arXiv subjects

Eric Miles

Publications and source records attributed to Eric Miles.

5 recordsLinked to original sources

arXiv2015

Projectivity of Bridgeland Moduli Spaces on Del Pezzo Surfaces of Picard Rank 2

We prove that, for a natural class of Bridgeland stability conditions on $\mathbb{P}^1\times\mathbb{P}^1$ and the blow-up of $\mathbb{P}^2$ at a point, the moduli spaces of Bridgeland semistable objects are projective. Our technique is to find suitable regions of stability conditions with hearts that are (after "rotation") equivalent to representations of a quiver. The helix and tilting theory is well-behaved on Del Pezzo surfaces and we conjecture that this program (begun in arXiv:1203.0316) runs successfully for all Del Pezzo surfaces, and the analogous Bridgeland moduli spaces are projective.

math.AG↗

arXiv2014

Local reductions

We reduce non-deterministic time $T \ge 2^n$ to a 3SAT instance $ϕ$ of quasilinear size $|ϕ| = T \cdot \log^{O(1)} T$ such that there is an explicit circuit $C$ that on input an index $i$ of $\log |ϕ|$ bits outputs the $i$th clause, and each output bit of $C$ depends on $O(1)$ input bits. The previous best result was $C$ in NC$^1$. Even in the simpler setting of polynomial size $|ϕ| = \poly(T)$ the previous best result was $C$ in AC$^0$. More generally, for any time $T \ge n$ and parameter $r \leq n$ we obtain $\log_2 |ϕ| = \max(\log T, n/r) + O(\log n) + O(\log\log T)$ and each output bit of $C$ is a decision tree of depth $O(\log r)$. As an application, we tighten Williams' connection between satisfiability algorithms and circuit lower bounds (STOC 2010; SIAM J. Comput. 2013).

cs.CC↗

arXiv2014

Bridgeland Stability of Line Bundles on Surfaces

We study the Bridgeland stability of line bundles on surfaces using Bridgeland stability conditions determined by divisors. We show that given a smooth projective surface $S$, a line bundle $L$ is always Bridgeland stable for those stability conditions if there are no curves $C\subseteq S$ of negative self-intersection. When a curve $C$ of negative self-intersection is present, $L$ is destabilized by $L(-C)$ for some stability conditions. We conjecture that line bundles of the form $L(-C)$ are the only objects that can destabilize $L$, and that torsion sheaves of the form $L(C)|_C$ are the only objects that can destabilize $L[1]$. We prove our conjecture in several cases, and in particular for Hirzebruch surfaces.

math.AG↗

arXiv2013

Iterated group products and leakage resilience against NC^1

We show that if NC$^1 \neq$ L, then for every element $α$ of the alternating group $A_t$, circuits of depth $O(\log t)$ cannot distinguish between a uniform vector over $(A_t)^t$ with product $= α$ and one with product $=$ identity. Combined with a recent construction by the author and Viola in the setting of leakage-resilient cryptography [STOC '13], this gives a compiler that produces circuits withstanding leakage from NC$^1$ (assuming NC$^1 \neq$ L). For context, leakage from NC$^1$ breaks nearly all previous constructions, and security against leakage from P is impossible. %In the multi-query setting, circuits produced by this compiler use a simple secure hardware component. We build on work by Cook and McKenzie [J.\ Algorithms '87] establishing the relationship between L $=$ logarithmic space and the symmetric group $S_t$. Our techniques include a novel algorithmic use of commutators to manipulate the cycle structure of permutations in $A_t$.

cs.CC↗

arXiv2008

Alternating Hierarchies for Time-Space Tradeoffs

Nepomnjascii's Theorem states that for all 0 <= ε< 1 and k > 0 the class of languages recognized in nondeterministic time n^k and space n^ε, NTISP[n^k, n^ε], is contained in the linear time hierarchy. By considering restrictions on the size of the universal quantifiers in the linear time hierarchy, this paper refines Nepomnjascii's result to give a sub- hierarchy, Eu-LinH, of the linear time hierarchy that is contained in NP and which contains NTISP[n^k, n^ε]. Hence, Eu-LinH contains NL and SC. This paper investigates basic structural properties of Eu-LinH. Then the relationships between Eu-LinH and the classes NL, SC, and NP are considered to see if they can shed light on the NL = NP or SC = NP questions. Finally, a new hierarchy, zeta -LinH, is defined to reduce the space requirements needed for the upper bound on Eu-LinH.

cs.CC↗