**All the mathematicians in the Atlanta area, at all stages in their career are welcome to participate! The talks are intended for a general audience.**

**Contacts**

**Yulia Babenko**

Kennesaw State University

ybabenko@kennesaw.edu

**Galyna Livshyts**

Georgia Institute of Technology

glivshyts6@math.gatech.edu

** Location
Georgia Institute of Technology
Skiles bldg room 006
686 Cherry St NW, Atlanta, GA 30313
**

**Schedule for Spring 2019**

**Event 1**

**Monday, March 11, 11 am-12 pm
**Skiles 006

686 Cherry St NW, Atlanta, GA 30311

**11:00 am: **Mariana Montiel, Georgia State University

Title: Mathematics of musical structures

Abstract: Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found for activities such as theoretical analysis, composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics on Words, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself. This talk provides a panoramic introduction in which we will see how mathematics and music have influenced each other throughout history and, in particular, how modern research has contributed with insights relevant to both disciplines, as well as to others such as computer science or physics. We will see how music connects with different areas of mathematics, both pure and applied. The talk will include some simple yet powerful examples of how abstract mathematical concepts can come to life through music, with its melodic and rhythmic components.

** 11:25 am:** Break–sandwiches

** 11:35 am:** Shahaf Nitzan, Georgia Institute of Technology

Title: Exponential systems over sets with a finite measure

Abstract: A fundamental result in Harmonic Analysis states that many functions defined over the interval [-\pi,\pi] can be decomposed as sums of sines and cosines with integer frequencies. This allows one to describe very complicated functions in a simple way, and therefore provides with a strong tool to study the properties of different families of functions. However, the above decomposition does not hold — or holds but is not efficient enough– if the functions are no longer defined over an interval, (e.g. if a function is defined over a union of two disjoint intervals). We will discuss the question of whether similar decompositions are possible also in such cases, with the frequencies of the sines and cosines possibly no longer being integers.

### Event 2:

**Monday, April 15, 2019 11 am-12 pm
**Skiles 006

686 Cherry St NW, Atlanta, GA 30311

**11:00 am:** Break–sandwiches

**11:10 am: **Mireille Boutin, Purdue University

Title: Deriving Nutrition Information Using Mathematics: a focus on phenylalanine for PKU management

Abstract: Diet plays an important role in the management of diseases. A striking example is phenylketonuria (PKU), an inherited metabolic disease that leads to severe mental retardation if left untreated. However, a life-long low-phenylalanine diet started shortly after birth enables patients to develop normally and lead productive lives. A key factor in successful diet management is having readily accessible nutrition information. Traditionally, PKU patients have relied on food nutrient databases, such as the US Department of Agriculture’s National Nutrient Database for Standard Reference, to estimate the phenylalanine (Phe) in the food they consume. But what to do when encountering a food that is not listed in any database? For commercial foods, one solution is to use mathematical reasoning, along with the information printed on the food label, to compute an estimate for the desired nutrition data. In this talk, we will present some lemmas to estimate Phe in candy, as well as more general numerical optimization techniques to estimate Phe in commercial foods.

This work is in collaboration with Jieun Kim and Amruthavarshini Talikoti.

**Schedule for Spring 2018**

**January 17**: cancelled due to snow;

**January 31**: Olivia Beckwith, Emory University.

Title: Arithmetic in quadratic fields |

Abstract: Quadratic fields arise naturally in the study of primes and in solving Diophantine equations. Their arithmetic is in many ways similar to the familiar arithmetic of whole numbers, but an important difference is that in quadratic fields, integers cannot be factored uniquely into primes. Class numbers measure the degree to which unique factorization fails, and it turns out that consequences of deep knowledge about these numbers percolate through virtually every important question in number theory. We’ll look at some of the rich history in the study of class numbers and at important questions that are the subject of current research. |

**February 14**: Rachel Kuske, Georgia Institute of Technology.

Title: New patterns in stochastic PDE’s with Pyragas control |

Abstract: We provide a multiple time scales analysis for the Swift-Hohenberg equation with delayed feedback via Pyragas control, focusing on Turing bifurcations with and without additive noise. In the deterministic case, a Ginzburg-Landau-type modulation equation is derived that inherits Pyragas control terms from the original equation. The Eckhaus stability criteria is obtained for the rolls, with the delay driving the appearance of an intermediate time scale observed in the transients. In the stochastic context, slow modulation equations are derived for the amplitudes of the primary modes that are coupled to a fast Ornstein-Uhlenbeck-type equation with delay for the zero mode driven by the additive noise. By deriving an averaging approximation for the amplitude of the primary mode, we show how the interaction of noise and delay influences the existence and stability range for the noisy roll-type patterns. Furthermore, approximations for the spectral densities of the primary and zero modes show that oscillations on the intermediate times scale are sustained through the phenomenon of coherence resonance. These dynamics on the intermediate time scale are sustained through the interaction of noise and delay, in contrast to the deterministic context where dynamics on the intermediate times scale are transient. |

**February 28**: Alexandra Smirnova, Georgia State University.

Title: Disease forecasting by new regularized Broyden-type optimization algorithms |

