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Dave Anderson

Publications and source records attributed to Dave Anderson.

At least 19 recordsLinked to original sources

Measuring and Eliminating Refusals in Military Large Language Models

Military Large Language Models (LLMs) must provide accurate information to the warfighter in time-critical and dangerous situations. However, today's LLMs are imbued with safety behaviors that cause the LLM to refuse many legitimate queries in the military domain, particularly those related to violence, terrorism, or military technology. Our gold benchmark for assessing refusal rates, which was developed by veterans of the US Army and special forces, is to our knowledge the first dataset of its kind. We present results for refusal and deflection rates on 31 public models and 3 military models. We observe hard rejection rates as high as 98.2% and soft deflection rates ranging from 0% to 21.3%. We also present results on two additional synthetic datasets and show their correlations with the gold dataset. Finally, we perform abliteration using the Heretic library on a military-tuned gpt-oss-20b model, showing an absolute increase in answer rate of 66.5 points but an average relative decrease of 2% on other military tasks. In our concluding remarks, we argue for deeper specialization, including with mid-training and end-to-end post-training, to achieve zero refusals and maximum military task accuracy for closed military models.

cs.CL

EdgeRunner 20B: Military Task Parity with GPT-5 while Running on the Edge

We present EdgeRunner 20B, a fine-tuned version of gpt-oss-20b optimized for military tasks. EdgeRunner 20B was trained on 1.6M high-quality records curated from military documentation and websites. We also present four new tests sets: (a) combat arms, (b) combat medic, (c) cyber operations, and (d) mil-bench-5k (general military knowledge). On these military test sets, EdgeRunner 20B matches or exceeds GPT-5 task performance with 95%+ statistical significance, except for the high reasoning setting on the combat medic test set and the low reasoning setting on the mil-bench-5k test set. Versus gpt-oss-20b, there is no statistically-significant regression on general-purpose benchmarks like ARC-C, GPQA Diamond, GSM8k, IFEval, MMLU Pro, or TruthfulQA, except for GSM8k in the low reasoning setting. We also present analyses on hyperparameter settings, cost, and throughput. These findings show that small, locally-hosted models are ideal solutions for data-sensitive operations such as in the military domain, allowing for deployment in air-gapped edge devices.

cs.AI

Toric arc schemes and $q$-enumeration of lattice points

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers-Szeg\H{o} polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behavior of the coefficients at $q = 1$ is Gaussian.

math.AG

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $\chi_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.

math.AG

Minuscule Schubert calculus and the geometric Satake correspondence

We describe a relationship between work of Laksov, Gatto, and their collaborators on realizations of (generalized) Schubert calculus of Grassmannians, and the geometric Satake correspondence of Lusztig, Ginzburg, and Mirkovi\'c and Vilonen. Along the way we obtain new proofs of equivariant Giambelli formulas for the ordinary and orthogonal Grassmannians, as well as a simple derivation of the "rim-hook" rule for computing in the equivariant quantum cohomology of the Grassmannian.

math.AG

Equivariant Grothendieck-Riemann-Roch and localization in operational K-theory

We produce a Grothendieck transformation from bivariant operational $K$-theory to Chow, with a Riemann-Roch formula that generalizes classical Grothendieck-Verdier-Riemann-Roch. We also produce Grothendieck transformations and Riemann-Roch formulas that generalize the classical Adams-Riemann-Roch and equivariant localization theorems. As applications, we exhibit a projective toric variety $X$ whose equivariant $K$-theory of vector bundles does not surject onto its ordinary $K$-theory, and describe the operational $K$-theory of spherical varieties in terms of fixed-point data. In an appendix, Vezzosi studies operational $K$-theory of derived schemes and constructs a Grothendieck transformation from bivariant algebraic $K$-theory of relatively perfect complexes to bivariant operational $K$-theory.

math.AG

K-classes of Brill-Noether loci and a determinantal formula

We compute the Euler characteristic of the structure sheaf of the Brill-Noether locus of linear series with special vanishing at up to two marked points. When the Brill-Noether number $\rho$ is zero, we recover the Castelnuovo formula for the number of special linear series on a general curve; when $\rho=1$, we recover the formulas of Eisenbud-Harris, Pirola, and Chan-Mart\'in-Pflueger-Teixidor for the arithmetic genus of a Brill-Noether curve of special divisors. These computations are obtained as applications of a new determinantal formula for the K-theory class of certain degeneracy loci. Our degeneracy locus formula also specializes to new determinantal expressions for the double Grothendieck polynomials corresponding to 321-avoiding permutations, and gives double versions of the flagged skew Grothendieck polynomials recently introduced by Matsumura. Our result extends the formula of Billey-Jockusch-Stanley expressing Schubert polynomials for 321-avoiding permutations as generating functions for flagged skew tableaux.

math.AG

Computing torus-equivariant K-theory of singular varieties

This expository note surveys some results on equivariant K-theory of varieties with a torus action, focusing on recent work with Sam Payne and Richard Gonzales. It is based on my contribution to the Clifford Lectures at Tulane University in March 2015.

