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## Analysis and its Applications Seminar

#### A Liouville-Riemann-Roch theorem on abelian coverings

### Abstract

A generalization by Nadirashvili, and then Gromov and Shubin of the classical Riemann-Roch theorem describes the index of an elliptic operator on a compact manifold with a divisor of prescribed zeros and allowed singularities. On the other hand, Liouville type theorems count the number of solutions of a given polynomial growth of the LaplaceBeltrami (or more general elliptic) equation on a non-compact manifold. The solution of a 1975 Yau’s conjecture by Colding and Minicozzi implies in particular, that such dimensions are finite for Laplace-Beltrami equation on a nilpotent co-compact covering. In the case of an abelian covering, much more complete Liouville theorems (including exact formulas for dimensions) have been obtained by Kuchment and Pinchover. One wonders whether such results have a combined generalization that would allow for a divisor that ”includes the infinity.” Surprisingly, combining the two types of results turns out being non-trivial. The talk will present such a result obtained in a joint work with Peter Kuchment.## Quantitative Biology Colloquium

#### Using Mathematics to Understand Influenza: A Comparison of Within-Host Models

### Abstract

Due to the public health concern of influenza, it is important to have models that can describe the mechanisms working within the body and ultimately create/predict the outcomes of treatments. In this talk, we provide an overview of some influenza models and discuss their construction, strengths, and weaknesses.## Modeling and Computation Seminar

#### Algorithmic approaches to identification and antibiotic susceptibility testing of pathogenic microbes

### Abstract

In this talk, I'll first provide some context regarding the challenges of hospital-acquired infections, antibiotic resistance, and sepsis, a disease estimated to kill millions worldwide, and one of the leading causes of death in hospitalized patients. I'll then review traditional laboratory techniques for identifying pathogenic microbes and performing antibiotic susceptibility testing (AST), and how antibiotic susceptibility is reported in terms of the Minimum Inhibitory Concentration (MIC). For the remainder of the talk, I'll describe the technologies and algorithms used by the Accelerate Pheno system for rapid identification and AST. Identification involves automated fluorescent in-situ hybridization, or FISH, coupled with algorithmic analysis of microscopic images. For AST, the Pheno system extracts a number of features from time-lapse images of growing bacterial colonies exposed to antibiotics (e.g. morphology, division rates, and growth curves), which become the inputs to a suite of machine learning algorithms for determining MIC values.## Brown Bag Seminar

#### Computational Hyperspectral Imaging: Math, Methods, and Mayhem

### Abstract

Hyperspectral imaging has many applications, including food quality and safety, cancer screening, crop assessments in agriculture, and fake currency detection, though hyperspectral imaging systems can be excessively complex and expensive. To simplify these optical systems, computational methods are used to calculate hyperspectral data from specially encoded optical signals. We will break down the mathematical structuring of a patented optical encoding system, explore some deconvolution methods to calculate hyperspectral information from encoded signals, and discuss the implications of different real-world training data sets.## Applied Math Colloquium

#### Modeling the brain as a dynamical network

### Abstract

Seeking to understand how the brain processes visual information, my collaborators and I have built a model of the monkey visual cortex, which is quite similar to our own. We have focused on an early part of the visual pathway called the primary visual cortex (V1), and have modeled it as a large network of neurons that interact dynamically to compute the model's response to stimuli, mimicking the way neurons compute in the real cortex. I will report on our model's capabilities thus far, including its response to drifting gratings varying in orientation, spatial frequency and contrast. Dynamical mechanisms will be discussed, as will emergent phenomena such as gamma-band rhythms, which are ubiquitous throughout cortex. This work was carried out jointly with Robert Shapley (Center for Neural Science, NYU) and Logan Chariker (CNS/Courant).