arXiv subjects

Milind Sohoni

Publications and source records attributed to Milind Sohoni.

14 recordsLinked to original sources

arXiv2025

Strategic Costs of Perceived Bias in Fair Selection

Meritocratic systems, from admissions to hiring, aim to impartially reward skill and effort. Yet persistent disparities across race, gender, and class challenge this ideal. Some attribute these gaps to structural inequality; others to individual choice. We develop a game-theoretic model in which candidates from different socioeconomic groups differ in their perceived post-selection value--shaped by social context and, increasingly, by AI-powered tools offering personalized career or salary guidance. Each candidate strategically chooses effort, balancing its cost against expected reward; effort translates into observable merit, and selection is based solely on merit. We characterize the unique Nash equilibrium in the large-agent limit and derive explicit formulas showing how valuation disparities and institutional selectivity jointly determine effort, representation, social welfare, and utility. We further propose a cost-sensitive optimization framework that quantifies how modifying selectivity or perceived value can reduce disparities without compromising institutional goals. Our analysis reveals a perception-driven bias: when perceptions of post-selection value differ across groups, these differences translate into rational differences in effort, propagating disparities backward through otherwise "fair" selection processes. While the model is static, it captures one stage of a broader feedback cycle linking perceptions, incentives, and outcome--bridging rational-choice and structural explanations of inequality by showing how techno-social environments shape individual incentives in meritocratic systems.

cs.GT↗

arXiv2023

Orbit closures, stabilizer limits and intermediate $G$-varieties

In this paper we study the orbit closure problem for a reductive group $G\subseteq GL(X)$ acting on a finite dimensional vector space $V$ over $\C$. We assume that the center of $GL(X)$ lies within $G$ and acts on $V$ through a fixed non-trivial character. We study points $y,z\in V$ where (i) $z$ is obtained as the leading term of the action of a 1-parameter subgroup $λ(t)\subseteq G$ on $y$, and (ii) $y$ and $z$ have large distinctive stabilizers $K,H \subseteq G$. Let $O(z)$ (resp. $O(y)$) denote the $G$-orbits of $z$ (resp. $y$), and $\overline{O(z)}$ (resp. $\overline{O(y)}$) their closures, then (i) implies that $z\in \overline{O(y)}$. We address the question: under what conditions can (i) and (ii) be simultaneously satisfied, i.e, there exists a 1-PS $λ\subseteq G$ for which $z$ is observed as a limit of $y$. Using $λ$, we develop a leading term analysis which applies to $V$ as well as to ${\cal G}= Lie(G)$ the Lie algebra of $G$ and its subalgebras ${\cal K}$ and ${\cal H}$, the Lie algebras of $K$ and $H$ respectively. Through this we construct the Lie algebra $\hat{\cal K} \subseteq {\cal H}$ which connects $y$ and $z$ through their Lie algebras. We develop the properties of $\hat{\cal K}$ and relate it to the action of ${\cal H}$ on $\overline{N}=V/T_z O(z)$, the normal slice to the orbit $O(z)$. We examine the case of {\em alignment} when a semisimple element belongs to both ${\cal H}$ and ${\cal K}$, and the conditions for the same. We illustrate some consequences of alignment. Next, we examine the possibility of {\em intermediate $G$-varieties} $W$ which lie between the orbit closures of $z$ and $y$, i.e. $\overline{O(z)} \subsetneq W \subsetneq O(y)$. These have a direct bearing on representation theoretic as well as geometric properties which connect $z$ and $y$.

math.RT↗

arXiv2022

Geometric Complexity Theory -- Lie Algebraic Methods for Projective Limits of Stable Points

