**Title:** * Bounds on the Number of Covers for Lattices and Related Posets *

**Abstract:** How many covers can there be in a lattice of order n? This question has gone unanswered for decades. In the pursuit of a couple of conjectures, one of which is nearly 40 years old, we have obtained novel results primarily related to asymptotic bounds for lattices and related posets. This talk will involve lattice theory, order theory, combinatorics, graph theory, and analysis.

Reconfiguration in Graph Coloring

In mathematics, as in life, there are often multiple solutions to a question.

Reconfiguration studies whether it is possible to move from one solution to

another following a given set of rules. Is it possible? How long will it take?

In this talk, we will consider reconfiguration of graph coloring.

A proper coloring of a graph is an assignment of a color to each vertex of the

graph so that neighboring vertices have different colors. Suppose we change the

color of just one vertex in a graph coloring. Can we get from one coloring to

another by a sequence of vertex changes so that each step along the way is a

proper coloring? The answer is yes, if we are allowed an unlimited number of

colors. But, what is the fewest colors we can have for this to work? How many

steps might it take? We will look at this, related questions and generalizations.

**Title: ***Mathematical Model of the Spread of Radical Ideas in the Context of General Elections and Conceptually Similar Manifestations of the Political Behavior*

**Abstract. **

In this project we develop and test the mathematical model that not only fits the dynamics of a radical political force or forces in the context of general elections, but also can be used for modeling of different outcomes of the political behavior of a society in elections and in other conceptually similar forms of political expression. For these purposes we use a particular example of the rise of Nazis in Germany in 1930’s of the last century, and then move to the general model which, if suitably adjusted, can be applied to any pertinent situation. While the results achieved in the project are conceptually valid within mathematical field, we will provide their brief explanations from the standpoint of view of political science.

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**Title: ** *THE LOG-PERIODIC POWER LAW MODEL: AN EXPLORATION*

**Abstract:**

Over the last two decades, a new financial model has emerged that might explain some of the rare instances of truly extreme volatility in asset prices. This model, known as the Log-Periodic Power Law (LPPL) model or the Johansen-Ledoit-Sornette (JLS) model attempts to diagnose, time, and predict the termination of these bubbles; we caution that there is no academic agreement about the existence or definition of a “bubble”. The creators of the model provide a motivation built upon some natural assumptions including risk-neutral assets, rational expectations, local self-reinforcing imitation, and probabilistic critical times. The model has evolved over time, partially in response to some sound criticism. This dissertation is focused on two criticisms that have not been fully addressed. First, it is unknown whether there exists unique best fits of the JLS model to log-price data. In this dissertation we explore the first level JLS model and analyze its relationship with extreme boundary vectors which serve as the building blocks for increasing concave up log-price paths. Second, it is unknown whether the current method for locating local minima is sufficient. Using numerical analysis, this dissertation uses the Cauchy-Schwarz Theorem and Taylor’s Theorem to find bounds for various moduli of continuity and first and second derivatives of the error of the JLS model fitted to appropriate log-price paths. In addition to discussing these criticisms, the question of the applicability of the JLS model is considered. Without committing ourselves to a definition of a bubble, this dissertation also presents the results of an ongoing test attempting to determine whether the JLS model can be used to generate systematic profits. An experiment using the JLS model on specifically chosen stocks filtered from the New York Stock Exchange (NYSE) is detailed along with a step-by-step procedure to determine specific dates on which to utilize a trading strategy.

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Antonella Perucca (University of Regensburg), Reductions of algebraic numbers

Number theory talk, accessible to all graduate students. Join us! Abstract: We start by considering the reductions of the integers modulo the various prime numbers. More precisely, we investigate the multiplicative order of the reductions of an integer.

We then generalize this setting to number fields and illustrate some results which are joint work with Christophe Debry.