Peter Feller

About me:

I am an assistant professor in the Geometry, groups and dynamics group at ETH Zurich. Previously, I was a postdoctoral fellow at the Max Plack Institute for Mathematics in Bonn and at Boston College. Before that I completed my PhD at the University of Bern. For more, consult my CV or contact me directly.

How to find or contact me:

Office: HG G 61.2
Address: ETH Zurich, Department of Mathematics, Raemistrasse 101, 8092 Zurich, Switzerland
email: peter.feller(you know the symbol)math.ethz.ch

Other geometric topologists at ETHZ:

Dr. Miguel Orbegozo Rodriguez.

Dr. Danica Kosanović.

Dr. Paula Truöl has obtained her PhD and has moved to MPIM in Bonn in fall 2023.

ETHZ Geometry Seminar:

You are in Zürich and you like Geometry and Topology? Check out ETHZ's Geometry Seminar.

Research Interests:

Within geometry and topology I am drawn to low-dimensional topology—the study of geometric objects of dimension four or less. One-dimensional objects that lie (in a potentially knotted way) in three-dimensional Euclidian space ℝ³—known as knots—fascinate me because their study relates to many other fields of mathematics. Often knot theory provides an approach towards visualizing more complicated objects such as 3-manifolds, 4-manifolds, and singularities of zero-sets of polynomials. Here is a brief report on recent results in low-dimensional topology that I find exciting.

Concerning 3-manifolds and their geometrization, a current focus of mine is the study of geometric properties, such as hyperbolizability, diameter, and systol length, via the mapping class group and the curve complex of surfaces.

In knot theory, terms such as knot concordance, algebraic knots and links, and notions of positivity for links often appear in my research. A focus of mine is to use the notion of concordance for knots to understand the differences between smooth and topological 4-manifolds.

In a different direction, I hope to apply knot theory in algebraic geometry to the study of complex plane curve singularities and their deformations using positive braids and tools from Heegaard-Floer theory.

Also concerning algberaic geometry, I wonder in how many ways complex algebraic varieties embed in affine space ℂⁿ and affine algebraic groups such as SL_n(ℂ). Maybe there are as many algebraic embeddings of the complex line ℂ in three-dimensional affine space ℂ³ as there are knots in Euclidean space ℝ³.
The question whether there are non-equivalent algebraic embeddings of ℂ in ℂ³ is the first instance of the embedding problem in complex affine algebraic geometry. While a connection to low-dimensional topology, such as having as many algebraic embeddings of ℂ in ℂ³ as knots in ℝ³, might be far-fetched, there is some precedent of results and proof strategies that work in both the setup of smooth manifolds and the setup of affine varieties. For example, Whitney's weak embedding and isotopy theorem for smooth closed manifolds has an analog for complex affine varieties known as the Holme-Kaliman-Srinivas theorem.

List of Publications:

For more details you can consult my detailed list of publications, where you find abstracts and pictures. My work is openly available on the ArXiv.

