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Title: Log-concavity of random Radon partitions

ÌýAbstract: Over one hundred years ago, Radon proved that any set of d+2 points in R^d can be partitioned into two sets whose convex hulls intersect. I will talk about Radon partitions when the points are selected randomly. In particular, if the points are independent normal random vectors, let p_k be the probability that the Radon partition has size (k, d+2-k). Answering a conjecture of Kalai and White, we show that the sequence (p_k) is ultra log-concave and that, in fact, a balanced partition is the most likely. Joint work with Swee Hong Chan, Gil Kalai, Bhargav Narayanan, and Moshe White.

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

Title:ÌýExact algorithms for combinatorial interdiction problems

Abstract: Typical optimization paradigms involve a single decision maker that can control all variables involved. However, several practical situations involve multiple (potentially adversarial) decision makers. Bilevel optimization is a field that involves two decision makers. The key paradigm is that an upper-level decision maker acts first and after observing the upper-level decisions, a lower level decision maker optimizes their own objective. One particular important instance of such problems are so-called interdiction problems, where the upper-level decision is to try and forbid access to some decisions by the lower level and the goal of the upper level is to make the most impact into the lower-level decisions. These problems are, in general $\Sigma^P_2$-hard (likely harder than NP-hard).Ìý

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In this talk I will present recent work on improving significantly on state-of-the-art exact algorithms to obtain optimal solutions to some combinatorial interdiction problems. Despite the hardness results, our algorithm can solve instances consistently in a short amount of time. We also generalize our algorithms to propose a framework that could be applied to other similar problems, by deriving relaxations based on looking at these problems as games and performing operations on such games.Ìý

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This is joint work with Noah Weninger.

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Title:ÌýQuantum Search with the Signless Laplacian

Abstract: Continuous-time quantum walks are typically effected by either the discrete Laplacian or the adjacency matrix. In this paper, we explore a third option: the signless Laplacian, which has applications in algebraic graph theory and may arise in layered antiferromagnetic materials. We explore spatial search on the complete bipartite graph, which is generally irregular and breaks the equivalence of the three quantum walks for regular graphs, and where the search oracle breaks the equivalence of the Laplacian and signless Laplacian quantum walks on bipartite graphs without the oracle. We prove that a uniform superposition over all the vertices of the graph partially evolves to the marked vertices in one partite set, with the choice of set depending on the jumping rate of the quantum walk. We boost this success probability to 1 by proving that a particular non-uniform initial state completely evolves to the marked vertices in one partite set, again depending on the jumping rate. For some parameter regimes, the signless Laplacian yields the fastest search algorithm of the three, suggesting that it could be a new tool for developing faster quantum algorithms.

This is joint work with Molly McLaughlin.

Title:Strongly connected orientations and integer lattices

Abstract: Let D = (V, A) be a digraph whose underlying graph is 2-edge-connected, and let P be the polytope whose vertices are the incidence vectors of arc sets whose reversal makes D strongly connected. We study the lattice theoretic properties of the integer points contained in a proper 'slanted' face F of P. We prove under a mild necessary condition that the 0,1 points in FÌýcontain anÌýintegralÌýbasisÌýB, i.e., B is linearly independent, and every integral vector in the linear of span of F is an integral linear combination of B. This result is surprising as the integer points in F do not necessarily form aÌýHilbert basis.

Our result has consequences for head-disjoint strong orientations in hypergraphs, and also to a famous conjecture by Woodall that the minimum size of a dicut of D, say k, is equal to the maximum number of disjoint dijoins. We prove a relaxation of this conjecture, by finding for any odd prime number p, a p-adic packing of dijoins of value k and of support size at most 2|A|. We also prove that the all-ones vector belongs to the lattice generated by the 0,1 points in F, where F is the face of P satisfying x(C) = 1 for every minimum dicut C.

This is based on joint work with Gerard Cornuejols, Siyue Liu, and Olha Silina.

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Title:ÌýExplicit Lossless Vertex Expanders

Abstract: We give the first construction of explicit constant-degree lossless vertex expanders. Specifically, for any ε>0 and sufficiently large d, we give an explicit construction of an infinite family of d-regular graphs where every small set S of vertices has (1−ε)d|S| neighbors (which implies (1−2ε)d|S| unique-neighbors). Our results also extend naturally to construct biregular bipartite graphs of any constant imbalance, where small sets on each side have strong expansion guarantees. The graphs we construct admit a free group action, and hence realize new families of quantum LDPC codes of Lin and M. Hsieh with a linear time decoding algorithm.

Our construction is based on taking an appropriate product of a constant-sized lossless expander with a base graph constructed from Ramanujan Cayley cubical complexes.

Based on joint work with Jun-Ting Hsieh, Alexander Lubotzky, Assaf Reiner, and Rachel Yun Zhang ()

Title:ÌýRandom analogues of Erd\H{o}s-Ko-Rado type results.

