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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Date/Time : 17 Aug 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Tran Chieu Minh, National University of Singapore

Abstract: (Joint with Jing Yifan) Suppose G is a compact simple Lie group, μ is the normalized Haar measure on G, and A, A^2 ⊆ G are measurable. I will discuss a measure growth gap result where we show that μ(A^2) ≥ (2 + η)μ(A) when μ(A) < c with η and c independent of the choice of G quantitatively determined. This is in contrast with the situation in compact connected solvable Lie group, where the Kemperman’s lower bound μ(A^2) ≥ min{2μ(A), 1} cannot be improved.

I will also survey the connections of this result to sumproduct phenomena, a conjecture by Breuillard and Green, the nonabelian Brunn—Minkowski conjecture, the effective Gleason—Yamabe conjecture, and the Kemperman inverse problem.

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Date/Time : 17 Aug 2022 17:00
Venue : S17 #04-05 (SR2)
Speaker : Wong Tin Lok, National University of Singapore

Abstract: I will present another quantifier-elimination result in arithmetic under negated induction. This gives new information about pigeonhole principles and expansions to second-order models.

Joint work with: David Belanger, CT Chong, Wei Li and Yue Yang.

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Date/Time : 16 Aug 2022 15:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Tianqi, National University of Singapore

Abstract: The notion of Anosov representations of a finitely generated group into a semisimple Lie group, introduced by Labourie, generalizes the notion of convex-cocompact representations to higher rank Lie groups. On the other hand, there are many non-discrete, and hence non-Anosov, representations that have good geometric properties, such as the primitive-stable representations introduced by Minsky. In this talk, we introduce a weakening of the notion of Anosov representations by requiring the Anosov condition to hold only on a chosen invariant closed geodesic subflows. This weakening allows for such non-discrete representations to arise as examples. We discuss several properties of this type of representations that are analogous to the properties of classical Anosov representations, such as stability and the limit maps being Hölder continuous in a weaker sense

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Date/Time : 16 Aug 2022 16:00
Venue : via Zoom
Speaker : Piyush Srivastava, Tata Institute of Fundamental Research

Join Zoom Meeting
https://nus-sg.zoom.us/j/83199111482?pwd=L09Yd054dDNTbDBpRkNtc0tmQnk5Zz09

Meeting ID: 831 9911 1482
Passcode: 569991

Abstract:  This series of lectures will survey various applications to algorithms of a collection of techniques that are broadly referred to as “geometry of polynomials”. Specific examples include:

1) Real stable polynomials and constructions of Ramanujan expanders
2) Lee-Yang theory and deterministic “approximate counting” algorithms”
3) Lorentzian or “strongly log concave” polynomials and applications to sampling

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