Date/Time : 26 Aug 2022 09:00
Venue : via Zoom
Speaker : Jinyoung Park, Stanford University

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https://nus-sg.zoom.us/j/84821387838?pwd=QnRaVGtNVFBscWNreG4yVjJ2WGM3QT09

Meeting ID: 848 2138 7838
Passcode: 969669

Abstract: Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its “expectation-threshold,” which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will present recent progress on this topic. Based on joint work with Huy Tuan Pham.

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Date/Time : 26 Aug 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jonathan Gruber, National University of Singapore

Abstract: In this talk, we explain how every object of a highest weight category can be associated with a \emph{minimal tilting complex}.

Minimal tilting complexes are well-behaved with respect to direct sums and short exact sequences, and they capture some important properties of the corresponding objects in the highest weight category.
After establishing these general properties, we will discuss the problem of finding the multiplicities of indecomposable tilting objects appearing in the terms of minimal tilting complexes. We develop some general techniques for computing these multiplicities and show that in many specific cases (including the category $\mathcal{O}$ for a complex simple Lie algebra or a Kac-Moody Lie algebra), they are governed by Kazhdan-Lusztig polynomials.

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Date/Time : 26 Aug 2022 10:00
Venue : S17 #04-04 (SR3)
Speaker : Michele Salvi, University of Tor Vergata

Abstract: Scale-free percolation is a spatial inhomogeneous random graph model which features three fundamental properties that are often present in real-world networks: (1) Scale-free: the degree of the nodes follows a power law; (2) Small-world: two far-away nodes have typically a small graph distance; (3) Positive clustering coefficient: two nodes with a common neighbour have a good chance to be linked. We study the mixing time of the simple random walk on this structure in one dimension and depict a rich phase diagram in the parameters of the model. In particular, we prove that the presence of hubs can speed up the mixing of the chain. Joint work with Alessandra Cipriani (UCL)

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Date/Time : 26 Aug 2022 10:00
Venue : S17 #06-11 (SR6)
Speaker : Zhang Junyan, National University of Singapore

Abstract: We consider 3D compressible Euler equations in a moving domain diffeomorphic to the lower-half space. The fluid is under the influence of gravity and surface tension, and not assumed to be irrotational. We prove the local well-posedness(LWP), incompressible limit(low Mach number limit) and zero surface tension limit. The proof is based on tangential-smoothing approximation and Lax-Phillips duality to establish LWP; and a carefully-chosen weighted energy to establish an estimate uniform in both Mach number and surface tension. This is the joint work with Chenyun Luo at CUHK.

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Date/Time : 25 Aug 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Timothée Bénard, University of Cambridge

Abstract:  A famous theorem due to Hopf-Tsuji-Sullivan-Kaimanovich asserts that the geodesic flow and the Brownian motion on a rank-one symmetric space of the non-compact type $\Lambda \backslash G/K$ are either both recurrent ergodic or both transient. In this talk, I will explain how this dichotomy can be extended to a much larger class of random walks, namely any right random walk on $\Lambda \backslash G$ induced by a Zariski-dense probability measure on $G$ with finite first moment.

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Date/Time : 24 Aug 2022 17:00
Venue : S17 #04-05 (SR2)
Speaker : Goh Jun Le, National University of Singapore

Abstract: We present ongoing work with Steffen Lempp, Ng Keng Meng and Mariya Soskova on the algebraic structure of the Sigma^0_2 e-degrees, i.e., the quotient structure on the Sigma^0_2 subsets of the natural numbers induced by enumeration reducibility. The Sigma^0_2 e-degrees are analogous to the r.e. Turing degrees in some ways, but an elementary difference between them was exhibited by Ahmad (1998): In the former structure there are incomparable degrees a and b such that if x < a, then x < b. (Such a and b cannot exist in the r.e. Turing degrees by the Sacks splitting theorem.) Ahmad also showed that this phenomenon cannot occur symmetrically, i.e., there cannot be incomparable degrees a and b such that x < a if and only if x < b. In contrast, we show that there are incomparable degrees a, b and c such that x < a if and only if (x < b and x < c). In other words, the ideal {x: x < a} admits an exact pair (b,c). This result constitutes progress towards our ultimate goal of algorithmically deciding the truth of all two-quantifier sentences in the Sigma^0_2 e-degrees.

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Date/Time : 19 Aug 2022 14:00
Venue : S17 #06-11 (SR6)
Speaker : Jonathan Gruber, National University of Singapore

Abstract:  Highest weight categories provide an axiomatic framework which encompasses many categories that naturally appear in Lie theoretic representation theory, including the BGG category $\mathcal{O}$ for complex simple Lie algebras, representations of quantum groups at roots of unity and rational representations of reductive algebraic groups.

In this talk, we give the definition of highest weight categories and establish some of their key properties.  Then we discuss the classification of \emph{tilting objects} in highest weight categories (due to C.M. Ringel) and explain how the subcategory of tilting objects in a given highest weight category $\mathcal{C}$ controls the homological properties of $\mathcal{C}$.

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Date/Time : 18 Aug 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Farhad Babaee, University of Bristol

Abstract: Tropical geometry is an effort to simplify algebraic subvarieties of the complex torus through the tropicalisation process to obtain tropical varieties. The upshot is that tropical varieties still retain a lot of information about the initial algebraic varieties. In this talk, I will explain basic notions in tropical geometry, and discuss how natural and useful it is to look at tropical geometry through the lens of complex geometry and holomorphic dynamics.

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Date/Time : 18 Aug 2022 10:00
Venue : via Zoom
Speaker : Fu Zhaohui, The University of British Columbia

Abstract:  Gradient flow models attract much attention these years. The energy naturally decreases along the direction of gradient flows. In order to preserve this property, various numerical schemes have been developed and among them, the exponential time differencing Runge–Kutta method (ETDRK) and the implicit-explicit (IMEX) Runge-Kutta method are two significant approaches. In this talk we introduce these two methods and prove them to unconditionally preserve the energy dissipation law for a family of phase field models, such as the Allen–Cahn equation, the Cahn–Hilliard equation and the molecular beam epitaxy (MBE) model. Furthermore, we present some numerical experiments to demonstrate the accuracy and stability of these schemes.

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