BSM-TDK — Research Opportunities — RES

COURSE DESCRIPTION

This course is designed in the style of the Hungarian "TDK" system, allowing advanced undergraduates to become acquainted with research methods and means in detail and acquire additional knowledge beyond their obligatory curriculum. (For a brief English description of the TDK system see a relevant ELTE University homepage.)

In this course, a student can choose from the topics/problems listed below and work with other students and the professor to solve the given problem. All work is summarized in a paper and during the semester there will be opportunities to present your work as well.

This may contribute to the successful beginning of a scientific career: depending on level, the results obtained can be presented at school, statewide or national undergraduate meetings ranging from a local Undergradute Seminar at your home school to MAA's MathFest. Papers may also be published in undergraduate research journals. such as The Rose-Hulman Undergraduate Mathematics Journal, Involve or many others.

In some PhD programs, fruitful undergraduate reserach activity is a prerequisite for admission.

Student research is supervised by professors. Research topics are offered by them, but students can also propose topics of their own interest.

You can view articles that were written under the auspices of the BSM program

COURSE LOGISTICS

The list of research topics and professors proposing them can be seen below. Contact the professor whose problem you are interested in.

• Who can participate? Most professors gave a list of problems and/or some reading and related tasks for those who are interested in working on their problem. If you are interested in participating, do as much of these as you can by the Welcome Party and discuss your progress with the professor. First enrollment will be based on work on these problem sets/reading assignments. Final enrollment will be decided by the third week (as explained below).
• Which topics will actually be offered ("stay alive")? Of the initially offered research topics below those will be offered eventually, for which a group of students (at least around 3) would like to sign up and are accepted by the Professor based on discussions at the Welcome party and/or the first week.
• Course work: weekly meetings.
  Class will meet twice weekly, two hours each. One class time is devoted to group work, when you discuss the problem and possible solutions with your student group (without the professor). The other class time is spent with your professor who will monitor your group's progress.
• Course work: presentations.
  Week 3 - Milestone 1: The first three weeks of the course are spent discussing, gathering and studying necessary background information for your problem. At the end of the 3rd week, you (and your group) will have to present the problem you are working on in 20 minutes at a "mini workshop" organized for all BSM-TDK participants, their professors and everyone else interested. At this point professors evaluate progress and final enrollment decision is made. Ideally, the size of research groups is low, at most three.
  Week 9 - Milestone 2:: Work continuous thrughout the semester. At the 9th week each group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.
  Week 9/10-14:: Write up of results continuous and papers are produced.

TOPICS PROPOSED — SPRING 2015

1. Title: Comparing random networks and human brain networks
   
   Description: Neuroscientists developed an artificial neural network that models the short term memory in the hippocampus and has a spiking activity similar to the real life brain. If one generates a random network with same/similar degree sequence, then the network will not show the spiking activity and will not properly model the short term memory. The conclusion of this experience is that the degree sequence is not a sufficient descriptor of the human brain network.
   
   We are going to compare the above mentioned trained artificial neural networks and random networks by randomly perturbing the artificial neural network. We want to know which perturbations destroy the dynamical behavior of the system and which ones do not. The main goal of the project is to understand how short term memory is stored in the brain and what kind of connectivity patterns are necessary to maintain a proper neural activity. Meanwhile, we are going to learn the most important randomization techniques and there is also a possibility to work on open questions on randomizations that already emerged in the previous semester.
   
   This project is a continuation of a previous semester's project, see this presentation .
   
   An ideal team will consist of one or more good programmers and also some students who are interested in computer science and combinatorics.
   
   Prerequisites: basic graph theory, basic probability theory, knowledge of any programming language, preferably Java
   Best for: students who are interested in applied mathematics, combinatorics, computer science and theoretical neural science
   Professor: Dr. István Miklós
   ASSIGNEMENT FOR THE FIRST WEEK: Read Chapter 12 from this electronic note:http://www.renyi.hu/~miklosi/AlgorithmsOfBioinformatics.pdf . This chapter gives the definition of degree sequences and also provides a randomization algorithm (the swap Markov chain) that generates a random network with prescribed degree sequence.
   