Abstract: In this talk, we will outline a new approach to the development of iteratively regularized Broyden-type algorithms for solving nonlinear ill-posed inverse problems in either finite or infinite dimensional spaces. The novel regularization methods are designed to solve large-scale unstable least squares problems, where the Jacobian of a discretized nonlinear operator is difficult or even infeasible to compute. To face this challenge, a family of Gauss-Newton and Levenberg-Marquardt algorithms with the Fréchet derivative operator recalculated iteratively by using Broyden-type single rank updates is considered. To balance accuracy and stability, the pseudo-inverse for the derivative-free Jacobian is regularized in a problem-specific manner at every step of the iteration process. The study has been motivated by applications of inverse problems in epidemiology and infectious disease modeling, where stable estimation of key epidemiological parameters at the onset of an emerging virus is paramount in assisting public health authorities to rapidly assess the situation in order to determine whether the pathogen in question is capable of generating sustained local or global outbreaks. To illustrate theoretical findings, numerical simulations for both parameter estimation and forecasting of future incidence cases will be presented. |

**March 14**: Amalia Culiuc, Georgia Institute of Technology.

Title: Sparse domination principles for singular integral operators |

Abstract: The estimation of singular integrals, which are generally non-local and non-positive operators, by sparse forms, which by contrast are positive and localized, has recently become a leading trend in Calderon-Zygmund theory and beyond. In this talk we will explore the concept of sparse domination and its implications for weighted norm inequalities. In particular, we will discuss several sparse domination results, including the study of the bilinear Hilbert transform and rough homogeneous singular integrals. |

**March 28**: Kirsten Wickelgren, Georgia Institute of Technology.

Title: How to count lines on a cubic surface

Abstract: I’ll introduce some enumerative geometry, algebraic topology, and A1 or motivic algebraic topology, and then discuss a joint result with Jesse Kass giving a generalization to all fields of characteristic not 2 of the classical enumerative result that there are 27 lines on a smooth cubic surface defined over the complex numbers

**April 11**: Samantha Petti, Georgia Institute of Technology.

Title: Graph Approximation: Szemer\’edi’s regularity lemma, graph limits, and the ROC model

Abstract: How can we approximate sparse graphs and sequences of sparse graphs (with average degree unbounded and o(n))? We consider convergence in the first k moments of the graph spectrum (equivalent to the numbers of closed k-walks) appropriately normalized. We introduce a simple, easy to sample, random graph model that captures the limiting spectra of many sequences of interest, including the sequence of hypercube graphs. The Random Overlapping Communities (ROC) model is specified by a distribution on pairs (s,q), s∈ℤ+,q∈(0,1]. A graph on n vertices with average degree d is generated by repeatedly picking pairs (s,q) from the distribution, adding an Erd\H{o}s-R\'{e}nyi random graph of edge density q on a subset of vertices chosen by including each vertex with probability s/n, and repeating this process so that the expected degree is d. Our proof of convergence to a ROC random graph is based on the Stieltjes moment condition. We also show that the model is an effective approximation for individual graphs. For almost all possible triangle-to-edge and four-cycle-to-edge ratios, there exists a pair (s,q) such that the ROC model with this single community type produces graphs with both desired ratios, a property that cannot be achieved by stochastic block models of bounded description size. Moreover, ROC graphs exhibit an inverse relationship between degree and clustering coefficient, a characteristic of many real-world networks.

**Schedule for Fall 2018:**

**Event 1: **

**Friday, September 21, Skiles 006, **

**686 Cherry St NW, Atlanta, GA 30311**

** 11:30 am**: refreshments

** 12 pm**: Laura Eslava Fernandez, Georgia Institute of Technology.

Title: Hubs in random trees: Fate or fluke?

Abstract: In social networks, we usually find nodes that are directly connected to a large portion of the network’s nodes. These nodes are called hubs. How do such hubs arise? Where can we find them?

In this talk, we consider two probabilistic toy-models of random networks that generate random trees. These are the precedent for the preferential attachment models introduced by Albert and Barabasi in 1999. We discuss how the choice of randomness prompts stark differences in the behavior of hubs, even in seemingly simple models such as random trees.

** 12:50 pm**: refreshments, break

** 1:10 pm**: Megan Bernstein, Georgia Institute of Technology.

Title: Algebraic Voting Theory for Committee Voting and the Representation Theory of Wreath Products

**Event 2: **

**Friday, October 26, Skiles 006, **

**686 Cherry St NW, Atlanta, GA 30311**

** 12:15 pm**: sandwiches

** 12:30 pm**: Christine Heitsch, Georgia Institute of Technology.

Title:

From Plato to Pasteur and Beyond: the Combinatorics of RNA Viruses

Abstract:

The interface of mathematics and biology has many facets, distinguished

by both the biological applications and the mathematical motivations.

We discuss here the problem of RNA folding which lies at the intersection

of discrete mathematics and molecular biology. As we will illustrate,

new theorems in combinatorics are helping to answer the question, “Is

there a cure for the common cold?” (This short talk will be accessible

to undergraduates.)

** 1 pm**: refreshments, break

** 1:10 pm**: Victoria Powers, Emory University.

Title:

The Mathematics and Statistics of Gerrymandering

Abstract:

Gerrymandering refers to drawing political boundary lines with an ulterior motive,

such as helping one political party or group of voters. In the US there is a history of

manipulating the shapes of legislative districts in order to obtain a preferred outcome.

In recent years there have been a number of court cases in which the plaintiffs have

used mathematical or statistical ideas to attempt to convince the courts that

gerrymandering has occurred. In this talk we will look at some of these methods and

explain how mathematicians, statisticians, and computer scientists are helping in

the legal fight against gerrymandering. (this talk will be suitable for undergraduates).