Math Slam

TBA

Optimization problems in material science often require quickly varying composites of materials. In this talk, basic homogenized properties of composites and laminates are discussed. The basic theory is applied to construct and optimize interesting structures, such as field rotators and conducting wheels. The talk will be accessible to everyone.

Abstract: Modular forms, a kind of SL2(Z)-invariant holomorphic functions defined on the upper half plane, are one of the most important objects studied in modern number theory. This talk will start from the basic definitions of modular forms and give some examples and important theorems associated with Eisenstein series. Finally we will use the power of modular forms to solve some classical problems on partitions of integers in additive number theory, including the Ramanujan congruence and sums of squares.

Many simple card tricks rely on mathematical principles and logic. I will be talking about some of these tricks, and the interesting ideas behind them. Hopefully I will have time to show you two or three tricks, and the proof to how they work. I will be using work from Mathematical Magic by Diaconis and Graham. The exciting thing about these tricks is they do not rely upon sleight of hand, and come out looking stunning nonetheless.

In the first part of this talk I will review the basic properties of the Yamabe problem on compact manifolds and recall some well-known results of LeBrun concerning the computation of the Yamabe invariant in dimension four. In the second part I will suggest some possible generalizations for finite volume 4-manifolds. Hopefully some grad student will want to resume this work were I left it...

Nowadays researchers encounter high dimensional data that arises in a variety of forms such as digital images, videos, and hyperspectral images. How to efficiently and effectively modeling such data sets has become an active research topic. A common model is to approximate such data by a mixture of affine subspaces. In this talk I will present a fast and accurate algorithm that can solve this problem in full generality, addressing both theoretical and applied issues. If time permits, I will also talk about the use of nonlinear models. This is joint work with Mauro Maggioni.

(Bio)chemical reaction networks are used to model processes that occur in cell biology. A fundamental problem is to characterize chemical reaction networks which admit multiple steady states. The existing literature has focused on identifying necessary conditions for multiple steady states. A natural (but significantly more difficult) problem is to determine sufficient conditions. In joint work with Anne Shiu, we suggest an approach for tackling this problem by defining certain minimal multistationary networks called 'atoms of multistationarity'. These 'atoms' are analogous to prime numbers in the theory of integers, in the sense that every multistationary network is either an atom or contains an atom. The talk will contain many examples from biology and will assume nothing more than calculus.

Cartan's theorem on maximal tori in compact Lie groups can be thought of as a generalization of the spectral theorem for unitary matrices. The goal of this talk will be to sketch the `topological' proof of this theorem, based on the Lefschetz fixed point theorem. Along the way, we'll encounter the flag variety, an interesting object whose geometry encodes the representation theory of the Lie group. Those who don't specialize in geometry or topology fear not, we will give examples showing that these concepts are very concrete objects familiar from linear algebra.

What do vector arrangements, discrete graphs, and perfect matchings have in common? These seemingly unrelated objects (and many others) have a very similar underlying structure, known as a matroid. As a result, studying matroids allows you to simultaneously study many different objects from all over mathematics. In addition, many properties and constructions from these various objects, such as loops, duals, bases, cycles, rank, polynomial invariants, and minors (subgraphs), generalize naturally to matroids. In this talk, we will give a general definition of a matroid, and motivate their study by examining some of these constructions in detail. The only prerequisite for this talk is basic linear algebra.

Skeletal structures, such as medial axis and curve skeleton, are a particular class of shape descriptors. They have numerous applications in shape recognition, shape retrieval, animation, morphing, registration, and virtual navigation. This talk will give a brief overview of the medial axis and the curve skeleton. The focus will be on the properties of the two objects crucial to applications. We shall show that the rigorous mathematical definition of the medial axis has allowed for an extensive and successful study of such properties. The curve skeleton, on the other hand, is typically defined by the set of properties it has to possess. As a result, numerous methods for computing the curve skeleton have been proposed, each providing mostly experimental verification of the required properties. If time permits, I will mention my work on defining shape skeleta via persistent homology, thus providing a powerful platform for investigating their properties.

Biology has given mathematicians many new problems to work on in the last half century and the role of mathematics in biology research is only increasing. Through a series of examples, ranging from coat pattern formation to the evolution of RNA viruses, I will illustrate the insight that a mathematical treatment can give to problems in biology and will also discuss the difficulties involved in doing mathematical biology.