math.AG

Effective divisors on Bott-Samelson varieties

We compute the cone of effective divisors on a Bott-Samelson variety corresponding to an arbitrary sequence of simple roots. The main tool is a general result concerning effective cones of certain $B$-equivariant $\mathbb{P}^1$ bundles. As an application, we compute the cone of effective codimension-two cycles on Bott-Samelson varieties corresponding to reduced words. We also obtain an auxiliary result giving criteria for a Bott-Samelson variety to contain a dense $B$-orbit, and we construct desingularizations of intersections of Schubert varieties. An appendix exhibits an explicit divisor showing that any Bott-Samelson variety is log Fano.

math.AG

The Lie algebra of type G_2 is rational over its quotient by the adjoint action

Let G be a split simple group of type G_2 over a field k, and let g be its Lie algebra. Answering a question of Colliot-Th\'el\`ene, Kunyavski\u{i}, Popov, and Reichstein, we show that the function field k(g) is generated by algebraically independent elements over the field of adjoint invariants k(g)^G. Soit G un groupe alg\'ebrique simple et d\'eploy\'e de type G_2 sur un corps k. Soit g son alg\`ebre de Lie. On d\'emontre que le corps des fonctions k(g) est transcendant pur sur le corps k(g)^G des invariants adjoints. Ceci r\'epond par l'affirmative \`a une question pos\'ee par Colliot-Th\'el\`ene, Kunyavski\u{i}, Popov et Reichstein.

math.AG

Operational K-theory

We study the operational bivariant theory associated to the covariant theory of Grothendieck groups of coherent sheaves, and prove that it has many geometric properties analogous to those of operational Chow theory. This operational K-theory agrees with Grothendieck groups of vector bundles on smooth varieties, admits a natural map from the Grothendieck group of perfect complexes on general varieties, satisfies descent for Chow envelopes, and is A^1-homotopy invariant. Furthermore, we show that the operational K-theory of a complete linear variety is dual to the Grothendieck group of coherent sheaves. As an application, we show that the K-theory of perfect complexes on any complete toric threefold surjects onto this group. Finally, we identify the equivariant operational K-theory of an arbitrary toric variety with the ring of integral piecewise exponential functions on the associated fan.

math.AG

Okounkov bodies of finitely generated divisors

We show that the Okounkov body of a big divisor with finitely generated section ring is a rational simplex, for an appropriate choice of flag; furthermore, when the ambient variety is a surface, the same holds for every big divisor. Under somewhat more restrictive hypotheses, we also show that the corresponding semigroup is finitely generated.

math.AG

Introduction to Equivariant Cohomology in Algebraic Geometry (IMPANGA 2010)

These are lecture notes from the IMPANGA 2010 Summer School. The lectures survey some of the main features of equivariant cohomology at an introductory level. The first part is an overview, including basic definitions and examples. In the second lecture, I discuss one of the most useful aspects of the theory: the possibility of localizing at fixed points without losing information. The third lecture focuses on Grassmannians, and describes some recent positivity results about their equivariant cohomology rings.

math.AG

Equivariant quantum Schubert polynomials

We establish an equivariant quantum Giambelli formula for partial flag varieties. The answer is given in terms of a specialization of universal double Schubert polynomials. Along the way, we give new proofs of the presentation of the equivariant quantum cohomology ring, as well as Graham-positivity of the structure constants in equivariant quantum Schubert calculus.

math.AG

Positivity of equivariant Gromov-Witten invariants

We show that the equivariant Gromov-Witten invariants of a projective homogeneous space G/P exhibit Graham-positivity: when expressed as polynomials in the positive roots, they have nonnegative coefficients.

math.AG

Okounkov bodies and toric degenerations

Let \Delta be the Okounkov body of a divisor D on a projective variety X. We describe a geometric criterion for \Delta to be a lattice polytope, and show that in this situation X admits a flat degeneration to the corresponding toric variety. This degeneration is functorial in an appropriate sense.

math.AG

Arc spaces and equivariant cohomology

We present a new geometric interpretation of equivariant cohomology in which one replaces a smooth, complex $G$-variety $X$ by its associated arc space $J_{\infty} X$, with its induced $G$-action. This not only allows us to obtain geometric classes in equivariant cohomology of arbitrarily high degree, but also provides more flexibility for equivariantly deforming classes and geometrically interpreting multiplication in the equivariant cohomology ring. Under appropriate hypotheses, we obtain explicit bijections between $\mathbb{Z}$-bases for the equivariant cohomology rings of smooth varieties related by an equivariant, proper birational map. We also show that self-intersection classes can be represented as classes of contact loci, under certain restrictions on singularities of subvarieties. We give several applications. Motivated by the relation between self-intersection and contact loci, we define higher-order equivariant multiplicities, generalizing the equivariant multiplicities of Brion and Rossmann; these are shown to be local singularity invariants, and computed in some cases. We also present geometric $\mathbb{Z}$-bases for the equivariant cohomology rings of a smooth toric variety (with respect to the dense torus) and a partial flag variety (with respect to the general linear group).

math.AG