Date/Time : 18 Dec 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Ding Lijun, University of California, San Diego

We revisit a formulation technique for inequality-constrained optimization problems that has been known for decades: the substitution of squared variables for nonnegative variables. Using this technique, inequality constraints are converted to equality constraints via the introduction of a squared-slack variable. Such formulations have the superficial advantage that inequality constraints can be dispensed with altogether. But there are clear disadvantages, not least being that first-order optimal points for the squared-variable reformulation may not correspond to first-order optimal points for the original problem, because the Lagrange multipliers may have the wrong sign. Extending previous results, this paper shows that points satisfying approximate second-order optimality conditions for the squared-variable reformulation also, under certain conditions, satisfy approximate second-order optimality conditions for the original formulation, and vice versa. Such results allow us to adapt complexity analysis of methods for equality-constrained optimization to account for inequality constraints. On the algorithmic side, we examine squared-variable formulations for several interesting problem classes, including bound-constrained quadratic programming, linear programming, and nonnegative matrix factorization. We show that algorithms built on these formulations are surprisingly competitive with standard methods.

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Date/Time : 18 Dec 2024 17:00
Venue : S17 #05-11 (SR5)
Speaker : Manlio Valenti, Swansea University

Recall that, in a partial order P, b is a minimal cover of b if the interval (a,b) is empty. A strengthening of this notion is the one of strong minimal cover, namely b is a strong minimal cover of a if c<b implies c\le a. In this talk, we provide a complete characterization of how minimal covers and strong minimal covers in the Weihrauch degrees look like. As a consequence, we obtain that the Weihrauch degree of the identity problem id can be defined as the highest top of a strong minimal cover, or the least degree whose upper cone is dense. In other words, “being (uniformly) computable” is a first-order definable property in (W,<). This implies that the first-order theory of the Weihrauch degrees (below id) is recursively isomorphic to the third-order theory of true arithmetic.

This is joint work with Steffen Lempp, Joe Miller, Arno Pauly, and Mariya Soskova.

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Date/Time : 12 Dec 2024 16:30
Venue : S17 #04-04 (SR3)
Speaker : Huang Jizu, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

In this talk, we propose a novel accuracy-preserved tensor-train (APTT) method to solve the Boltzmann-BGK equation. We use the Crank-Nicolson Leap Frog (CNLF) scheme for temporal discretization, based on which the collision term can be evaluated only once at each time step. A second-order scheme based on the upwind rule is employed for spatial discretization to match the accuracy of the CNLF scheme. At each time step, the linear system is constructed with the tensor-train (TT) format, where the matrix, the right-hand side, and the collision term involved in the linear system are all represented using the low-rank TT format. Based on such a representation, an efficient and effective iterative solver can be implemented for solving the linear system, which can reduce two orders of magnitude in terms of both time and memory costs compared with classical methods. We present a theoretical analysis to show that the proposed APTT method can maintain accuracy and an extensive set of 3D3V Boltzmann-BGK test examples to demonstrate its effectiveness.

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Date/Time : 12 Dec 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yihong Du, University of New England, Australia

A new phenomenon in nonlocal diffusion models is that accelerated propagation may happen, that is, the propagation speed could be infinite, which never occurs in the corresponding local diffusion model with compactly supported initial data. In this talk, we will first briefly review the history of the KPP model used to describe the propagation of biological/chemical species, and then look at some very recent results on the KPP equation with nonlocal diffusion and free boundaries. For several natural classes of kernel functions appearing in the nonlocal diffusion term, we will show how the exact rate of acceleration can be determined. The talk is based on joint works with several collaborators.

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Date/Time : 11 Dec 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jiaqi Liu, University of Southern California

The Guderley problem describes the behavior of a strong shock wave propagating radially in an ideal gas. A spherical shock converges radially inwards to the spatial origin, strengthening as it collapses. At the collapse point, the shock’s strength becomes infinite, leading to the formation of a new outgoing shock wave of finite strength, which then propagates outwards to infinity. In this talk, I will present recent work on the rigorous construction of the Guderley solution for $\gamma \in (1,3]$. These solutions are analytic away from the shock interfaces and the blow-up point. The proof relies on continuity arguments, nonlinear invariances, and barrier functions.

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Date/Time : 10 Dec 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Gianluca Faraco, University of Milano Bicocca

A translation surface is the datum of an abelian differential on a Riemann surface. Every such pair determines a representation called absolute period representation or period character. In this talk we discuss the realization of a given representation as the period character of some translation surface.

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Date/Time : 09 Dec 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Daniel Mathews, Monash University

Work of Penrose and Rindler in the 1980s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors as flags in Minkowski space. We present some extensions of this work involving hyperbolic geometry. In particular, we give a correspondence between between spinors and horospheres with certain decorations, and show how spinors can be used to calculate a complex-valued, 3-dimensional generalisation of Penner’s lambda lengths. Time permitting, we may discuss some further applications, including inter-cusp distances in knot complements, higher-dimensional generalisations, and circle packings.

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Date/Time : 05 Dec 2024 15:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Igor Balla, Masaryk University

Despite being elementary objects in geometry, large sets of equiangular lines appear naturally in a wide variety of fields including elliptic geometry, frame theory, Banach space theory, spectral graph theory, and quantum information theory with connections to algebraic number theory and Hilbert’s twelfth problem. In 1973, Lemmens and Seidel posed the problem of determining the maximum number of equiangular lines in an r-dimensional Euclidean space with a given angle. In this talk, we will present new upper bounds on this problem that unify and significantly extend or improve essentially all previously known results, as well as being best possible either exactly or up to a factor of 2. These recent results were obtained independently or together with Matija Bucic.

Roughly speaking, our approach relies on new lower bounds on the second eigenvalue of a graph as well as new upper bounds on its multiplicity. In particular, we obtain the first extension of the Alon–Boppana theorem to dense graphs, with equality for strongly regular graphs corresponding to families of r+1 choose 2 equiangular lines in r dimensions. Time permitting, we will also mention some results in the complex setting.

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Date/Time : 05 Dec 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Hong Liu, Institute for Basic Science, South Korea

A well-known theorem of Nikiforov asserts that any graph with a positive $K_r$-density contains a logarithmic blowup of $K_r$. We explore variants of Nikiforov’s result and investigate when positive clique density condition implies the existence of a significantly larger blowup of a clique. Our results study such problems for families of ordered graphs with forbidden induced monotone path, obtaining optimal bounds. As corollaries, we strengthen a result of Pach and Tomon, and resolve a conjecture of Tomon in a strong form. To find a large blowup, we reduce the embedding problem to a certain Ramsey problem. For optimal lower bound constructions, we make use of concentration of measure and the isodiametric inequality on high dimensional spheres.

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