Abstract:ÌýThe classical Erd\H{o}s-Ko-Rado (EKR) theorem and its variants can be translated into characterizing maximum co-cliques of graphs in Association schemes. For instance, the classical Erd\H{o}s-Ko-Rado characterizes maximum co-cliques in the Kneser graph. Given a graph $G$, by $G_{p}$, we denote the random subgraph of $G$ in which edges appear independently, each with a probability $p$. In this talk, we consider the following question: for which probabilities is the independence number of $G_{p}$ equal to that of $G$? Bollabas-Narayanan-Raigorodskii investigated the independence numbers of random subgraphs of the Kneser graph. In this talk, we will investigate the independence numbers of random subgraphs of (i) the derangement graph on permutations; and (ii) the perfect matching graphs. The derangement graph is associated with the EKR type result on permutations and the perfect matching graph is associated with EKR type result on perfect matchings. This is joint work with the members of the PIMS Collaborative Research Group on Movement and Symmetry in graphs. ÌýÌý

Title:Arrangements and Likelihood

Abstract: In this talk, we establish connections between hypersurface arrangements and likelihood geometry. The central object is the likelihood correspondence which captures the dependence between data and critical points of the likelihood function of a statistical model parametrized by the polynomials defining the arrangement. In particle physics, this same object is known as the scattering correspondence. The connection to hypersurface arrangements leads to a new description of the prime ideal of the likelihood correspondence, which is often computationally advantageous. This description is based on the Rees algebra of the likelihood module of the arrangement, a module closely related to the module of logarithmic derivations. We present results for generic and graphic arrangements.

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

Title:Faster linear algebra using treewidth

Abstract:

: We look at the complexity of solving sparse linear systems as a function of the treewidth of the instance. That is, the sparse matrix is associated with a sparse graph, and solutions can be found faster when that graph has low treewidth. We give a parameterized algorithm in system size and treewidth achieving the conjectured optimal performance.

This is joint work with Daniel Grier.

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Title:ÌýAlice, Bob, and colours: An algebraic approach to quantum advantage

Abstract:ÌýTwo-player one-round games exhibit quantum advantage if the availability of quantum resources results in non-classical correlations between the players' answers, which allow outperforming any classical strategy. The first part of the talk illustrates a link between (i) the occurrence of quantum advantage in a game, and (ii) the complexity of the corresponding classical computational problem. In particular, I will show that both (i) and (ii) are determined by the same algebraic invariants of the game, known as polymorphisms. This allows transferring methods from the universal-algebraic theory of constraint satisfaction to the setting of quantum advantage. In particular, this approach yields a complete characterisation of the occurrence of quantum advantage for games played on graphs. ÌýÌý

The second part of the talk outlines recent work on the quantum chromatic number introduced in [Cameron--Montanaro--Newman--Severini--Winter'07]. The gap between this parameter and its classical counterpart is a measure of quantum advantage for the graph colouring game. Using the polymorphic approach, I will show that the maximum gap is linear when the quantum chromatic number makes use of entangled strategies on a constant number of qubits. In contrast, in the case of unlimited entanglement resources, I will prove the existence of graphs whose quantum chromatic number is 3 and whose classical chromatic number is arbitrarily large, conditional to the quantum variants of Khot's d-to-1 Conjecture [Khot'02] and HÃ¥stad's inapproximability of linear equations [HÃ¥stad'01].

Title:Source characterization of the hypegraphic posets

Abstract: For a hypergraph $\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\mathbb{H}$ is the transitive closure of the oriented $1$-skeleton of the hypergraphic polytope $\Delta_\mathbb{H}$, which is the Minkowski sum of the standard simplices $\Delta_H$ for each hyperedge $H \in \mathbb{H}$. In 2019, C. Benedetti, N. Bergeron, and J. Machacek established a remarkable correspondence between the transitive closure of the oriented $1$-skeleton of $\Delta_\mathbb{H}$ and the flip graph on acyclic orientations of $\mathbb{H}$. Viewing an orientation of $\mathbb{H}$ as a map $A$ from $\mathbb{H}$ to $[n]$, we define the sources of the acyclic orientations as the values $A(H)$ for each hyperedge $H \in \mathbb{H}$. In a recent paper, N. Bergeron and V.

Pilaud provided a characterization of $P_\mathbb{H}$ based on the sources of acyclic orientations for interval hypergraphs. Specifically, two distinct acyclic orientations $A$ and $B$ of $\mathbb{H}$ are comparable in $P_\mathbb{H}$ if and only if their sources satisfy $A(H) \leq B(H)$ for all hyperedges $H\in \HH$. The goal of this work is to extend this source characterization of $P_\mathbb{H}$ to arbitrary hypergraphs on $[n]$.

There will be a pre-seminar presenting relevant background at the beginning graduate level starting at 1:30pm,