Knotting and linking in spatial graph embeddings

12/07/2016 - 4:15pm
12/07/2016 - 5:15pm
Speaker:
Kenji Kozai (HMC)
Abstract:

An abstract graph can be realized (embedded) in 3-dimensional space by associating vertices to a point in space and edges between vertices as an arc between the associated points. A given graph has infinitely many embeddings, and some embeddings may be more complicated than others. One way of measuring how complicated an embedding is is to consider the knotting and linking of cycles in the graph embedding. I will give an introduction to some elementary knot and link invariants, and then show how they can be used to prove that certain graphs are intrinsically linked or knotted, that is every embedding has a non-trivial link or knot. In addition, I will discuss random knot and graph embedding models as well as what can be said about "typical" embedding of graphs.

Where:
Kravis Lower Court, Claremont McKenna College

10/05/2016 - 6:05am
10/05/2016 - 7:05am
Speaker:
Matt Rathbun (Cal State Fullerton)
Abstract:

This talk will give a brief introduction to knot theory, and some of the applications to understanding the behaviors and mechanisms of DNA and interactions with proteins. No background will be assumed.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

How to roll a Fibonacci-sided die

11/09/2016 - 4:15pm
11/09/2016 - 5:15pm
Speaker:
Mark Huber (CMC)
Abstract:

The Fibonacci numbers come from a famous recursive sequence: $F_n = F_{n-1} + F_{n-2}$. They also count the number of {\em isolated sequences} of length $n$, where an isolated sequence is a sequence of 0's and 1's where no two 1's are adjacent. For instance, 0101 and 1000 are isolated, while 0110 is not. In this talk, I'll present a method for drawing uniformly from the set of isolated sequences of length $n$ using linear expected time. The algorithm also gives a way of extending the probability distribution on finite sequences out to infinite sequences. Along the way we'll use this algorithm to prove some asymptotic facts about the Fibonacci numbers, and derive the exact solution. This method is wide ranging, and can be used for arbitrary order $k$ recursive sequences with positive coefficients.

Where:
Kravis Center Lower Court 62, Claremont McKenna College
Misc. Information:

This talk is a replacement of the previously scheduled talk by Mark Alber.

Reflecting Sequences and Universal Traversal Sequences for Graphs

11/02/2016 - 4:15pm
11/02/2016 - 5:15pm
Speaker:
H. K. Dai (CMC)
Abstract:

Graph traversal is of fundamental importance to computing.  The study of universal traversal sequences and reflecting sequences is motivated by the complexity of graph traversal.  A universal traversal sequence for the family of all connected d-regular graphs with an edge-labeling is a sequence of symbols of

{0, 1, ..., d-1} that traverses every graph of the family starting at every vertex of the graph.  Reflecting sequences are variants of universal traversal sequences.  This presentation introduces the combinatorial nature of universal traversal sequences and reflecting sequences and their length lower-bound arguments.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

Better together: Swarming, information and strange interactions.

10/19/2016 - 4:15pm
10/19/2016 - 5:15pm
Speaker:
Lou Rossi (UDel)
Abstract:

Animal groups often exhibit social behaviors that lead to aggregations.  Popular examples include schools of fish, flocks of birds and cohorts of pedestrians.  These aggregations serve a variety of functions including navigational guidance, protection, foraging and so forth.  These behaviors provide inspiration for the control of autonomous robots and other entities.  The advantage to this approach to control is that it is decentralized and trivially scales to large numbers of individuals.  Mathematical theory and analysis plays a crucial role in understanding how local interactions map to the motion and dynamics of the entire swarm.  One popular proposed signaling mechanism for swarming individuals uses three zones.  In three-zone swarming, individual behavior is driven by the position and orientation of neighboring individuals in each of three concentric zones, corresponding to repulsion, orientation and attraction respectively.  We will review some results on stability, information transfer and leadership with three zone swarming using a continuum model.  Finally, we will explore very different swarming interactions arising in certain plankton species which can signal one another using toxins or by shading one another.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

Structured signal recovery without the shackles of convexity

09/21/2016 - 4:15pm
09/21/2016 - 5:15pm
Speaker:
Mahdi Soltanolkotabi (USC)
Abstract:

Many problems in science and engineering ask for solutions to underdetermined systems of linear equations. The last decade has witnessed a flurry of activity in understanding when and how it is possible to solve such problems using convex/greedy schemes. Structured signal recovery via convex methods has arguably revolutionized signal acquisition, enabling signals to be measured with remarkable fidelity using a small number of measurements. Despite many success stories, in this talk I will argue that over insistence on convex methods has stymied progress in some application domains. I will discuss my ongoing research efforts to “unshackle” structured signal recovery from the confines of convexity opening the door for new applications. In particular, I will present a unified theoretical framework for sharply characterizing the convergence rates of various (non-)convex iterative schemes for solving such problems. Time permitting, I will also discuss problem domains where carefully designed non-convex techniques are effective but convex counterparts are known to fail or yield suboptimal results. This is based on joint work with collaborators who shall be properly introduced during the talk.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

What is Mathematical Biology and is it useful?

11/16/2016 - 4:15pm
11/16/2016 - 5:15pm
Speaker:
Avner Friedman（OSU）
Abstract:

In this talk I will explain why biology is far more complicated than any of the physical sciences. I will then proceed to describe the process of doing research in mathematical biology, what are the aims, and what are the challenges. I will give two examples. The first one is modeling of tuberculosis (by ODEs) and the second one is modeling the risk of high cholesterols for heart attack or stroke. These examples address bio-medical questions. But I will also show how simplified version of the models introduce new mathematical questions both in ODEs and PDEs and lead to new mathematical results in analysis.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

What is a formula?

09/28/2016 - 4:15pm
09/28/2016 - 5:15pm
Speaker:
Igor Pak (UCLA)
Abstract:

Integer sequences arise in a large variety of combinatorial problems as a way to count combinatorial objects.  Some of them have nice formulas, some have elegant recurrences, and some have nothing interesting about them at all.  Can we characterize when?  Can we even formalize what is a "formula"?  I will give a mini-survey aiming to answer these question with many examples.  At the end, I will present some recent results counting certain permutation classes, and finish with open problems.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

Codes, Curves, and Configurations of Points

09/07/2016 - 4:15pm
09/07/2016 - 5:15pm
Speaker:
Nathan Kaplan (UCI)
Abstract:

We give an introduction to error-correcting codes focusing on connections to combinatorics, linear algebra, and geometry.  No previous familiarity with coding theory will be assumed.

A code C is a subset of (F_q)^n, where F_q denotes the finite field of q elements.  The Hamming distance between two elements of (F_q)^n is the number of coordinates in which they are different.  For fixed q and n what is the maximum size of a code such that any two of its elements have Hamming distance at least d?  This problem is easy to state but leads to surprisingly complicated mathematics.  We will focus on one extremal family, the Maximum Distance Separable (MDS) codes.  Understanding these codes (when they exist, how many there are, etc.) is a major problem in coding theory that has natural connections to classical algebraic geometry.

Where:
Kravis Center Lower Court 62, Claremont McKenna College

Conductivity imaging from minimal interior measurements

03/30/2016 - 4:15pm
03/30/2016 - 5:15pm
Speaker: