Mathematical science is the human recognition on the evolution laws of things that we can understand with the principle of logical consistency by mathematics, i.e., mathematical reality. So, is the mathematical reality equal to the reality of thing? The answer is not because there always exists contradiction between things in the eyes of human, which is only a local or conditional conclusion. Such a situation enables us to extend the mathematics further by combinatorics for the reality of thing from the local reality and then, to get a combinatorial reality of thing. This is the combinatorial conjecture for mathematical science, i.e., CC conjecture that I put forward in my postdoctoral report for Chinese Acade- my of Sciences in 2005, namely any mathematical science can be reconstructed from or made by combinatorialization. After discovering its relation with Smarandache multi-spaces, it is then be applied to generalize mathematics over 1-dimensional topological graphs, namely the mathematical combinatorics that I promoted on science internationally for more than 20 years. This paper surveys how I proposed this conjecture from combinatorial topology, how to use it for characterizing the non-uniform groups or contradictory systems and furthermore, why I introduce the continuity ow GL as a mathematical element, i.e., vectors in Banach space over topological graphs with operations and then, how to apply it to generalize a few of important conclusions in functional analysis for providing the human recognition on the reality of things, including the subdivision of substance into elementary particles or quarks in theoretical physics with a mathematical supporting.

The Mathematical Combinatorics (International Book Series) is a fully refereed international book series, quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

A thing is complex, and hybrid with other things sometimes. Then, what is the reality of a thing? The reality of a thing is its state of existed, exists, or will exist in the world, independent on the understanding of human beings, which implies that the reality holds on by human beings maybe local or gradual, not the reality of a thing. Hence, to hold on the reality of things is the main objective of science in the history of human development.

The International J. Mathematical Combinatorics is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly, which publishes original research papers and survey articles in all aspects of mathematical combinatorics, Smarandache multi-spaces, Smarandache geometries, non-Euclidean geometry, topology and their applications to other sciences.

The mathematical combinatorics is a subject that applying combinatorial notion to all mathematics and all sciences for understanding the reality of things in the universe, motivated by CC Conjecture of Dr.Linfan MAO on mathematical sciences. TheMathematical Combinatorics (International Book Series) is a fully refereed international book series with an ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly, which publishes original research papers and survey articles in all aspects of mathematical combinatorics, Smarandachemulti-spaces, Smarandache geometries, non-Euclidean geometry, topology and their applications to other sciences.

Descriptions of summer research programs: The AIM REU: Individual projects with a common theme by D. W. Farmer The Applied Mathematical Sciences Summer Institute by E. T. Camacho and S. A. Wirkus Promoting research and minority participantion via undergraduate research in the mathematical sciences. MTBI/SUMS-Arizona State University by C. Castillo-Chavez, C. Castillo-Garsow, G. Chowell, D. Murillo, and M. Pshaenich Summer mathematics research experience for undergraduates (REU) at Brigham Young University by M. Dorff Introducing undergraduates for underrepresented minorities to mathematical research: The CSU Channel Islands/California Lutheran University REU, 2004-2006 by C. Wyels The REUT and NREUP programs at California State University, Chico by C. M. Gallagher and T. W. Mattman Undergraduate research at Canisius. Geometry and physics on graphs, summer 2006 by S. Prassidis The NSF REU at Central Michigan University by S. Narayan and K. Smith Claremont Colleges REU, 2005-07 by J. Hoste The first summer undergraduate research program at Clayton State University by A. Lanz Clemson REU in computational number theory and combinatorics by N. Calkin and K. James Research with pre-mathematicians by C. R. Johnson Traditional roots, new beginnings: Transitions in undergraduate research in mathematics at ETSU by A. P. Godbole Undergraduate research in mathematics at Grand Valley State University by S. Schlicker The Hope College REU program by T. Pennings The REU experience at Iowa State University by L. Hogben Lafayette College's REU by G. Gordon LSU REU: Graphs, knots, & Dessins in topology, number theory & geometry by N. W. Stoltzfus, R. V. Perlis, and J. W. Hoffman Mount Holyoke College mathematics summer research institute by M. M. Robinson The director's summer program at the NSA by T. White REU in mathematical biology at Penn State Erie, The Behrend College by J. P. Previte, M. A. Rutter, and S. A. Stevens The Rice University Summer Institute of Statistics (RUSIS) by J. Rojo The Rose-Hulman REU in mathematics by K. Bryan The REU program at DIMACS/Rutgers University by B. J. Latka and F. S. Roberts The SUNY Potsdam-Clarkson University REU program by J. Foisy The Trinity University research experiences for undergraduates in mathematics program by S. Chapman Undergraduate research in mathematics at the University of Akron by J. D. Adler The Duluth undergraduate research program 1977-2006 by J. A. Gallian Promoting undergraduate research in mathematics at the University of Nebraska-Lincoln by J. L. Walker, W. Ledder, R. Rebarber, and G. Woodward REU site: Algorithmic combinatorics on words by F. Blanchet-Sadri Promoting undergraduate research by T. Aktosun Research experiences for undergraduates inverse problems for electrical networks by J. A. Morrow Valparaiso experiences in research for undergraduates in mathematics by R. Gillman and Z. Szaniszlo Wabash Summer Institute in Algebra (WSIA) by M. Axtell, J. D. Phillips, and W. Turner THe SMALL program at Williams College by C. E. Silva and F. Morgan Industrial mathematics and statistics research for undergraduates at WPI by A. C. Heinricher and S. L. Weekes Descriptions of summer enrichment programs: Twelve years of summer program for women in mathematics-What works and why? by M. M. Gupta Research experience for undergraduates in numerical analysis and scientific computing: An international program by G. Fairweather and B. M. Moskal Articles: The Long-Term Undergraduate Research (LURE) model by S. S. Adams, J. A. Davis, N. Eugene, K. Hoke, S. Narayan, and K. Smith Research with students from underrepresented groups by R. Ashley, A. Ayela-Uwangue, F. Cabrera, C. Callesano, and D. A. Narayan Research classes at Gettysburg College by B. Bajnok Research in industrial projects for students: A unique undergraduate experience by S. Beggs What students say about their REU experience by F. Connolly and J. A. Gallian Diversity issues in undergraduate research by R. Cortez, D. Davenport, H