# Get To Know the Math Prof: Lutz Warnke

## Hoping to transmit to Georgia Tech students his zest for mathematics

Lutz Warnke is an assistant professor in the School of Mathematics. Before he joined Georgia Tech in January 2017, he received the 2016 Dénes König Prize from the Society of Industrial and Applied Mathematics. The Hungarian mathematician Dénes König (1884-1944) was a pioneer of discrete mathematics.

The main mathematical objects I study are called graphs, which are also known as networks in everyday language. In simple terms, a graph is a collection of points with links (or line-segments) connecting them. Besides being of theoretical interest, graphs have a number of different applications. For example, in network science, they are used to gain insights into concrete networks such as food chains and communication networks. Certain models of graphs are also used to simulate or predict properties of real-world networks, such as Facebook (where points correspond to members, and links represent friendships).

Random graphs are my core area of research. These graphs are constructed by adding the links or points in a random way (for example, by flipping coins to decide which ones to add). While some people consider random graphs to be toy models for real-world networks, I like to think of them as idealized reference models that can, for example, guide the study of concrete networks (to help identify which parameters and phenomena are interesting/worth studying, and which are not).

I have extensively studied the connectivity properties of random graphs. Perhaps surprisingly, random graphs can undergo “phase transitions,” similar to how water changes its state. Indeed, as we add more and more links to the graph, the size and structure of the connected sets of points undergo quite a dramatic change. They can go from small sets made of relatively few points to a situation where there is a big set containing 1% of all points (and a bunch of smaller sets). To put this into context, if the underlying random graph represented a model for the paths along which diseases spread amongst a population, then the aforementioned sizes of the connected sets could, for example, be vital in predicting the possibility of a pandemic.

Nowadays researchers from a variety of different backgrounds study random graphs, ranging from applied mathematics to physics and to biology. Motivated by applications, such interdisciplinary effort often leads to new viewpoints and questions, which I find particularly exciting from the perspective of pure mathematics.

What is the work for which you received the Dénes König Prize?

I was awarded the Dénes König Prize for my contributions to the study of phase transitions in random graphs that change over time. Achlioptas processes are interesting variants of classical random graph models that are easy to define, but hard to analyze. In a nutshell, the standard mathematical techniques in the area did not apply to these models due to a number of technical difficulties. In joint work with Oliver Riordan I was able to overcome some of these difficulties and prove a number of properties which were hitherto out of reach. Our new techniques have recently also found further applications, enabling us to prove some very precise results for a large class of random graph models.

What attracted you to mathematics?

As an undergraduate student I was initially interested in theoretical computer science and was fascinated by the idea that random choices can make algorithms faster (and simpler). At first this might sound quite counterintuitive, but one of the basic ideas is surprisingly simple.

Indeed, suppose that we have a bag of real and fake coins, of which at least half are real. If we then pick one coin at random from the bag (which roughly corresponds to “blindly” grabbing from the bag), then we fail to find a real coin with probability at most 1/2. Similarly, after two attempts we fail to find a real coin with probability at most 1/2 × 1/2 = 1/4, and so on.

In other words, by repeating this random experiment a few times we are very likely to find a real coin (in a very simple way). These and related examples motivated my subsequent course choices. In classes on probabilistic combinatorics and random graphs, I learned about further remarkable phenomena in random discrete structures and a number of beautiful proof techniques. In my opinion the mix of both is quite attractive, and since my graduate studies I have been doing research in this exciting area of mathematics.

What has been the highlight of your professional career so far?

One highlight has been my work on the so-called “explosive percolation” phenomenon. Loosely speaking, a large number of simulations and heuristics suggested that an important parameter “jumps” (grows very abruptly) in a number of intriguing random graph processes, which was considered a big surprise by many mathematicians and physicists.

However, in joint work with Oliver Riordan I was able to mathematically prove that the simulations were all misleading; that is, the parameter always grows in a “smooth” way. This result was published in Science, and it attracted a great deal of interdisciplinary attention (more than 150 citations). For me it was particularly encouraging to see that a number of people outside of pure mathematics were interested in our findings and arguments.

What do you bring to Georgia Tech as a mathematics professor and researcher?

My research complements the existing strengths in graph theory and the Algorithms, Combinatorics and Optimization program (ACO). Naturally, I thus hope to contribute some new topics to the curriculum. For example, I envision an advanced course in the area of probabilistic combinatorics, which might benefit the research of graduate students in the area of discrete mathematics, theoretical computer science, and the ACO program, say. Such a course could discuss some modern tools and techniques in the area, and I hope to transfer some of my own enthusiasm to the students. Of course, it would be great if some of them develop an interest in pursuing research in the area.

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