# CMU Mathematics Department Colloquia

## Spring 2017

### Organizer

• Department of Mathematics Colloquium Speaker Committee
• Because there might be a time delay in updating the webpage, please always check with the committee for the available dates.

### Meeting Times

Refreshments take place 30 minutes prior to the talk.
Typical Colloquium Talks are Thursday, 4:00–4:50pm, in Pearce 227. Refreshments are in Pearce 216.
There might be some talks scheduled on a different day and time. The following table gives the most accurate information for each event.

### Schedule

 Date Speaker Title (Scroll down for Abstract) Remark 1/31/2017 Tuesday Robert Megginson (University of Michigan) Special Colloquium in Mathematics Education "Alternative facts": Are we doing our part in arming our students to spot them, particularly when people are using mathematics to construct them? 4:00 -- 5:00 pm PE 227 2/23/2017 Department Meeting Department Meeting 2/28/2017 Tuesday Dmitry Zakharov (Courant Institute, NYU) The Tautological Ring of the Moduli Space of Riemann Surfaces 4:00 -- 5:00 pm PE 227 3/2/2017 Kasso Okoudjour (University of Maryland) On the HRT Conjecture 3/9/2017 Spring Recess Spring Recess 3/14/2017 Tuesday Dan Goldston (San Jose State University) Special Colloquium in Mathematics Differences Between Primes 4:00 -- 5:00 pm PE 227 3/23/2017 Department Meeting Department Meeting 3/30/2017 No Classes No Classes 4/6/2017 Kyunghee Lee (Recreation, Parks and Leisure Services, CEHS, CMU) Spatial Analysis and Its Applications 4/13/2017 Loredana Lanzani (Syracuse University) Harmonic Analysis Techniques in Several Complex Variables 4/20/2017 Award Ceremony Award Ceremony 4/22/2017 Saturday Cedric Villani (Universite de Lyon and Institut Henri Poincare) Rich Fleming General Public Lecture On Particles, Stars and Eternity (This talk is accessible to a wide audience with curious minds.) 10:30 am Bioscience 1010 4/22/2017 Saturday Cedric Villani (Universite de Lyon and Institut Henri Poincare) Rich Fleming Mathematics Lecture When Planification and Statistical Mechanics Meet in the non-Euclidean World (This talk is accessible to undergraduate students.) 3:00 pm Bioscience 1010 4/27/2017 Department Meeting Department Meeting

### Abstracts

Speaker: Robert Megginson (January 31)
Title: "Alternative facts": Are we doing our part in arming our students to spot them, particularly when people are using mathematics to construct them?
Abstract: "Mathematics will always be a key element of liberal education, since it promotes logical reasoning." Mathematicians have often made this claim, and I confess that I have done so myself. But with the flowering of "alternative facts", it is timely for the disciplines of mathematics and mathematics education to consider how well we are preparing our students to examine critically the reasoning, mathematical and otherwise, of those who are pushing alternative "truth" upon us. In this presentation I will describe a classification scheme for types of fallacious arguments that have been used to try to convince the public of the wisdom of a policy decision or the safety of a new product, and provide some examples. Several will have their roots in applications of mathematics to climate science, one of my interests. For those involved in teacher preparation or who are K-12 teachers themselves, some connections to the Common Core standards will be given.

Speaker: Dmitry Zakharov (February 28)
Title: The Tautological Ring of the Moduli Space of Riemann Surfaces
Abstract: The moduli space M_g of compact Riemann surfaces of genus g has been one of the most important objects of study in modern algebraic geometry. The intersection theory of M_g, however, remains far from being completely understood. In 1987, Mumford initiated the study of the tautological ring of M_g, which is the subring of the cohomology ring generated by a collection of natural geometric classes.
In recent years there has been remarkable progress in finding relations in the tautological ring of M_g and its compactifications. Most of these relations have been constructed using very modern methods having their ultimate origin in string theory. I will explain in detail one particular set of relations, which in contrast is obtained via a very classical construction: by considering the embedding of a Riemann surface in its Jacobian. I will also survey some of the other relations that have been proved, and the relationships between them.

Speaker: Kasso Okoudjou (March 2)
Title: On the HRT Conjecture
Abstract: Given a non-zero square integrable function g and S = {(a_k, b_k): k = 1,..., N} in a two-dimensional vector space over real numbers, let G(g, S) = {exp{2 \pi i b_k} g( . -- a_k): k = 1,..., N}. The Heil-Ramanathan-Topiwala (HRT) Conjecture is the question of whether G(g,S) is linearly independent. For the last two decades, very little progress has been made in setting the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture. I will then describe some recent developments in settling the conjecture in the special case where g is real-valued and N = 4.

