Events

Title: Concrete analysis of a few aspects of lattice-based cryptography

Abstract: A seminal 2013 paper by Lyubashevsky, Peikert, and Regev proposed using ideal lattices as a foundation for post-quantum cryptography, supported by a polynomial-time security reduction from the approximate Shortest Independent Vectors Problem (SIVP) to the Decision Learning With Errors (DLWE) problem in ideal lattices. In our concrete analysis of this multi-step reduction, we find that the reduction’s tightness gap is so significant that it undermines any meaningful security guarantees. Additionally, we have concerns about the feasibility of the quantum aspect of the reduction in the near future. Moreover, when making the reduction concrete, the approximation factor for the SIVP problem turns out to be much larger than anticipated, suggesting that the approximate SIVP problem may not be hard for the proposed cryptosystem parameters.

Title:Smooth points on positroid varieties and planar N=4 supersymmetric Yang-Mills theory

 Abstract: Positroid varieties are subvarieties in the Grassmannian defined by cyclic rank conditions and which are related to Schubert varieties. We will provide a criterion for whether positroid varieties are smooth at certain distinguished points, and we will show that this information is sufficient to determine smoothness for the entire positroid variety. This will involve looking at combinatorial diagrams called "affine pipe dreams." We can also form a partial order on positroid varieties given by deletion and contraction, such that there is closure for smooth positroid varieties, and we will characterize the minimal singular elements in this order. Finally, we will discuss a couple of connections between the techniques of this work and planar N=4

SYM: the BCFW bridge decomposition and inverse soft factors.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title:Descents for Border Strip Tableaux

 Abstract: Lusztig's fake degree is the generating polynomial for the major index of standard Young tableaux of a given shape. Results of Springer and James & Kerber imply that, mysteriously, its evaluation at a d-th primitive root of unity yields the number of border strip tableaux with all strips of size d, up to sign. This is essentially the special case of the Murnaghan-Nakayama rule for rectangular partitions as cycle type. We refine this result to standard Young tableaux and border strip tableaux with a given number of descents. To do so, we introduce a new descent statistic for border strip tableaux, extending the classical definition for standard Young tableaux.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Problem Decomposition in Optimization:  Algorithmic Advances Beyond ADMM

Abstract:

Decomposition schemes like those coming from ADMM typically start by posing a separable-type problem in the Fenchel duality format.  They then pass to an augmented Lagrangian, which however can interfere with the separability and cause a slow-down.  Progressive decoupling offers a more flexible approach which can utilize augmented Lagrangians while maintaining decomposability.  Based on a variable metric extension of the proximal point algorithm that's applied in a twisted sort of way, progressive decoupling benefits from stopping criteria which can guarantee convergence despite inexact minimization in each iteration.   The convergence is generically at a linear rate, and for convex problems, is global. But the method also works for nonconvex problems when initiated close enough to a point that satisfies a natural extension of the strong sufficient condition for local optimality known from nonlinear programming.

This talk is held as part of the 26^th Annual Midwest Optimization Meeting (“MOM26”).

Title:Vanishing of Schubert coefficients

 Abstract: Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, they are in #P. In this talk we discuss the closely related problem of the vanishing of Schubert coefficients. We prove that this vanishing problem is rather low in the polynomial hierarchy and discuss implications of this result.

This is joint work with Igor Pak.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Proximal methods for nonsmooth and nonconvex fractional programs: when sparse optimization meets fractional programs

Abstract:Nonsmooth and nonconvex fractional programs are ubiquitous and also highly challenging. It includes the composite optimization problems studied extensively lately, and encompasses many important modern optimization problems arising from diverse areas such as the recent proposed scale invariant sparse signal reconstruction problem in signal processing, the robust Sharpe ratio optimization problems in finance and the sparse generalized eigenvalue problem in discrimination analysis.

In this talk, we will introduce extrapolated proximal methods for solving nonsmooth and nonconvex fractional programs and analyse their convergence behaviour. Interestingly, we will show that the proposed algorithm exhibits linear convergence for the scale invariant sparse signal reconstruction model,  and the sparse generalized eigenvalue problem with either cardinality regularization or sparsity constraints. This is achieved by identifying the explicit desingularization function of the Kurdyka-Ł ojasiewicz inequality for the merit function of the fractional optimization models. Finally, if time permits, we will present some preliminary encouraging numerical results for the proposed methods for sparse signal reconstruction and sparse Fisher discriminant analysis

The talk is based on joint work with R.I. Bo ̧t, M. Dao, T.K. Pong and P. Yu.

