Martin George Scharlemann (born 6 December 1948) is an American

topologist In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

who is a professor at the

University of California, Santa Barbara The University of California, Santa Barbara (UC Santa Barbara or UCSB) is a Public university, public Land-grant university, land-grant research university in Santa Barbara County, California, Santa Barbara, California with 23,196 undergraduate ...

. He obtained his

Ph.D. A Doctor of Philosophy (PhD, Ph.D., or DPhil; Latin: or ') is the most common degree at the highest academic level awarded following a course of study. PhDs are awarded for programs across the whole breadth of academic fields. Because it is a ...

from the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California) is a public land-grant research university in Berkeley, California. Established in 1868 as the University of California, it is the state's first land-grant u ...

under the guidance of

Robion Kirby Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he invented the Kirby–Siebenmann invariant ...

in 1974. A conference in his honor was held in 2009 at the

University of California, Davis The University of California, Davis (UC Davis, UCD, or Davis) is a public land-grant research university near Davis, California. Named a Public Ivy, it is the northernmost of the ten campuses of the University of California system. The institut ...

. He is a Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

, for his "contributions to low-dimensional topology and knot theory."

Abigail Thompson Abigail A. Thompson (born 1958 in Norwalk, Connecticut) is an American mathematician. She works as a professor of mathematics at the University of California, Davis, where she specializes in knot theory and low-dimensional topology. Education a ...

was a student of his. Together they solved the graph planarity problem: There is an algorithm to decide whether a finite graph in 3-space can be moved in 3-space into a plane. He gave the first proof of the classical theorem that

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...

s with

unknotting number In the mathematical area of knot theory, the unknotting number of a knot is the minimum number of times the knot must be passed through itself (crossing switch) to untie it. If a knot has unknotting number n, then there exists a diagram of the ...

one are

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. He used hard combinatorial arguments for this. Simpler proofs are now known.

Selected publications

*"Producing reducible 3-manifolds by surgery on a knot" ''

Topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

'' 29 (1990), no. 4, 481–500. *with

, "Heegaard splittings of (surface) x I are standard" ''

Mathematische Annalen ''Mathematische Annalen'' (abbreviated as ''Math. Ann.'' or, formerly, ''Math. Annal.'') is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann. Subsequent managing editors were Felix Klein, David Hilbert, ...

'' 295 (1993), no. 3, 549–564. *"Sutured manifolds and generalized Thurston norms", ''

Journal of Differential Geometry The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book ...

'' 29 (1989), no. 3, 557–614. *with J. Hyam Rubinstein, "Comparing Heegaard splittings of non-Haken 3-manifolds" ''Topology'' 35 (1996), no. 4, 1005–1026 *"Unknotting number one knots are prime", ''

Inventiones mathematicae ''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors ...

'' 82 (1985), no. 1, 37–55. *with Maggy Tomova, "Alternate Heegaard genus bounds distance" ''

Geometry & Topology ''Geometry & Topology'' is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sci ...

'' 10 (2006), 593–617. *"Local detection of strongly irreducible Heegaard splittings" '' Topology and its Applications'', 1998 *with Abigail Thompson – "Link genus and the Conway moves" ''

Commentarii Mathematici Helvetici The ''Commentarii Mathematici Helvetici'' is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society sti ...

'', 1989 *"Smooth spheres in

\mathbb^4

with four critical points are standard" ''

'', 1985 *"Tunnel number one knots satisfy the Poenaru conjecture" '' Topology and its Applications'', 1984 *with A Thompson – "Detecting unknotted graphs in 3-space" ''

'', 1991 *with A Thompson – "Thin position and Heegaard splittings of the 3-sphere" ''J. Differential Geom'', 1994

References

{{DEFAULTSORT:Scharlemann, Martin Topologists 1948 births Living people University of California, Santa Barbara faculty Fellows of the American Mathematical Society 20th-century American mathematicians 21st-century American mathematicians University of California, Berkeley alumni