Date/Time : 24 Oct 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Meng-Che Ho, California State University Northridge

Classically, the word problem of a group is the set of words equal to the identity of the group, and we analyze them using Turing reductions. In this talk, we consider the word problem of a group as a computably enumerable equivalence relation (ceer), namely, two words are equivalent if and only if they are equal in the group. We compare ceers using the computable reduction: $E$ is reducible to $F$ if there is a computable function $f$ so that $iEj$ if and only if $f(i)Ff(j)$.

We will discuss some recent results and see that the landscape of word problems as ceers is very different from the classical theory. For instance, in the classical setting, any Turing degree can be realized as a word problem by first constructing a countable group and then embedding it into a finitely presented group via the Higman embedding theorem. However, we prove that in the ceer setting, there is a group $G$ whose word problem is not universal, but for any nontrivial $H$, the free product $G*H$ has a universal word problem.

This is a joint work with Uri Andrews and Luca San Mauro.

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Date/Time : 20 Oct 2023 15:00
Venue : S17 #05-11 (SR5)
Speaker : Apoorva Khare, Indian Institute of Science, Bangalore

We explain recent results involving Schur polynomials – viewed as functions on the positive orthant – which have led to progress along several classical directions of study:

(1) Understanding functions that preserve positive semidefiniteness when applied entrywise to matrices. This journey takes us from work of Schur, Schoenberg, Rudin, and Loewner (1900s) to modern-day results.

(2) The proofs of these recent results lead to characterizing (weak) majorization of real tuples using Schur polynomials; this has since been generalized to all Weyl groups. Majorization inequalities have been studied since Maclaurin and Newton (1700s).

(3) We also explain how all Schur polynomials are lurking within each smooth function, by completing a 1960s determinantal computation of Loewner. This helps extend determinantal identities by Cauchy and Frobenius (1800s) involving one or two geometric series, to all power series, over all unital commutative rings.

(Partly based on joint works with Alexander Belton, Dominique Guillot, Mihai Putinar, and with Terence Tao.)

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Date/Time : 19 Oct 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

The studies of Poincaré-Einstein manifolds originate from the AdS/CFT correspondence in mathematical physics, which relates quantum gravity theory and conformal field theory. On the mathematics side, numerous conformal invariants were successfully constructed via the microlocal analysis on Poincaré-Einstein spaces, so that it has become a central area in conformal geometry.

A primary object in this talk is the so-called fractional GJMS operators, which provide a uniform treatment of conformally covariant operators with different orders. From the analytic point of view, the degenerate and nonlocal equations arising in this setting are particularly interesting. We will mainly exhibit a series of sharp dimension bounds and Hausdorff measure estimates for the singular set of those solutions. The proof is based on the quantitative differentiation theory for degenerate elliptic operators and involves subtle regularity issues on the Poincaré-Einstein metrics.

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Date/Time : 18 Oct 2023 13:00
Venue : S17 #05-11 (SR5)
Speaker : Chandrima Kayal, Indian Statistical Insitute, Kolkata

Suppose we want to cover all the vertices of the n-dimensional Boolean cube using a minimum number of hyperplanes. Observe that this can be done using only two hyperplanes. Moreover, no single hyperplane can cover all the vertices. Now what if we want to cover only a subset of the Boolean cube? For example, suppose we want to cover all the vertices except one, viz. the origin. One can observe that n hyperplanes are sufficient. But can we do better? The celebrated covering result by Alon and Füredi shows that at least n hyperplanes will be required to cover all the vertices of the n-dimensional cube leaving out exactly one vertex.

We shall discuss different versions of this covering problem, and give a generalization of Alon and Füredi’s covering result for any symmetric subset of the Boolean cube. Also, we shall show a strict separation between the size of a polynomial cover and a hyperplane cover.

This work was jointly done with Arijit Ghosh, Soumi Nandi and S. Venkitesh.

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Date/Time : 18 Oct 2023 15:00
Venue : S14 #06-19
Speaker : Apoorva Khare, Indian Institute of Science, Bangalore

The goal of this (informal) talk is to provide a gentle introduction to the space of limits of dense graph sequences. This will include multiple equivalent ways in which this space of graph limits can be metrized, and will discuss other basic properties (including the compactness and separability of graphon space, akin to [0,1]^k).

I will then discuss why the homomorphism densities on graphons are a natural analogue of multivariate monomials on [0,1]^k, in that they too form a countable multiplicatively closed set whose span is a point-separating algebra of continuous functions that contains the constants. Thus, the span of this set is a natural “Stone-Weierstrass counterpart” to the multivariate polynomials, on graphon space.

Finally, I will discuss a result (JCT-A 2015, joint with Diao, Guillot, and Rajaratnam) that says that these analogues of polynomials, with at most N edges, are the only functions whose (N+1)st mixed Gateaux derivatives are all zero – again parallel to multivariate polynomials on [0,1]^k.

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Date/Time : 18 Oct 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Mahsa Allahbakhshi, Pontificia Universidad Católica de Chile

This presentation explores the creation of a course package, designed for newcomers to the Bachelor’s degree in Mathematics and Statistics. The primary objective is to establish an environment that enhances the ability to self-assess strengths and weaknesses in one’s study journey, a key component of self-regulated learning. As students become familiar with the required quality standards, the initiative strives to maintain their motivation and resilience amidst challenges, thereby facilitating a smooth transition into university life. Moreover, this initiative aims to enhance communication and argumentation skills, while also increasing both individual and collective awareness of the cognitive, emotional, and social challenges involved in collaborative tasks.

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Date/Time : 18 Oct 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ruobing Zhang, Princeton University

This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics.

My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress and propose several open questions in the field.

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Date/Time : 17 Oct 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Frank Stephan, National University of Singapore

Floyd and Knuth studied in their 1990 paper addition machines which are machines which can add, subtract and compare integers (<,=,>) in unit time; also the reading or writing of an integer is in unit time. They showed that multiplying and dividing can be done in linear time with six registers and asked in their Open Problem (2) whether this bound can be broken; furthermore, they asked in Open Problem (5) of their paper whether there is a register machine which can output in subquadratic time the powers of two occurring in the binary representation of an integer.

The talk answers both questions affirmatively and presents the key ideas and programs to witness this.

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Date/Time : 16 Oct 2023 16:00
Venue : S16 #04-35
Speaker : Lin Yi-Hsuan, National Yang Ming Chiao Tung University

We investigate inverse problems in the determination of leading coefficients for nonlocal parabolic operators, by knowing the corresponding Cauchy data in the exterior space-time domain. The key contribution is that we reduce the nonlocal parabolic inverse problems to the corresponding local inverse problems by using the extension problem for nonlocal parabolic equations. This talk is based on joint works with Ching-Lung Lin and Gunther Uhlmann.

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