Mathematics Problem Archive

Find a 3-dimensional topological interpretation of the Jon es polynomial of links....

Understand Khovanov’s categorification of the Jones polyno - mial....

(H. Murakami) For a torus knot K, calculate CS(S3− K) (giving an appropriate definition of it) and calculate lim log JN (K) N (fixing an appropriate c...

[154] Every Vassiliev invariant of classical knots can be extended to a finite type invariant of long virtual knots. (Se e also Problem 3.9.) 2.8 Fini...

(K. Habiro [165], see also [153, “Theorem 5”]) Two m- strand string links L and L′ are Cd -equivalent if and only if v(L) = v(L′) for any A-valued fin...

(J. Roberts) What is graph cohomology the cohomology of?...

(R. Fenn, C. Rourke, B. Sanderson) H3(Rp)∼ = Z⊕ Z/pZ for p prime....

Is the Burau representation of B4 faithful?...

(3-move conjecture, Y. Nakanishi [305]) Any link can be related to a trivial link by a sequence of 3-moves....

Characterize hyperbolic $3$-manifolds with infinitely generated fundamental group. In particular, is there a $3$-manifold that is locally hyperbolic, ...

Let $M$ be a finite-volume hyperbolic $3$-manifold. If $M$ contains infinitely many immersed totally geodesic surfaces, must $M$ be arithmetic?...

What is the growth rate of dW (tn, 1)? One would expect that either the growth is linear, or dW (tn, 1) = O(logn). In the arithmetic groups case, the ...

. Is Z×BAut(F∞)+ homotopy equivalent to Ω ∞S∞? 4. The interplay between homotopy theory and the mapping class group is inspired by the study of confor...

For a closed surface of genus g≥ 3, is the Torelli subgroup of Mg,m undistorted? More generally, find a distorted finitely generated subgroup of Mg,m....

(Ergodic measures). Classify the ergodic measures for the action of SL(2, R) on H1(α) andQ1(β). McMullen [ McM3] has solved Problems 9 and 10 in the c...

(Analog of Ratner theorem)....

What are the Dehn functions of Aut(Fn) and Out(Fn) for n> 3?...

Produce non-trivial rational (co)homology classes of OutFn. Next we consider the group IOut n. In [ 38] Igusa defined higher Franz-Reidemeister torsio...

The stable rational cohomology of OutFn is trivial. Namely lim n→∞ ˜H ∗(OutFn; Q) = 0. We can aslo ask how the cohomology of Out Fn with twisted coeffic...

Elucidate the connection between the Morava stabilizer groups and the K(n)-local category. The first such problem, which is certainly not very hard an...

Presumably one should be able to form a category of E-S module spectra; spectra with an action of the ring spectrum E and a compatible action of the g...

Show that there is only a set of localizing subcategories. It is known that there is only a set of Bousfield classes (Ohkawa; Strickland simplified hi...

The safest sort of problem to work on with model categories is building one of interest in applications. The essential idea is: whenever someone uses ...

Every stable homotopy category I know of comes from a model category. Well, that used to be true, but it is no longer. Given a flat Hopf algebroid, St...

Given a symmetric monoidal model category C, Schwede and Shipley have given conditions under which the category of monoids in C is again a model categ...

The second step: show that the category of algebras over a cofibrant operad admits a model structure, where the fibrations and weak equivalences are t...

Find conditions under which algebras over a noncofibrant operad admit a model structure that generalize the monoid axiom of Schwede-Shipley. This woul...

Let A be a cofibrant operad as above. Use the above results to construct spectral sequences that converge to the homotopy groups of the space of A-alg...

Charles Rezk has a homotopy theory of homotopy theories. This is just a category, though it is large. The objects are generalizations of categories wh...

This one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-in...