# Fall 2018 Colloquia

### September 7, 2018

#### Zhengwei Liu, Harvard University

##### Quantum Fourier Analysis

**Abstract: **We first recall some classical inequalities and uncertainty principles in Fourier analysis. Then we discuss our recent work on Fourier analysis in various subjects, including subfactors, planar algebras, Kac algebras, locally compact quantum groups, modular tensor categories. Moreover, we provide a 2D picture language to study Fourier analysis. Finally, we discuss some applications and open questions.

### September 14, 2018

No colloquium this week.

### September 21, 2018

#### Christina Sormani, CUNY

**Abstract:** The spacelike universe is curved by gravity forming deep wells around massive objects. A black hole is formed when it is curved so strongly that a neck forms and the apparent horizon is the minimal sphere around that neck. The ADM mass of an asymptotically flat region in space is measured by the decay of the curvature near infinity. Shing-Tung Yau and Richard Schoen proved that in such spaces the ADM mass must be nonnegative, and if the ADM mass is 0 then the space is flat Euclidean space with no curvature at all. Here we present recent joint work with Dan Lee, Lan-Hsuan Huang, and Iva Stavrov proving that in special settings, spaces with small ADM mass are almost Euclidean space. All students who have completed vector calculus are welcome to attend.

### September 28, 2018

#### Anna Haensch, Duquesne University

##### 17 Facts About Science Writing That Will Totally Blow Your Mind

**Abstract:** Scientists are always doing research. Occasionally, they do something catchy and it gets covered by the mainstream media. I'm going to talk about how that science gets from the lab bench to the Twitter feed, and trace the evolution of facts as science becomes journalism and what gets lost and gained along the way. Next, I'll show you all the ways that math and science actually show up in mainstream journalism even when the stories have nothing to do with science! Finally, I'll make the case for scientific and numerical literacy as a necessary skill for understanding the news, promoting social justice and participating in the democratic process.

### October 5, 2018

#### Shing-tung Yau, Harvard University

##### Quasilocal Mass in General Relativity

**Abstract:** I will talk about the problem of defining conserved quantities in general relativity and explain their properties.

### October 12, 2018

*Student Presentations from the Directed Reading Program*

#### Eva Sachar (graduate student mentor: Casey Cavanaugh)

##### An Application of Clustering to Socioeconomic Data

**Abstract:** When analyzing socioeconomic data we wish to uncover its inherent and underlying structure. We will be presenting a few approaches to clustering and discussing their advantages and disadvantages when applied to a housing dataset, and see if the results of clustering on property characteristics and census block demographics accurately reflect tiering in housing prices.

#### Carter Silvey (graduate student mentor: Matthew Friedrichsen)

##### Fractal Geometry

**Abstract:** Fractals are some of the most beautiful and mysterious things to come out of mathematics. I’m going to discuss the geometry behind these fractals, such as how they are created and their dimensions. Specifically, I will talk about the Middle Third Cantor Set, Julia Sets, and the Mandelbrot Set as well as some applications that fractal geometry has in both the realm of mathematics and the real world.

### October 19, 2018

#### Michael Geline, Northern Illinois University

##### The conjectures of Brauer's block theory, and the role of integral representations

**Abstract:** Frobenius's local to global principle for finite groups asserts that properties of G, related to a prime p, should be controlled by analogous properties of normalizers of proper p-subgroups of G. Examples of properties of interest include the existence of normal subgroups with index p and the existence of irreducible representations with dimension divisible by a fixed power of p. Brauer, Alperin, and Broue have given quite a few specific conjectures of this nature which remain open to the present day, partly because no one knows whether to expect proofs to depend on the classification of finite simple groups. I will state several of these conjectures and summarize how they have influenced my work on p-adic representations.

### October 26, 2018

#### Dubi Kelmer, Boston College

##### Shrinking target problems, homogenous dynamics and Diophantine approximations

**Abstract:** The shrinking target problem for a dynamical system tries to answer the question of how fast can a sequence of targets shrink so that a typical orbit will keep hitting them indefinitely. I will describe some new and old results on this problem for flows on homogenous spaces, with various applications to problems in Diophantine approximations.

