of the area functional in M, and they are just the mathematical models for soap films. The investigation of FBMHs dates back at least to Courant and Lewy in 1940s. We will first survey some recent developments for free boundary minimal surfaces. If we do not assume any boundary convexity of the ambient manifold, then the FBMH may be improper, i.e., the interior of the FBMH may touch the boundary of the ambient manifold. In this general setting, we will discuss the compactness for stable FBMHs, or more generally, FBMHs with bounded Morse index. One important ingredient of the compactness result is the curvature estimate for stable FBMHs.