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

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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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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 : 11 Aug 2022 15:00
Venue : via Zoom
Speaker : Alexander Stoimenow, KAIST

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https://nus-sg.zoom.us/j/87511476637?pwd=VURaUDVzaHpMYmduN2Fod0h0Y29sQT09

Meeting ID: 875 1147 6637
Passcode: 468399

Abstract: The braid groups $B_n$ were introduced in the 1930s in the work of Artin. An element in the braid group $B_n$ is called an $n$-braid. Alexander related braids to knots and links in 3-dimensional space, by means of a closure operation. In that realm, it became important to understand the braid representatives of a given link. Markov’s theorem relates these representatives by two moves, conjugacy in the braid group, and (de)stabilization, which passes between braid groups. Markov’s moves and braid group algebra have become fundamental in Jones’ pioneering work and its later continuation towards quantum invariants. Conjugacy is, starting with Garside’s, and later many others’ work, now relatively well group-theoretically understood. In contrast, the effect of (de)stabilization on conjugacy classes of braid representatives of a given link is in general difficult to control. Only in very special situations can these conjugacy classes be well described.

We are concerned with the question when infinitely many conjugacy classes of $n$-braid representatives of a given link occur. Birman and Menasco introduced a move called exchange move, and proved that it necessarily underlies the switch between many conjugacy classes of $n$-braid representatives of a given link.

After some very brief general introduction to knots and braids and their applications, we will discuss several results when the exchange move is also sufficient for generating infinitely many conjugacy classes of braid representatives.

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Date/Time : 10 Aug 2022 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yifan Jing, University of Oxford

Abstract:  Given natural numbers $s$ and $k$, we say that a finite set $X$ of integers is an additive $B_{s}[k]$ set if for any integer $n$, the number of solutions to the equation \[ n = x_1 + … + x_s, \] with $x_1, …, x_s$ lying in $X$, is at most $k$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{s}[k]$ set analogously. These sets have been studied thoroughly from various different perspectives in combinatorial and additive number theory. In this talk, we will discuss sum-product phenomena for these sets. We show that there is either an exceptionally large additive $B_{s}[k]$ set, or an exceptionally large multiplicative $B_{s}[k]$ set, with $k \ll s$. This is joint work with Akshat Mudgal.

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Date/Time : 10 Aug 2022 17:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Tran Chieu-Minh, National University of Singapore

Abstract:  For a bivariate P(x,y) in R[x,y] – (R[x] union R[y]), we show that for all finite subsets A of R,|P(A,A)| is greater or equal alpha |A|^(5/4) with alpha = alpha(deg P) being a real number greater than 0, P(x,y)=f(gamma u(x)+delta u(y)) or P(x,y)=f(u^m(x)u^n(y)) for some univariate f, u R[t] – </b>, constants gamma, delta in R – {0}</b> and m, n in {1,2,3,…}.

This resolves the symmetric nonexpanders classification problem proposed by de Zeeuw and is a step towards the analog for polynomials of the Erdoes-Szemeredi sumproduct Conjecture. The proofs of our results use tools from semialgebraic / o-minimal geometry.

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Date/Time : 27 Jul 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Paolo Cascini, Imperial college

Abstract:  I will survey some recent results on the study of the birational geometry of foliations over complex projective varieties, focusing, in particular, on the case of algebraically integrable foliations.

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Date/Time : 26 Jul 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Antonio De Rosa, University of Maryland

Abstract: We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3-dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard [Invent. Math.,1983] in dimension 3. Joint work with Guido De Philippis.

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Date/Time : 26 Jul 2022 16:00
Venue : via Zoom
Speaker : Piyush Srivastava, Tata Institute of Fundamental Research

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

Meeting ID: 831 9911 1482
Passcode: 569991

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.

Add to calendar :