Let $G$ be a connected reductive group acting on a complex vector space $V$ and projective space ${\mathbb P}V$. Let $x\in V$ and ${\cal H}\subseteq {\cal G}$ be the Lie algebra of its stabilizer. Our objective is to understand points $[y]$, and their stabilizers which occur in the vicinity of $[x]$. We construct an explicit ${\cal G}$-action on a suitable neighbourhood of $x$, which we call the local model at $x$. We show that Lie algebras of stabilizers of points in the vicinity of $x$ are parameterized by subspaces of ${\cal H}$. When ${\cal H}$ is reductive these are Lie subalgebras of ${\cal H}$. If the orbit of $x$ is closed this also follows from Luna's theorem. Our construction involves a map connected to the local curvature form at $x$. We apply the local model to forms, when the form $g$ is obtained from the form $f$ as the leading term of a one parameter family acting on $f$. We show that there is a flattening ${\cal K}_0$ of ${\cal K}$, the stabilizer of $f$ which sits as a subalgebra of ${\cal H}$, the stabilizer $g$. We specialize to the case of forms $f$ whose $SL(X)$-orbits are affine, and the orbit of $g$ is of co-dimension $1$. We show that (i) either ${\cal H}$ has a very simple structure, or (ii) conjugates of the elements of ${\cal K}$ also stabilize $g$ and the tangent of exit. Next, we apply this to the adjoint action. We show that for a general matrix $X$, the signatures of nilpotent matrices in its projective orbit closure (under conjugation) are determined by the multiplicity data of the spectrum of $X$. Finally, we formulate the path problem of finding paths with specific properties from $y$ to its limit points $x$ as an optimization problem using local differential geometry. Our study is motivated by Geometric Complexity Theory proposed by the second author and Ketan Mulmuley.

math.RT↗

arXiv2019

Fast Algorithms for Rank-1 Bimatrix Games

The rank of a bimatrix game is the matrix rank of the sum of the two payoff matrices. This paper comprehensively analyzes games of rank one, and shows the following: (1) For a game of rank r, the set of its Nash equilibria is the intersection of a generically one-dimensional set of equilibria of parameterized games of rank r-1 with a hyperplane. (2) One equilibrium of a rank-1 game can be found in polynomial time. (3) All equilibria of a rank-1 game can be found by following a piecewise linear path. In contrast, such a path-following method finds only one equilibrium of a bimatrix game. (4) The number of equilibria of a rank-1 game may be exponential. (5) There is a homeomorphism between the space of bimatrix games and their equilibrium correspondence that preserves rank. It is a variation of the homeomorphism used for the concept of strategic stability of an equilibrium component.

cs.GT↗

arXiv2014

Incorporating Sharp Features in the General Solid Sweep Framework

This paper extends a recently proposed robust computational framework for constructing the boundary representation (brep) of the volume swept by a given smooth solid moving along a one parameter family $h$ of rigid motions. Our extension allows the input solid to have sharp features, i.e., to be of class G0 wherein, the unit outward normal to the solid may be discontinuous. In the earlier framework, the solid to be swept was restricted to be G1, and thus this is a significant and useful extension of that work. This naturally requires a precise description of the geometry of the surface generated by the sweep of a sharp edge supported by two intersecting smooth faces. We uncover the geometry along with the related issues like parametrization, self-intersection and singularities via a novel mathematical analysis. Correct trimming of such a surface is achieved by a delicate analysis of the interplay between the cone of normals at a sharp point and its trajectory under $h$. The overall topology is explicated by a key lifting theorem which allows us to compute the adjacency relations amongst entities in the swept volume by relating them to corresponding adjacencies in the input solid. Moreover, global issues related to body-check such as orientation are efficiently resolved. Many examples from a pilot implementation illustrate the efficiency and effectiveness of our framework.

cs.GR↗

arXiv2014

A Computational Framework for Boundary Representation of Solid Sweeps

This paper proposes a robust algorithmic and computational framework to address the problem of modeling the volume obtained by sweeping a solid along a trajectory of rigid motions. The boundary representation (simply brep) of the input solid naturally induces a brep of the swept volume. We show that it is locally similar to the input brep and this serves as the basis of the framework. All the same, it admits several intricacies: (i) geometric, in terms of parametrizations and, (ii) topological, in terms of orientations. We provide a novel analysis for their resolution. More specifically, we prove a non-trivial lifting theorem which allows to locally orient the output using the orientation of the input. We illustrate the framework by providing many examples from a pilot implementation.

cs.CG↗

arXiv2013

Geometric Complexity Theory IV: nonstandard quantum group for the Kronecker problem

The Kronecker coefficient g_{λμν} is the multiplicity of the GL(V)\times GL(W)-irreducible V_λ\otimes W_μin the restriction of the GL(X)-irreducible X_νvia the natural map GL(V)\times GL(W) \to GL(V \otimes W), where V, W are \mathbb{C}-vector spaces and X = V \otimes W. A fundamental open problem in algebraic combinatorics is to find a positive combinatorial formula for these coefficients. We construct two quantum objects for this problem, which we call the nonstandard quantum group and nonstandard Hecke algebra. We show that the nonstandard quantum group has a compact real form and its representations are completely reducible, that the nonstandard Hecke algebra is semisimple, and that they satisfy an analog of quantum Schur-Weyl duality. Using these nonstandard objects as a guide, we follow the approach of Adsul, Sohoni, and Subrahmanyam to construct, in the case dim(V) = dim(W) =2, a representation \check{X}_νof the nonstandard quantum group that specializes to Res_{GL(V) \times GL(W)} X_νat q=1. We then define a global crystal basis +HNSTC(ν) of \check{X}_νthat solves the two-row Kronecker problem: the number of highest weight elements of +HNSTC(ν) of weight (λ,μ) is the Kronecker coefficient g_{λμν}. We go on to develop the beginnings of a graphical calculus for this basis, along the lines of the U_q(\sl_2) graphical calculus, and use this to organize the crystal components of +HNSTC(ν) into eight families. This yields a fairly simple, explicit and positive formula for two-row Kronecker coefficients, generalizing a formula of Brown, van Willigenburg, and Zabrocki. As a byproduct of the approach, we also obtain a rule for the decomposition of Res_{GL_2 \times GL_2 \rtimes §_2} X_νinto irreducibles.

cs.CC↗

arXiv2013

Local and Global Analysis of Parametric Solid Sweeps

In this work, we propose a detailed computational framework for modelling the envelope of the swept volume, that is the boundary of the volume obtained by sweeping an input solid along a trajectory of rigid motions. Our framework is adapted to the well-established industry-standard brep format to enable its implementation in modern CAD systems. This is achieved via a "local analysis", which covers parametrization and singularities, as well as a "global theory" which tackles face-boundaries, self-intersections and trim curves. Central to the local analysis is the "funnel" which serves as a natural parameter space for the basic surfaces constituting the sweep. The trimming problem is reduced to the problem of surface-surface intersections of these basic surfaces. Based on the complexity of these intersections, we introduce a novel classification of sweeps as either decomposable or non-decomposable. Further, we construct an {\em invariant} function $θ$ on the funnel which efficiently separates decomposable and non-decomposable sweeps. Through a geometric theorem we also show intimate connections between $θ$, local curvatures and the inverse trajectory used in earlier works as an approach towards trimming. In contrast to the inverse trajectory approach, $θ$ is robust and is the key to a complete structural understanding, and an efficient computation of both, the singular locus and the trim curves, which are central to a stable implementation. Several illustrative outputs of a pilot implementation are included.

cs.CG↗

arXiv2012

A procedural framework and mathematical analysis for solid sweeps

Sweeping is a powerful and versatile method of designing objects. Boundary of volumes (henceforth envelope) obtained by sweeping solids have been extensively investigated in the past, though, obtaining an accurate parametrization of the envelope remained computationally hard. The present work reports our approach to this problem as well as the important problem of identifying self-intersections within the envelope. Parametrization of the envelope is, of course, necessary for its use in most current CAD systems. We take the more interesting case when the solid is composed of several faces meeting smoothly. We show that the face structure of the envelope mimics locally that of the solid. We adopt the procedural approach at defining the geometry in this work which has the advantage of being accurate as well as computationally efficient. The problem of detecting local self-intersections is central to a robust implementation of the solid sweep. This has been addressed by computing a subtle mathematical invariant which detects self-intersections, and which is computationally benign and requires only point queries.

cs.OH↗

arXiv2010

Rank-1 Bi-matrix Games: A Homeomorphism and a Polynomial Time Algorithm

Given a rank-1 bimatrix game (A,B), i.e., where rank(A+B)=1, we construct a suitable linear subspace of the rank-1 game space and show that this subspace is homeomorphic to its Nash equilibrium correspondence. Using this homeomorphism, we give the first polynomial time algorithm for computing an exact Nash equilibrium of a rank-1 bimatrix game. This settles an open question posed in Kannan and Theobald (SODA 2007) and Theobald (2007). In addition, we give a novel algorithm to enumerate all the Nash equilibria of a rank-1 game and show that a similar technique may also be applied for finding a Nash equilibrium of any bimatrix game. This technique also proves the existence, oddness and the index theorem of Nash equilibria in a bimatrix game. Further, we extend the rank-1 homeomorphism result to a fixed rank game space, and give a fixed point formulation on $[0,1]^k$ for solving a rank-k game. The homeomorphism and the fixed point formulation are piece-wise linear and considerably simpler than the classical constructions.

cs.GT↗

arXiv2010

Nash equilibria in Fisher market

Much work has been done on the computation of market equilibria. However due to strategic play by buyers, it is not clear whether these are actually observed in the market. Motivated by the observation that a buyer may derive a better payoff by feigning a different utility function and thereby manipulating the Fisher market equilibrium, we formulate the {\em Fisher market game} in which buyers strategize by posing different utility functions. We show that existence of a {\em conflict-free allocation} is a necessary condition for the Nash equilibria (NE) and also sufficient for the symmetric NE in this game. There are many NE with very different payoffs, and the Fisher equilibrium payoff is captured at a symmetric NE. We provide a complete polyhedral characterization of all the NE for the two-buyer market game. Surprisingly, all the NE of this game turn out to be symmetric and the corresponding payoffs constitute a piecewise linear concave curve. We also study the correlated equilibria of this game and show that third-party mediation does not help to achieve a better payoff than NE payoffs.

cs.GT↗

arXiv2007

Geometric Complexity Theory: Introduction

These are lectures notes for the introductory graduate courses on geometric complexity theory (GCT) in the computer science department, the university of Chicago. Part I consists of the lecture notes for the course given by the first author in the spring quarter, 2007. It gives introduction to the basic structure of GCT. Part II consists of the lecture notes for the course given by the second author in the spring quarter, 2003. It gives introduction to invariant theory with a view towards GCT. No background in algebraic geometry or representation theory is assumed. These lecture notes in conjunction with the article \cite{GCTflip1}, which describes in detail the basic plan of GCT based on the principle called the flip, should provide a high level picture of GCT assuming familiarity with only basic notions of algebra, such as groups, rings, fields etc.

cs.CC↗

arXiv2006

Geometric Complexity Theory II: Towards explicit obstructions for embeddings among class varieties

In part I we reduced the arithmetic (characteristic zero) version of the P \not \subseteq NP conjecture to the problem of showing that a variety associated with the complexity class NP cannot be embedded in the variety associated the complexity class P. We call these class varieties. In this paper, this approach is developed further, reducing the nonexistence problems, such as the P vs. NP and related lower bound problems, to existence problems: specifically to proving existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such obstructions. The first result is a generalization of the Borel-Weil theorem to a class of orbit closures, which include class varieties. The recond result is a weaker form of a conjectured analogue of the second fundamental theorem of invariant theory for the class variety associated with the complexity class NC. These results indicate that the fundamental lower bound problems in complexity theory are intimately linked with explicit construction problems in algebraic geometry and representation theory.

cs.CC↗

arXiv2005

Geometric Complexity III: on deciding positivity of Littlewood-Richardson coefficients

We point out that the remarkable Knutson and Tao Saturation Theorem and polynomial time algorithms for LP have together an important and immediate consequence in Geometric Complexity Theory. The problem of deciding positivity of Littlewood-Richardson coefficients for GLn(C) belongs to P. Furthermore, the algorithm is strongly polynomial. The main goal of this article is to explain the significance of this result in the context of Geometric Complexity Theory. Furthermore, it is also conjectured that an analogous result holds for arbitrary symmetrizable Kac-Moody algebras.

cs.CC↗