Speaker: Dan Goldston (March 14)
Title: TBA
Abstract: An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common (i.e. frequently occurring) gap between consecutive primes up to $x$. In 1999, Odlyzko, Rubinstein and Wolf conjectured that the jumping champions greater than 1 are 4 and the primorials 2, 6, 30, 210, 2310,... A few years ago Goldston and Ledoan proved that an appropriate form of the Hardy-Littlewood conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and as $x$ increases the jumping champions increase through the sequence of primorials except for possible transistion ranges where two consecutive primorials compete with each other. On the other hand, absolutely nothing is known unconditionally about jumping champions. Numerically 6 is the jumping champion for $947 \le x \le 10^{15}$, and it is likely numerical calculations will never find a larger jumping champion than 6 - Odlyzko, Rubinstein, and Wolf conjectured 30 will first become the jumping champion at around $1.7427 \cdot 10^{35}$.
Consider now all the primes $\le x$ and form all the positive differences between these primes. We examine in this talk the question of finding the most common difference, which we call the Prime Difference Champion(s) (PDC). Just like for jumping champions, we believe the PDC's run through the primorials, and assuming a Hardy-Littlewood prime pair conjecture we can prove this for sufficiently large $x$. Numerical results for $x \le 2 \times 10^8$ show the PDC's running through the first 8 primorials as well as other interesting phenomena associated with prime differences. Unconditionally we prove that the PDCs go to infinity and further have asymptotically the same number of prime factors when counted logarithmically as the primorials.
We also discuss the total error in the Hardy-Littlewood prime pair conjecture for all differences up to $x$, and relate this to a conjecture of Vaughan for the $L^1$ norm of an exponential sum over primes.
This is joint work with Scott Funkhouser, Dipendra Sengupta, and Jharna Sengupta.

Speaker: Kyunghee Lee (April 6)
Title: Spatial Analysis and Its Applications
Abstract: The purpose of this presentation is twofold. The first purpose is to introduce a concept of spatial analysis and recently developed spatial regression modeling technique. The second purpose is to show how to manipulate geospatial big data with various public and private level of secondary data. The specific context and models of the previous case study will be used for definition of the method.
The primary purpose of spatial analysis is to provide sophisticated and detailed results of analysis while demonstrating spatial relationships and interactions. Spatial statistics were designed specifically for use with spatial data. Generally, spatial statistics have been used in past research because traditional statistics such as ordinary least squares (OLS) cannot control spatial relationships and interactions. Recently, GWR (geographically weighted regression) has begun to make its mark as an effective regional level analysis approach to detect varying relationships. For example, GWR, not OLS technique, should be used for regional scale spatial analysis because it is able to account for local effects and shows geographical variation in the strength of the relationship.
GWR models can capture the heterogenic nature of social changes by allowing the equation to alter over space to reflect the structure within the data. The typical output from a GWR model is a set of parameters that can be mapped in the geographic space to represent spatial heterogeneity (non-stationarity). This is the essence of GWR in a regional analysis. GWR allows variations in relationships over space to be measured within a single modeling framework.

Speaker: Loredana Lanzani (April 13)
Title: TBA
Abstract: This talk concerns the applications of relatively classical tools from real harmonic analysis (namely, the $T(1)$-theorem for spaces of homogenous type) to the novel context of several complex variables. Specifically, I will present recent joint work with E. M. Stein on the extension to higher dimension of Calder\'on's and Coifman-McIntosh-Meyer's seminal results about the Cauchy integral for a Lipschitz planar curve (interpreted as the boundary of a Lipschitz domain $D\subset\mathbb C$). From the point of view of complex analysis, a fundamental feature of the 1-dimensional Cauchy kernel: $$H(w, z) = \frac{1}{2\pi i}\frac{dw}{w-z}$$
is that it is holomorphic (that is, analytic) as a function of $z\in D$. In great contrast with the one-dimensional theory, in higher dimension there is no obvious holomorphic analogue of $H(w, z)$. This is because of geometric obstructions (the Levi problem) that in dimension 1 are irrelevant.
A good candidate kernel for the higher dimensional setting was first identified by Jean Leray in the context of a $C^\infty$-smooth, convex domain $D$: while these conditions on $D$ can be relaxed a bit, if the domain is less than $C^2$-smooth (never mind Lipschitz!) Leray's construction becomes conceptually problematic.
In this talk I will present (a), the construction of the Cauchy-Leray kernel and (b), the $L^p(bD)$-boundedness of the induced singular integral operator under the weakest currently known assumptions on the domain's regularity -- in the case of a planar domain these are akin to Lipschitz boundary, but in our higher-dimensional context the assumptions we make are in fact optimal. The proofs rely in a fundamental way on a suitably adapted version of the so-called $T(1)$-theorem technique'' from real harmonic analysis.
Time permitting, I will describe applications of this work to complex function theory -- specifically, to the Szeg\:o and Bergman projections (that is, the orthogonal projections of $L^2$ onto, respectively, the Hardy and Bergman spaces of holomorphic functions).