Mathematics Problem Archive

Is every Laplacian algebra of polynomials maximal?...

Does every curve bounding a surface of positive curvature in 3-space have (at least) four points where the torsion vanishes?...

Does connectedness of the shadows imply that f(M) is convex?...

Deligne's conjecture on Hochschild cohomology about the operadic structure on Hochschild cochain complex....

Does every convex polyhedron have Rupert's property?...

The asymptotic behavior of a variety of combinatorial problems has now been analyzed in great detail. Here we have in mind Ulam's problem for the leng...

For a full-rank integer lattice, is finding a nonzero vector of minimum Euclidean norm NP-hard?...

Under the promise that $b^2-4ac$ is not a square, is deciding whether $ax^2+bxy+cy^2+dx+ey+f=0$ has an integral solution NP-hard?...

Is the same binary quadratic Diophantine solvability problem NP-hard under randomized reductions?...

Develop a fast algorithm to compute the Tate–Lichtenbaum pairing $T_n$....

Factor $N=pqr$ from fewer known bits of its prime factors....

Given integers $n$ and a prime power $q$, construct, when possible, an elliptic curve $E/\mathbb{F}_q$ with $\#E(\mathbb{F}_q)=n$....

Speed up the computation of cryptographic pairings....

Give the cited pairing-based signature constructions and their security proofs without relying on distortion maps....

Construct efficient pseudorandom generators from ideal-lattice problems....

Construct efficient digital-signature schemes from ideal-lattice problems....

Base cryptographic constructions directly on worst-case hardness assumptions from algebraic number theory....

Construct an efficient ideal-lattice version of Regev's quantum-SVP-based cryptosystem....

For the classical-group pairs and generic local $L$-parameters in the Gan–Gross–Prasad setting, is there exactly one relevant representation $\pi$ in ...

Can every integer be written as a sum of four perfect cubes?...

Let $S_n=S_0+X_1+\cdots+X_n$ be a renewal process whose i.i.d. strictly positive recurrence times have infinite mean and a non-lattice distribution. I...

Prove that every bounded-degree transient hyperbolic graph contains a transient subtree....

For simple random walk $(X_n)$ on the uniform infinite planar triangulation, prove that the graph distance from the starting point has exponent $1/4$,...

Take a uniform quadrangulation with $N$ faces conditioned to have genus $CN$, where $0<C<1/4$. Prove that its rooted local limit is the stochastic hyp...

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...