### November 2, 2018

#### Daryl DeFord, MIT/Tufts University

##### Matched Products and Stirling Numbers of Graphs

**Abstract:** In this talk I will introduce the matched product for graphs, motivated by a popular construction for modeling multiplex networks. The matched product depends on consistent labelings of the nodes in the component graphs and recovers the Cartesian, rooted, and hierarchical products as special cases. There are natural conditions for the product to be planar, Hamiltonian, and Eulerian in terms of the corresponding properties on the laers and we will also consider the related problem of computing the probability that a random relabeling of a given graph preserves each property. In addition to these traditional graph-theoretic properties, the matched product naturally defines several families of graphs whose Stirling numbers of the first kind can be enumerated in terms of the component values. We will see some explicit examples of these families with combinatorial proofs in terms of the Pell numbers and discuss a connection between this enumeration problem and gerrymandering.

### November 9, 2018

#### Jennifer Balakrishnan, Boston University

##### Rational points on the cursed curve

**Abstract: **How do we compute rational points on curves? I'll present a selection of techniques and motivating examples, from antiquity to modern times. One particularly interesting example is the split Cartan modular curve of level 13, also known as the "cursed curve," a genus 3 curve defined over the rationals. By Faltings' proof of Mordell's conjecture, we know that it has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points. We discuss how to determine rational points on this curve using "quadratic Chabauty," part of Kim's nonabelian Chabauty program. This is joint work with Netan Dogra, Steffen Mueller, Jan Tuitman, and Jan Vonk.

November 16, 2018

Bob Holt, University of Florida

Title: Niche conservatism, evolution, and applied ecology: Theoretical perspectives

Abstract: The Hutchinsonian niche of a species is defined to be that set of abiotic and biotic conditions allowing it to persist. Much of the diversity of life reflects evolution in species' niches. The evolutionary record reveals a spectrum of rates of change in species' and clade niches, from rapid niche evolution to profound niche conservatism. Understanding the determinants of evolution vs. conservatism in niches is of key importance in many vital applied arenas, ranging from controlling the evolution of resistance to pesticides and antibiotics, to facilitating evolutionary rescue in species facing extinction in changed environments. Theoretical studies of niche evolution with explicit demography and genetics in spatially and temporally heterogeneous environments can help illuminate when one might expect niche conservatism, vs. evolution. This talk will provide an overview of such studies. The mathematical topics touched upon will include how demographic stochasticity can alter niche quantification, the use of branching processes to illuminate niche evolution, and surprising effects that emerge from the interplay of spatial processes and temporal variability.

### November 30, 2018

#### Carolyn Abbott, UC Berkeley

##### Random walks on groups acting on hyperbolic spaces

**Abstract:** Imagine you are standing at the point 0 on a number line, and you take a step forward or a step backwards, each with probability 1/2. If you take a large number of steps, is it likely that you will end up back where you started? What if you are standing at a vertex of an 4-valent tree, and you take a step in each of the 4 possible directions with probability 1/4? This process is special case of what is called a random walk on a space. If the space you choose is the Cayley graph of a group (as these examples are), then a random walk allows you to choose a "random" or "generic" element of the group by taking a large number of steps and considering the label of the vertex where you end up. One can ask what properties a generic element of the group is likely to have: for example, is it likely that the element you land on has infinite order? In this talk, I will discuss the algebraic and geometric properties of generic elements of groups which act "nicely" on hyperbolic metric spaces, with a focus on how such elements interact with certain subgroups of the group. These results will apply to generic elements of hyperbolic groups, relatively hyperbolic groups, mapping class groups, many fundamental groups of 3—manifolds, the outer automorphism group of a free group of rank at least two, and CAT(0) groups with a rank one element, among many others. This is joint work with Michael Hull.

### December 7, 2018

#### Gianluca Caterina, Endicott College

##### The diagrammatic logic of C.S. Peirce: An approach via generic figures

**Abstract:** At the turn of the 20th century, the American philosopher and logician Charles Sanders Peirce developed a logical system based on diagrams ("existential graphs") that capture the essential features of what is currently know as first-order logic. We will present an introduction to Peirce's work in logic, along with a tentative model aimed to represent the existential graphs within a category-theory framework.

The talk will be accessible to undergraduates in both Mathematics and Philosophy.