This talk will discuss some of the basic ideas pervasive in computational topology, especially persistent homology. Then, we will look at some currently active areas of application for these ideas, including large data sets, sensor networks, protein docking, plant root systems, and natural images. I assume no topology background, so this talk will be a good introduction to anyone interested in seeing what is going on in the field.

A toric variety is a special class of algebraic variety, which can be expressed by a picture that makes some of it's properties much easier to analyze. These varieties are an area of ongoing research in algebraic geometry, due in part to their applications in many different fields (such as mathematical physics). In this talk I will discuss how we can use a fan to construct a toric variety as a quotient. This talk will be accessible to all graduate students and will focus on examples, to illustrate the usefulness of this method.

Einstein's General Relativity is a geometric theory of space, time, and gravitation. In some sense, it is the successor of Newton's famous theory of gravitation -- the theory Newton is said to have come up with when an apple fell onto his head. But although Einstein's theory is much better at predicting gravitational effects in our universe, Newton's theory is not at all outdated or even obsolete. In fact, many astrophysical measurements and simulations still heavily rely on Newtonian intuitions, calculations, and concepts. In the talk, I will explain how and to what extent this usage of Newtonian theory in astrophysics and related fields is motivated and mathematically justified. This will lead us to the notion of Newtonian limit. We will also see some examples for the behavior of relativistic quantities like mass and center of mass under this Newtonian limit.

Many machine learning algorithms today rely on finding the nearest neighbors of all points in a data set. In data sets which are too large and too complex, we cannot ask for finding the true nearest neighbors without paying the price of a full pairwise search. Therefore, we look to using approximate nearest neighbor searches to quickly construct rough approximations of the nearest neighbor graph. I will talk about many of the methods used today in practice.

We'll look at a random process on the integer lattice $/mathbb{Z}^2$ known as the voter model. Let's suppose that each point on the lattice represents a single household with one voter who holds one of two possible opinions, 0 or 1 (e.g. Republican or Democrat). Starting with an initial configuration of 0's and 1's on $/mathbb{Z}^2$, a voter changes their opinion at a rate proportional to the number of neighbors holding a different opinion. I'll demonstrate a clever and useful approach to analysing the voter model via the dual process. We'll be able to determine whether or not our process has any nontrivial stationary distributions. I'll also briefly discuss the newest research on variants of the voter model.

At some point in every mathematician's life they have seen the paradoxical 'proof' that 1=0 obtained by different groupings of the infinite sum 1-1+1-1+... As we learn, the issue is that this series does not converge. The Eilenberg-Mazur swindle is a twist on this argument which shows that A+B+A+B+... = 0 implies that A=0=B in certain situations where we can make sense of the infinite sum. In this talk, we will explore these swindles, touching on many interesting areas of mathematics along the way.

In a chemical reaction system, the concentrations of chemical species evolve in time, governed by the polynomial differential equations of mass-action kinetics. This talk provides an introduction to chemical reaction network theory, and gives an overview of algebraic and combinatorial approaches to determining whether a chemical reaction network admits multiple steady states. In general, establishing the existence of (multiple) steady states is challenging, as it requires the solution of a large system of polynomials with unknown coefficients. However, for networks that have special structure, various easy criteria can be applied. This talk will highlight results from Deficiency Theory (due to Feinberg), criteria for multistationarity for chemical reaction systems whose steady states are defined by binomial equations (in joint work with Carsten Conradi, Mercedes Pérez Millán, and Alicia Dickenstein), and a classification of small multistationary chemical reaction networks (in joint work with Badal Joshi).

A polytope is a subset of R^d which is the convex hull of a finite set of vertices. Given a polytope P, we can consider integer dilations of P, and ask how many integer points are contained in each dilation, as a function of the dilation factor. A theorem by Eugene Ehrhart tells us that, under the right conditions, this counting function is a polynomial, with some very interesting and unexpected properties. To demonstrate the usefulness of these results, we will give alternative proofs to some well known results from far outside the realm of geometry, including some basic facts about the chromatic polynomial of a graph. This talk will contain a little geometry, a little analysis, a little algebra, and a little combinatorics, and will be accessible to anyone who enjoys at least one of these topics.

I will first explain with a few simple examples a beautiful picture in geometric invariant theory which relates quotient constructions in symplectic and algebraic geometry. Then we will naively apply this picture in a suitable infinite dimensional setting, leading us to the notion of extremal Kahler metrics and the Yau-Tian-Donaldson conjecture, which is an important problem in Kahler geometry today.