1. Seifert surfaces in the four-ball and composition of binary quadratic forms. [ArXiv:2311.17746]
   With Menny Aka, Alison B. Miller, and Andreas Wieser.
2. The Dehn twist coefficient for big and small mapping class groups. [ArXiv:2308.06214]
   With Diana Hubbard and Hannah Turner.
3. Hyperbolic Heegaard splittings and Dehn twists. [ArXiv:2304.05990]
   With Alessandro Sisto and Gabriele Viaggi.
4. Asymptotics of the smooth An-realization problem. [ArXiv:2303.15434]
   With Sebastian Baader.
5. The slice-Bennequin inequality for the fractional Dehn twist coefficient. [ArXiv:2204.05288]
6. On the values taken by slice torus invariants. [ArXiv:2202.13818], accepted for publication in Proc. Cambridge Philos. Soc.
   With Lukas Lewark and Andrew Lobb.
7. Squeezed knots. [ArXiv:2202.12289], accepted for publication in Quantum Topol.
   With Lukas Lewark and Andrew Lobb.
8. The ℤ-genus of boundary links. [ArXiv:2012.14367], accepted for publication in Rev. Mat. Complut.
   With JungHwan Park and Mark Powell.
9. Examples of non-minimal open books with high fractional Dehn twist coefficient. [ArXiv:2010.07869], accepted for publication in New York J. Math.
   With Diana Hubbard.
10. A note on the four-dimensional clasp number of knots. [ArXiv:2009.01815] [Math. Proc. Cambridge Philos. Soc.]
    With JungHwan Park.
11. Existence of Embeddings of Smooth Varieties into Linear Algebraic Groups. [ArXiv:2007.16164] [J. Algebraic Geom.]
    With Immanuel van Santen.
12. Bouquets of curves in surfaces. [ArXiv:2007.10429], accepted for publication in Glasg. Math. J.
    With Sebastian Baader and Levi Ryffel.
13. Uniform models and short curves for random 3-manifolds. [ArXiv:1910.09486]
    With Alessandro Sisto and Gabriele Viaggi.
14. Embedding spheres in knot traces. [ArXiv:2004.04204] [Compos. Math.]
    With Allison N. Miller, Matthias Nagel, Patrick Orson, Mark Powell, and Arunima Ray.
15. Non-orientable slice surfaces and inscribed rectangles. [ArXiv:2003.01590], accepted for publication in Ann. Sc. Norm. Super. Pisa Cl. Sci.
    With Marco Golla.
16. Genus one cobordisms between torus knots. [ArXiv:1910.01672] [Int. Math. Res. Not.]
    With JungHwan Park.
17. A note on the topological slice genus of satellite knots. [ArXiv:1908.03760] [Algebr. Geom. Topol.]
    With Allison N. Miller and Juanita Pinzon-Caicedo.
18. Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space. [ArXiv:1905.08305], accepted for publication in J. Differential Geom.
    With Lukas Lewark.
19. Almost positive links are strongly quasipositive. [ArXiv:1809.06692] [Math. Ann.]
    With Lukas Lewark and Andrew Lobb.
20. Up to topological concordance links are strongly quasipositive. [ArXiv:1802.02493] [J. Math. Pures Appl.]
    With Maciej Borodzik.
21. Calculating the homology and intersection form of a 4-manifold from a trisection diagram. [ArXiv:1711.04762] [Proc. Natl. Acad. Sci. USA]
    With Michael Klug, Trent Schirmer, and Drew Zemke.
22. Braids with as many full twists as strands realize the braid index. [ArXiv:1708.04998] [J. Topol.]
    With Diana Hubbard.
23. On classical upper bounds for slice genera. [ArXiv:1611.02679] [Selecta Math.]
    With Lukas Lewark.
24. Khovanov width and dealternation number of positive braid links. [ArXiv:1610.04534] [Math. Res. Lett.]
    With Sebastian Baader, Lukas Lewark, and Raphael Zentner.
25. Uniqueness of Embeddings of the Affine Line into Algebraic Groups. [ArXiv:1609.02113] [J. Algebraic Geom.]
    With Immanuel van Santen.
26. On the Upsilon invariant and satellite knots. [ArXiv:1604.04901] [Math. Z.]
    With JungHwan Park and Arunima Ray.
27. On cobordisms between knots, braid index, and the Upsilon-invariant. [ArXiv:1602.02637] [Math. Ann.]
    With David Krcatovich.
28. On the topological 4-genus of torus knots. [ArXiv:1509.07634] [Trans. Amer. Math. Soc.]
    With Sebastian Baader, Lukas Lewark, and Livio Liechti.
29. Alternating numbers of torus knots with small braid index. [ArXiv:1508.05825] [Indiana Univ. Math. J.]
    With Simon Pohlmann and Raphael Zentner.
30. On 2-bridge knots with differing smooth and topological slice genera. [ArXiv:1508.01431] [Proc. Amer. Math. Soc.]
    With Duncan McCoy.
31. A sharp signature bound for positive four-braids. [ArXiv:1508.00418] [Q. J. Math.]
32. The degree of the Alexander polynomial is an upper bound for the topological slice genus. [ArXiv:1504.01064] [Geom. Topol.]
33. Optimal Cobordisms between Torus Knots. [ArXiv:1501.00483] [Comm. Anal. Geom.]
34. Holomorphically Equivalent Algebraic Embeddings. [ArXiv:1409.7319]
    With Immanuel van Santen.
35. Signature and the Alexander polynomial. (An appendix to Livio Liechti's `Signature, positive Hopf plumbing and the Coxeter transformation.' [ArXiv:1401.5336] [Osaka J. Math.])
    With Livio Liechti.
36. The signature of positive braids is linearly bounded by their first Betti number. [ArXiv:1311.1242] [Internat. J. Math.]
37. Gordian adjacency for torus knots. [ArXiv:1301.5248] [Algebr. Geom. Topol.]