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Humanity's Last Exam (with Long Phan, et al.)
[ arxiv ▪
abstract ]
Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities.
However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90% accuracy on popular benchmarks
like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last
Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic
benchmark of its kind with broad subject coverage. HLE consists of 3,000 questions across dozens of subjects,
including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and
consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known
solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval.
State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between
current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and
policymaking upon a clear understanding of model capabilities, we publicly release HLE at this https URL.
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Explicit polynomial bounds on Dehn functions of subgroups of hyperbolic groups. (with
R. Kropholler
and C. Llosa Isenrich, submitted).
[ pdf ▪
arxiv ▪
abstract ]
In 1999 Brady constructed the first example of a non-hyperbolic finitely presented subgroup of a hyperbolic
group by fibering a non-positively curved cube complex over the circle. We show that his example has Dehn function
bounded above by n96. This provides the first explicit polynomial upper bound on the Dehn function
of a finitely presented non-hyperbolic subgroup of a hyperbolic group. We also determine the precise hyperbolicity
constant for the 1-skeleton of the universal cover of the cube complex in Brady's construction with respect to
the 4-point condition for hyperbolicity.
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Property R∞ for new classes of Artin groups (with N. Vaskou)
[ pdf ▪
arxiv ▪
abstract ]
We establish property R∞ for Artin groups of spherical type Dn, n≥6,
their central quotients, and also for large hyperbolic-type free-of-infinity Artin groups and some other classes
of large-type Artin groups. The key ingredients are recent descriptions of the automorphism groups for these Artin groups
and their action on suitable Gromov-hyperbolic spaces.
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Endomorphisms of Artin groups of type Bn. (with L. Paris,
accepted in Journal of Algebra)
[ pdf ▪
arxiv ▪
abstract ]
We determine a classification of the endomorphisms of the Artin groups of spherical type
Bn for n≥5, and of their quotients by the center.
- Endomorphisms of Artin groups of type Ãn. (with L. Paris, submitted)
[ pdf ▪
arxiv ▪
abstract ]
We determine a classification of the endomorphisms of the Artin group of affine type Ãn for n≥4.
- Divergence, thickness and hypergraph index for general Coxeter groups. (with P. Dani,
Y. Naqvi and A. Thomas,
accepted in Israel Journal of Mathematics)
[ pdf ▪
arxiv ▪
GAP code ▪
slides ▪
video ▪
abstract ]
We introduce a computable combinatorial invariant, hypergraph index, for arbitrary Coxeter systems, which generalizes
the construction of Levcovitz for right-angled Coxeter groups. We use it to obtain an upper bound on the order of divergence
of general Coxeter groups. This upper bound is sharp for some infinite families of non-right-angled Coxeter groups,
and conjecturally, for all Coxeter groups.
- Property R∞ for some spherical and affine Artin-Tits groups.
(with M. Calvez)
Journal of Group Theory 25, 6 (2022), 1045-1054
[ pdf ▪
arXiv ▪
doi ▪
MathReviews ▪
zbMATH ▪
slides ▪
abstract ]
We give a short uniform proof of property R∞
for the Artin-Tits groups of spherical types An, Bn, D4, I2(m),
their pure subgroups, and for the Artin-Tits groups of affine types Ãn-1 and C̃n
for n≥2.
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Homological Dehn functions of groups of type FP2.
(with N. Brady and R. Kropholler, submitted)
[ pdf ▪
arXiv ▪
video ▪
abstract ]
We study the properties of homological Dehn functions of groups of type FP2. We show how to build uncountably many quasi-isometry classes of such groups with a given homological Dehn function. As an application we prove that there exists a group of type FP2 with quartic homological Dehn function and unsolvable word problem.
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Artin groups of types F4 and H4 are not commensurable with that of type D4.
Topology and its Applications 300 (2021), 107770
[ pdf ▪
arXiv ▪
doi ▪
MathReviews ▪
slides ▪
video ▪
abstract ]
We resolve two out of six cases left undecided in a recent article of Cumplido and Paris. We also determine the automorphism group of Art(D4) and describe torsion elements, their orders and conjugacy classes in all Artin groups of spherical type modulo their centers.
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Linearity of some low-complexity mapping class groups. Forum Mathematicum 32 (2020), no. 2, 279-286
[ pdf ▪
arXiv ▪
doi ▪
MathReviews ▪
zbMATH ▪
abstract ]
We show that the pure mapping class group of the orientable surface of genus g with b boundary components and n punctures is linear for the following values of (g,b,n): (0,m,n), (1,2,0), (1,1,1), (1,0,2), (1,3,0), (1,2,1), (1,1,2), (1,0,3).
A (longer) earlier version with an alternative computation "from first principles": [ pdf ]
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Realizable ranks of joins and intersections of subgroups in free groups.
International Journal of Algebra and Computation
30 (2020), no. 3, 625-666
[ pdf ▪
arXiv ▪
doi ▪
MathReviews ▪
zbMATH ▪
slides ▪
video ▪
abstract ]
We describe the locus of possible ranks ( rk(H∨K), rk(H∩K) ) for any given subgroups H, K of a free group. In particular, we resolve the remaining open case (m=4) of R.Guzman's "Group-Theoretic Conjecture" in the affirmative.
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Uncountably many quasi-isometry classes of groups of type FP.
(with R. Kropholler and
I. Leary)
American Journal of
Mathematics 142, 6 (2020), 1931-1944
[ pdf ▪
arXiv ▪
doi ▪
MathReviews ▪
slides ▪
abstract ]
We prove that among I. Leary's groups of type FP there exist uncountably many non-quasi-isometric ones. We also prove that for each n≥4 there exist uncountably many quasi-isometry classes of non-finitely presented n-dimensional Poincare duality groups.
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Genus bounds in right-angled Artin groups.
(with M. Forester and J. Tao)
Publicacions Matemàtiques 64 (2020), no. 1, 233-253
[ pdf ▪
arXiv ▪
doi ▪
MathReviews ▪
zbMATH ▪
slides ▪
comments ▪
abstract ]
We generalize Culler's proof for the lower bound for the stable commutator length in free groups to the case of right-angled Artin groups.
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Dehn functions of subgroups of right-angled Artin groups. (with N. Brady)
Geometriae Dedicata 200 (2019), 197-239
[ pdf ▪
arXiv (old version) ▪
doi ▪
MathReviews ▪
comments ▪
abstract ]
We show that polynomials of arbitrary integer degree are realizable as Dehn functions of subgroups in right-angled Artin groups. In the Appendix we prove that no finite index subgroup of the famous Gersten's free-by-cyclic group can be embedded into a right-angled Artin group.
