# Colloquia

## Professor Gheorghe Craciun 5/2/2014

Date: 5/2/2014

Time: 2:30PM-3:30PM

Place: 315 Armstrong Hall

Gheorghe Craciun

Abstract: Complex interaction networks are present in all areas of biology, and

manifest themselves at very different spatial and temporal scales. Persistence,

permanence and global stability are emergent properties of complex networks, and

play key roles in the dynamics of living systems.

Mathematically, a dynamical system is called persistent if, for all positive

solutions, no variable approaches zero. In addition, for a permanent system, all

variables are uniformly bounded. We describe criteria for persistence and permanence

of solutions, and for global convergence of solutions to an unique equilibrium, in a

manner that is robust with respect to initial conditions and parameter values.

We will also point out some connections to classical problems about general

dynamical systems, such as the construction of invariant sets and Hilbert's 16th

problem.

## Professor Xujing Wang 5/1/14

Date: 5/1/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Xujing Wang

Abstract:

Living systems are characterized by complexity in structure and emergent

dynamic orders. In many aspects the onset of a chronic disease resembles a phase

transition in a complex dynamic system: quantitative changes accumulate largely

unnoticed until a critical threshold is reached, which causes abrupt qualitative changes

of the system. This insight can help us identify the key factors driving the

disease, and the critical parameters that are predictive of disease onset. In this

study we investigated this ideas in a real example, the insulin-producing

pancreatic islet ?-cells and the onset of type 1 diabetes. Within each islet, the ?-cells are

electrically coupled to each other. This intercellular coupling enables

the ?-cells to synchronize their insulin release, thereby generating the multi-scale

temporal rhythms in blood insulin that are critical to maintaining blood glucose

homeostasis. Using percolation theory we show how normal islet function is

intrinsically linked to network connectivity. Percolation is a geometric phase transition of

network structure that has profound impact to the function and dynamics of the

network. We found that the critical amount of ?-cell death at which the islet cellular

network loses site percolation, is consistent with laboratory and clinical

observations of the threshold loss of ?-cells that causes islet functional failure. In

addition, numerical simulations confirm that the islet cellular network needs to be

percolated for ?-cells to synchronize. Furthermore, the interplay between site

percolation and bond strength predicts the existence of a transient phase of islet functional

recovery after onset and introduction of treatment, potentially explaining

a long time mystery in the clinical study of type 1 diabetes: the honeymoon

phenomenon. Based on these results, we hypothesized that the onset of T1D may be the

result of a phase transition of the islet ?-cell network. We will further discuss the

potential applications, as well as the importance of such approaches in the study of

complex networks in living systems.

## Professor Martha Alibabli 4/25/2014

Date: 4/25/2014

Time: 3:30PM-4:30PM

Place: 422 Armstrong Hall

Martha Alibabli

Abstract:

Teachers use a range of modalities to communicate in the classroom. In this talk, I focus on how teachers use gestures during instructional communication in mathematics. The bulk of the data are drawn from a corpus of middle school mathematics lessons covering a range of topics. I focus specifically on the role of teachers’ gestures in their communication in two discourse contexts: (1) segments of the lessons in which teachers attempt to connect ideas, concepts, or procedures, and (2) teachers’ responses to "trouble spots" in the classroom discourse (i.e., points where students do not understand or misunderstand the instructional material). The data reveal that gesture is an integral part of classroom communication, and that teachers adjust their gestures depending on lesson content and students’ needs.

## Professor Emanuel Indrei 4/24/2014

Date: 4/24/2014

Time: 3:30PM-4:30PM

Place: 313 Armstrong Hall

Emanuel Indrei

Abstract: Obstacle-type problems appear in various branches of minimal

surface theory, potential theory, and optimal control. In this talk, we

discuss the optimal regularity of solutions to fully nonlinear

obstacle-type free boundary problems. This represents joint work with

Andreas Minne.

## Professor Jie Ma 4/16/2014

The maximum number of proper

colorings in graphs with fixed

numbers of vertices and edges

Jie Ma

Date: 4/16/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract:

We study an old problem of Linial and Wilf to find the graphs

with n vertices and m edges which maximize the number of proper q-colorings

of their vertices. In a breakthrough paper, Loh, Pikhurko and

Sudakov asymptotically reduced the problem to an optimization problem. We

prove the following result which tells us how the optimal solution must

look like: for any instance, each solution of the optimization

problem corresponds to either a complete multipartite graph or a graph

obtained from a complete multipartite graph by removing certain edges. We

then apply this result to general instances, including a conjecture of

Lazebnik from 1989 which asserts that for any q>=s>= 2, the Turan graph

T_s(n) has the maximum number of q-colorings among all graphs with the same

number of vertices and edges. We disprove this conjecture by providing

infinity many counterexamples in the interval s+7 <= q <= O(s^{3/2}). On

the positive side, we show that when q= \Omega(s^2) the Turan graph indeed

achieves the maximum number of q-colorings. Joint work with Humberto Naves.

## Professor Martin Feinberg 4/7/2014

An Introduction to Chemical

Reaction Network Theory

Professor Martin Feinberg

Date: 4/7/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract: Here

## Professor John Tyson 3/6/2014

Irreversible Transitions,Bistability and Checkpoints in the Eukaryotic

Cell Cycle

Professor John Tyson

Date: 3/6/2014

Time: 3:30PM-4:30PM

Place: 315 Armstrong Hall

Abstract: Here

## Professor Zach Etienne 2/28/2014

Throwing in the Kitchen Sink: Adding Mixed Type PDEs to Better Solve Einstein's Equations

Professor Zach Etienne

Date: 2/28/2014

Time: 2:30PM-3:30PM

Place: 315 Armstrong Hall

Abstract:

With the first direct observations of gravitational waves (GWs) only a few years away, an exciting new window on the Universe is about to be opened. But our interpretation of these observations will be limited by our understanding of how information about the sources generating these waves is encoded in the waves themselves. The parameter space of likely sources is large, and filling the space of corresponding theoretical GWs will require a large number of computationally expensive numerical relativity simulations.

Numerical relativity (NR) solves Einstein's equations of general relativity on supercomputers, and with the computational challenge of generating thoeretical GWs comes an arguably even greater mathematical one: finding an optimal formulation of Einstein's equations for NR. These formulations generally decompose the intrinsically 4D machine of Einstein's equations into a set of time evolution and constraint equations---similar to Maxwell's equations of electromagnetism. Once data on the initial 3D spatial hypersurface are specified, the time evolution equations are evaluated, gradually building the four-dimensional spacetime one 3D hypersurface at a time. We are free to choose coordinates however we like in this 4D manifold, which are generally specified on each 3D hypersurface via a set of coordinate gauge evolution equations.

Robust coordinate gauge evolution equations are very hard to come by and are critically important to the stability and accuracy of NR simulations. Most of the NR community continues to use the highly-robust---though nearly decade-old---"moving-puncture gauge conditions" for such simulations. We present dramatic improvements to this hyperbolic PDE gauge condition, which include the addition of parabolic and elliptic terms. The net result is a reduction of numerical errors by an order of magnitude without added computational expense.

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