# Boston Graduate Math Colloquium

**UPDATE: We will be eating lunch OUTSIDE. We are going to walk to OTTO PIZZA which is located in HARVARD SQUARE.**

This is a joint collaboration between graduate students and PhD candidates in Boston who happen to know each other. The aim is to provide a venue for grad students at all levels to give math talks accessible to a general audience, and meet fellow researchers who are right down the street or across the river. BU, Northeastern, BC, and Harvard will host four events throughout the academic year. Each Saturday will consist of 4-5 speakers discussing their research.

## Date & Location

*Date:* Saturday, 28 April, 2018

*Location:* Hall E, Science Center (in the basement)

## RSVP

Please use this so we can order enough food!

## Schedule

*11:00-11:15 : *Coffee and Tea

*11:15-11:45 : *Elizabeth Upton

*11:45-12:45 : *Monika Pichler

*12:45-1:30 :* Lunch - Walking to Otto and eating OUTSIDE near Harvard Sq.

*1:30-2:30 :* Siddhi Krishna

*2:30-3:00 :* Jessica Nadalin

*3:00-3:30 :* Afternoon Tea

*3:30-4:30 :* Ying Zhang

## Speakers, Titles and Abstracts

Speakers:

Siddhi Krishna (BC) - Low-dim Topology

Jessica Nadalin (BU) - Mathematical Neuroscience

Monika Pichler (NEU) - PDE

Elizabeth Upton (BU) - Networks

Ying Zhang (BU) - Numerical PDE

Siddhi Krishna

Title: Taut Foliations, Positive 3-Braids, and the L-Space Conjecture

Abstract: The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold M. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology iff Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll discuss how to build taut foliations for manifolds obtained by surgery on positive 3-braid closures. No background in Heegaard-Floer, foliation, or surgery theories will be assumed.

Jessica Nadalin

Title: A New Method to Assess Cross-Frequency Coupling: Applications to Human Epilepsy Data

Abstract: Cross-frequency coupling (CFC) has been proposed to play an important role in information processing and memory in both humans and animal models. It is also thought to be involved with pathological brain activity. Here, we examine instances of CFC from microelectrode array recordings in four human patients with pharmacoresistant epilepsy. During a seizure, voltage fluctuations appear across a broad range of frequencies, from very slow (< 1Hz) to very fast (>100 Hz). How these rhythms coordinate is not well understood. To address this, we develop a new modeling framework to characterize the extent of cross-frequency coupling (CFC) observed at seizure onset. Through this modeling framework, we assess the standard relationships between high frequency amplitude and low frequency phase, as well as the relationship between high and low frequency amplitude. In this way, the method allows us to address an important confound in the characterization of CFC from noisy data in which the low frequency amplitude may vary. Through simulations, we show that the proposed modeling framework is more sensitive to increases in CFC strength than methods that do not account for changes in the low frequency amplitude, and detects changes in CFC where other methods may not. Our preliminary results show that, at seizure onset, the amplitude of fast (100-140 Hz) activity is modulated by the phase of a slow (4-7 Hz) oscillation. A deeper understanding of CFC at human seizure onset may provide new insight into the mechanisms involved in seizure initiation and propagation, and suggest improved therapeutic strategies to control seizures.

Monika Pickler

Title: An inverse boundary value problem for Maxwell’s equations

Abstract: Inverse boundary value problems arise naturally in many physical situations where one wishes to study the properties of an object using measurements on its surface. This amounts to recovering coefficients of a partial differential equation from boundary data of its solutions. A pioneer contribution in the mathematical study of such problems was a paper by A. P. Calderón published in 1980, posing an inverse problem for the conductivity equation. This work motivated many developments in the field, first concerning inverse problems for the conductivity equation, and subsequently also for other partial differential equations. I will give a brief history of the study of this prototypical inverse problem, and then discuss an inverse problem for Maxwell’s equations, how it relates to that for the conductivity equation, and what tools are needed to show unique solvability of this inverse problem.

Elizabeth Upton

Title: Bayesian Network Regularized Regression for Modeling Urban Crime Occurrences

Abstract: We consider the problem of statistical inference and prediction for processes defined on networks. We assume that the network is known and measures similarity, and our goal is to learn about an attribute associated with its vertices. Classical regression methods are not immediately applicable to this setting, as we would like our model to incorporate information from both network structure and pertinent covariates. Our proposed model consists of a generalized linear model with vertex indexed predictors and a basis expansion of their coefficients, allowing the coefficients to vary over the network. We employ a regularization procedure, cast as a prior distribution on the regression coefficients under a Bayesian setup, so that the predicted responses vary smoothly according to the topology of the network. We motivate the need for this model by examining occurrences of residential burglary in Boston, Massachusetts. Noting that crime rates are not spatially homogeneous, and that the rates appear to vary sharply across regions in the city, we construct a hierarchical model that addresses these issues and gives insight into spatial patterns of crime occurrences.

Ying Zhang

Title: A Stochastic Reaction-Diffusion Model for Tethered Enzymatic Reactions

Abstract: Tethered enzymatic reactions are a key component in signaling transduction pathways. It is found that many surface receptors rely on the tethereing of cytoplasmic kinases to initiate and integrate signaling. Over the past decade a large number of compartment-based ODE and stochastic models have been developed to investigate properties of such pathways, among which lattice-based stochastic reaction-diffusion models are a popular approach. Chemical reactions in the most widely used lattice-based reaction-diffusion model, the reaction-diffusion master equation (RDME), are generally limited to molecules within the same voxel. With this restriction, the RDME has several drawbacks in accurately resolving tethered interactions at cellular scales. To more accurately represent tethered interactions, we develop a particle-based convergent reaction-diffusion master equation (CRDME) model for the reaction and diffusion of individual receptors, kinases and phosphatases in the cell membrane and cytosol.

## Past Colloquiua

Boston University, Oct 21 2017. *Speakers:* Brian Hepler, Spencer Leslie, Jackson Walters, Yusheng Luo.

Northeastern University, Dec 2 2017. *Speakers: *Angus McAndrew, Eric Chang, Maria Fox, Emre Sen.

Boston College, Feb 10 2018. *Speakers: *Brian Choi, Roderic Guigo, Rahul Singh, Ben Thompson, Shucheng Yu