   
2. Title: Density of sets with distance restrictions
   
   Description: The motivating problem is the following well know open problem:
   A 1-avoiding set is a subset of R^n that does not contain pairs of points at distance 1. What is the maximal proportion of the plane that a 1-avoiding plane set can fill up? Right now the best upper estimate is 0.2588, the best example gives lower estimate 0.2293, and Erdos conjectured that it is less than 1/4.
   All reasonable constructions of 1-avoiding sets are countable union of "blocks" such that every distance within each block is less than 1 and every distance between points from distinct blocks is bigger than 1. So the blocks have diameter at most 1 and the distance between any two blocks is at least one. In our brand new paper http://arxiv.org/pdf/1501.00168v1.pdf we proved Erdos conjecture for sets of these structure, and more generally we proved that in R^n these sets have density less then 1/2^n.
   
   We plan to attack the following questions:
   1. How can we generalize the above mentioned density estimates for sets of more general block structure, when for given d and delta the diameter of each block is at most d and the distance of any two blocks is at least delta?
   2. How can we show that 1-avoding sets of large density must have block structure?
   
   Prerequisities: Analysis, Geometry
   Best for: students who like analysis, combinatorics and would like to try research
   
   Professor: Dr. Tamás Keleti
   ASSIGNEMENT FOR THE FIRST WEEK: Right after the statement of Theorem 2.2 of http://arxiv.org/pdf/1501.00168v1.pdf you can find an argument that gives a slightly weaker result than the theorem. As a warm up (and to get a first answer to the above question 1) generalize this argument to get an upper estimate for sets that are countable union of blocks such that the diameter of each block is at most d and the distance of any two blocks is at least delta. (Should be done preferably for the welcome party, or even better - before that - submit by e-mail, my e-mail address is tamas.keleti at gmail.)
   
   
3. Title: (Small) Forbidden Configurations
   
   Description: The topic is extremal hypergraph theory formulated mostly in the language of 0-1 matrices, since that is the most convenient way. We say that a 0-1 matrix A has F as a configuration if there is a submatrix ofA, which is a row and column permutation of F. This concept is also called a trace or subhypergraph depending upon whether the context is set systems or hypergraphs.
   
   The fundamental question is that for given F (or sometimes for given collection F of configurations) what is the largest possible number of columns of a simple mxn 0-1 matrix A on given number of rows without having F (or any member of F as configuration, in notation forb(m,F) or forb(m,F). A 0-1 matrix is simple if it has no repeated columns. Considering columns as characteristic vectors of subsets of an m element set matrix A describes a hypergraph, or set system.
   
   In this project you will get acquainted with basic proof ideas of the area, such as ``standard induction'', shifting, applications of graph theory and linear algebra. The goal is to get results on particular forbidden configurations. For more info check
   http://www.math.ubc.ca/~anstee/FCsurvey12.pdf

   Prerequisites: basic combinatorics, possibly linear algebra;
   Best for: students who intend to do research in combinatorics
   Professor: Dr. Attila Sali
   ASSIGNMENT FOR THE FIRST WEEK: Click here



4. Title: 2-colorable triple-systems
   
   Description: click here for the description of the problem
   
   Prerequisites: basic combinatorics;
   Best for: students who intend to do research in combinatorics
   Professor: Dr. András Gyárfás
   ASSIGNEMENT FOR THE FIRST WEEK:Start working on the exercises given in the description.


5. Title: Spectral Clustering: Is the multiway discrepancy monotonous?
   
   Description: Networks can be modeled by edge-weighted graphs or, more generally, by rectangular arrays of nonnegative entries (e.g., keyword-document matrices or microarrays). We want to find clusters of the nodes or biclusters of the rows and columns so that the information flow within the clusters and between the cluster pairs be as homogeneous as possible. For this purpose, we minimize discrepancy and relate it to the spectral properties of the normalized adjacency matrix or contingency table.
   
   So far, in the framework of BSM research courses in this topic, we have managed to characterize spectra of these matrices (our joint paper is to appear in the Linear Algebra and Its Applications) and made experiments on real life data that helped in formulating estimates between these spectra and discrepancies.
   
   The question to be answered in this semester is whether the multiway discrepancy is monotonous. A positive answer would simplify many related statements. With the notation of the arXiv paper below: disc_{k} (C) \le \disc_{k-1} (C)? First consider simple graphs and the k=2 case.
   
   Simulation aided proofs and experiments on real-life data are also welcome.
   
   Prerequisites: Basic combinatorics and linear algebra
   Best for: students who intend to do research in networks
   Professor: Dr. Marianna Bolla
   
   Assignment for the first week: read http://arxiv.org/abs/1408.6443 (need not read details of Section 2) and some references therein.