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Mushfeq Khan

khan@math.hawaii.edu
305 Physical Sciences Building
University of Hawai'i at Mānoa

My area of interest is logic, specifically computability theory. I did my PhD at the University of Wisconsin–Madison under the supervision of Joseph S. Miller.

Bjørn Kjos-Hanssen is my postdoctoral mentor at the University of Hawai'i.

Teaching: Fall 2017

Math 241

Math 301

Research

Research statement

Shift-complex sequences. Bulletin of Symbolic Logic, 2013.

Forcing with bushy trees. With J. S. Miller. Bulletin of Symbolic Logic, to appear.

Lebesgue density and \(\Pi^0_1\) classes. Journal of Symbolic Logic, 2016.

The strength of the Grätzer-Schmidt Theorem. With Katie Brodhead, Bjørn Kjos-Hanssen, William A. Lampe, Paul Kim Long V. Nguyen, and Richard A. Shore. Archive of Mathematical Logic, 2016.

Effective bi-immunity and randomness. With Achilles A. Beros and Bjørn Kjos-Hanssen. Festschrift in honor of Rod Downey, 2016.

Strongly nonrecursive functions. With Achilles A. Beros, Bjørn Kjos-Hanssen, and André Nies. In preparation.

My thesis

Slides

Shift-complex sequences. ASL Meeting, UC Berkeley, 2011.

Density-one points of \(\Pi^0_1\) classes. CCR, Buenos Aires, 2013.

Density-one points of \(\Pi^0_1\) classes. Midwest Computability Seminar XIII, University of Chicago, 2013.

Lebesgue density and \(\Pi^0_1\) classes. Algorithmic Randomness and Complexity, Shonan Village Center, Japan, 2014.

Note: Many of the questions in the slides about density-one points have been answered since I gave these talks. See the published version.

Mass problems and recursively bounded DNR functions. ASL Meeting, University of Illinois at Urbana-Champaign, 2015.

Effective bi-immunity and randomness. ASL Meeting, University of Connecticut-Storrs, 2016.

Effective bi-immunity and randomness. Algorithmic Randomness Interacts with Analysis and Ergodic Theory, Casa Matemática Oaxaca, 2016.

The Computability Menagerie

Joseph Miller and I maintain an interactive web-based version of his "Computability Menagerie". It depicts relationships between downward closed classes of Turing degrees, and is inspired by Bjørn Kjos-Hanssen's original version, which can be found here.

The Menagerie generates its graph based on a database file that contains some key facts about the classes. It can make a few basic types of deductions from these facts and fill in missing information.

You can tell the Menagerie to display only the classes that you care about. For example, this is a part of the diagram showing some questions that might be open.

The code is available here.