Title: Induced subgraphs of graphs of large $K_{r}$-free chromatic number

Abstract:For an integer $r\geq 2$, the $K_{r}$-free chromatic number of a graph $G$, denoted by $\chi_{r}(G)$, is the minimum size of a partition of the set of vertices of $G$ into parts each of which induces a $K_{r}$-free graph. In this setting, the $K_{2}$-free chromatic number is the usual chromatic number. Which are the unavoidable induced subgraphs of graphs of large $K_{r}$-free chromatic number? Generalizing the notion of $\chi$-boundedness, we say that a hereditary class of graphs is $\chi_{r}$-bounded if there exists a function which provides an upper bound for the $K_{r}$-free chromatic number of each graph of the class in terms of the graph's clique number. With an emphasis on a generalization of the Gy\'arf\'as-Sumner conjecture for $\chi_{r}$-bounded classes of graphs and on polynomial $\chi$-boundedness, I will discuss some recent developments on $\chi_{r}$-boundedness and related open problems. Based on joint work with Mathieu Rundstr\"om and Sophie Spirkl.

Title:Coloring the integers while avoiding monochromatic arithmetic

progressions

 Abstract: Consider coloring the positive integers either red or blue one at a time in order.  Van der Waerden's classical theorem states that no matter how you color the integers, you will eventually have k equally spaced integers all colored the same for any k.  But, how can we minimize the number of times k equally spaced integers are colored the same?  Even for k = 3, this question is unsolved.  We will discuss progress towards proving an existing conjecture by leveraging a connection to coloring the continuous interval [0,1]. Our strategy relies on identifying classes of colorings with permutations and then using mixed integer linear programming.  Joint work with Jonathan Kariv and Noah Williams.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1pm,

Title: Soft Condorcet Optimization

Abstract:

A common way to drive the progress of AI models and agents is to compare their performance on standardized benchmarks. This often involves aggregating individual performances across a potentially wide variety of tasks and benchmarks and many of the leaderboards that draw greatest attention are Elo-based. 

In this paper, we describe a novel ranking scheme inspired by social choice frameworks, called Soft Condorcet Optimization (SCO), to compute the optimal ranking of agents: the one that makes the fewest mistakes in predicting the agent comparisons in the evaluation data. This optimal ranking is the maximum likelihood estimate when evaluation data (which we view as votes) are interpreted as noisy samples from a ground truth ranking, a solution to Condorcet's original voting system criteria and inherits desirable social-choice inspired properties since SCO ratings are maximal for Condorcet winners when they exist, which we show is not necessarily true for the classical rating system Elo.

We propose three optimization algorithms to compute SCO ratings and evaluate their empirical performance across a variety of synthetic and real-world datasets, to illustrate different properties.

With Marc Lanctot, Ian Gemp, Quentin Berthet, Yoram Bachrach, Manfred Diaz, Roberto-Rafael Maura-Rivero,  Anna Koop, and Doina Precup

Title: On the relation among $\Delta$, $\chi$ and $\omega$

Abstract:I will present some work from my MMath thesis, which is on the relation among the maximum degree $\Delta(G)$, the chromatic number $\chi(G)$ and the clique number $\omega(G)$ of a graph $G$. In particular, we focus on two important and long-standing conjectures on this subject, the Borodin-Kostochka Conjecture and Reed's Conjecture. In 1977, Borodin and Kostochka conjectured that given a graph $G$ with $\Delta(G) \ge 9$, if $\chi(G) = \Delta(G)$, then $\omega(G) = \Delta(G)$. This is a step toward strengthening the well-known Brooks' Theorem. In 1998, Reed proposed a more general conjecture, which states that $\chi(G) \le \lceil \frac{1}{2} (\Delta(G)+\omega(G)+1) \rceil$ for any graph $G$.

In this talk, we show a weaker but more general Borodin-Kostochka-type result. That is, given a nonnegative integer $t$, for every graph $G$ with $\Delta(G) \ge 4t^2+11t+7$ and $\chi(G) = \Delta(G)-t$, the graph $G$ contains a clique of size $\Delta(G)-2t^2-7t-4$. We introduce the technique of Mozhan partitions and give a high-level overview of the proof. This generalizes some previous work on this topic. Then, we prove that both conjectures hold for odd-hole-free graphs. Lastly, we discuss a few constructions of classes of graphs for which Reed's Conjecture holds with equality, including a new family of irregular tight examples.