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 2016

1. Title: Equivalences between generalized quasi-random properties
   
   Description: At the turn of the millennium, thanks to the spread of the World Wide Web and the human genome project, the study of expanding networks was extended to random situations different from the classical Erdos-Rényi model, and to large deterministic networks showing so-called quasi-random properties. The multiclass generalization is the generalized random graph (popularly, stochastic block model), the deterministic counterpart of which is the generalized quasi-random graph (defined by L. Lovász and V. T-Sós). We have characterized spectra and spectral subspaces of graph based matrices, and established relations to the multiway discrepancy, a measure of a good clustering. Based on the spectra, multiway discrepancy, and degree sequences, in Section 3 of the arXiv paper below, we have defined so-called generalized quasi-random properties, the implications of which are true for certain deterministic graph sequences, irrespective of stochastic models. The research aims at proving the missing equivalences, mainly the P0-PIV one, where elementary calculations and classical results of Thomason, Chung, Graham, and Wilson could be used (these apply to the one-class situation).
   
   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 this arXiv paper (it is important that you use version 3).
   You may have a look at the available parts of the following book too: Bolla, M., Spectral Clustering and Biclustering. Learning Large Graphs and Contingency Tables. Wiley, 2013.
   
   
2. Title: Normal forms and eigenvalues of perturbations of endomorphisms
   
   Description: The theory of endomorphisms of a finite-dimensional vector space V over the complex numbers is well-understood and classical: up to conjugacy these endomorphisms are in one-to-one correspondence with Jordan normal forms. A lot is known in general about perturbations of such endomorphisms (i.e. endomorphisms parametrized analytically by a complex variable z) too. Nevertheless for some applications it would be interesting to work out quantitative results concerning such perturbations. As a first step, it is interesting to study the possible normal forms of perturbations up to conjugacy and the dependence of the eigenvalues of the endomorphism on the variable z. It is known that these eigenvalues are in general multi-valued analytic functions of z. In an ongoing joint work with A. Stipsicz and P. Ivanics, in the case dim(V) = 2 we write down a normal form for such matrices up to conjugacy, then spell out explicit relations between the parameters appearing in the first few terms of this normal form and the first few coefficients of the expansion of the eigenvalues. The case where the initial endomorphism has distinct eigenvalues being relatively simple, we focus on the case with non-trivial nilpotent part.
   
   The aim of this project is to study the analogous questions first in dimension 3, then in arbitrary higher dimension: given any series \varphi (z) with values in the endomorphisms of a complex vector space V of dimension >= 3 such that the Jordan normal form of \varphi (0) consists of a single Jordan block, first derive a normal form of such series up to conjugacy, then establish a relationship between the parameters appearing in the first few terms of this normal form and the first couple of coefficients in the series-expansion of the eigenvalues.
   
   Prerequisites: linear algebra, rudiments of analysis and complex function theory, possibly some skills in a scientific computing software.
   Best for: students interested in analysis and linear algebra with an ability to do computations.
   Professor: Dr. Szilárd Szabó
   
   Assignment for the first week: read and check the details
   
   
3. Title: The combinatorics and dynamics of a discrete k winners take all network model
   
   Description: We consider a modification of the classical perceptron model of neural networks inspired by neuroscience research on hippocampus. In this model, those k neurons fire whose weighted inputs are maximal. The model tries to mimic the dynamics of the neural activity in the hippocampus, where only a few percentages of the neurons fire at the same time, however, all the neurons eventually fire in a longer time frame. Neuroscientists conjecture that this dynamics is caused by the structured connections of excitatory neurons (pyramidal cells) and the random connections of inhibitory neurons (basket cells). The structured connections of the pyramidal cells are responsible for the periodic dynamics of the firing activity, while the inhibitory neurons are responsible to prevent the system from blowing up (all or almost all neurons firing the same time). In the simplified model, this inhibition is modeled by letting only the k neurons with the highest weighted input fire.
   
   The main goal of the research is to characterize what kinds of connections of the pyramidal cells are necessary to obtain a periodic firing dynamics.
   
   Prerequisites: basic combinatorics, elementary graph theory, basic calculus, elementary probability theory
   Best for: students who intend to do research in applied combinatorics and/or theoretical neuroscience
   Professor: Dr. István Miklós
   ASSIGNEMENT FOR THE FIRST WEEK: download the more detailed description and start working on the exercises:from here
   
   
4. Title: What is unavoidable - Forbidden Configurations

   Description: Click here
   Professor: Dr. Attila Sali
   ASSIGNMENT FOR THE FIRST WEEK: Click here