Date/Time : 18 Nov 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Kenji Nakanishi, Kyoto University

Since the pioneering work of Kenig and Merle, there have been numerous results on global behavior of large general solutions to nonlinear dispersive equations that vary depending on the initial data, from asymptotically linear or scattering solutions to solitons and blow-up. The main question is to distinguish the types of solutions or to predict them from the initial data, without any asymptotic solution formula such as those provided by the inverse scattering theory in the special case of completely integrable systems. Despite of some spectacular progress such as the soliton resolution without size restriction, the range of applicable equations has been rather limited, such as the nonlinear wave equation or Schrodinger with simple nonlinearity, because certain monotonicity or invariance properties play crucial roles in the analysis without small factor. The Zakharov system is a model of the Langmuir turbulence in plasma, coupling the Schrodinger and the wave equations, and may be regarded also as a mathematical model to extend the methods from the simple scalar equations. It has been quite successful in the last decade for space dimensions 3 and 4, in spite of a few difficulties such as the derivative loss, the lower degree of nonlinear interactions, and the loss of invariance for scaling and for momentum or coordinate velocity. This talk will review the state of art, together with some detail on most recent results and open problems.

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Date/Time : 28 Oct 2026 14:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Guillaume Tahar, Guangdong Technion – Israel Institute of Technology

On a topological sphere endowed with a flat metric with conical singularities, the curvature gap quantifies the obstruction to realize a partition of the set of conical singularities into two sets of equal total angle defect. Unless its curvature gap is equal to zero, such a flat sphere cannot contain a simple closed geodesic. Drawing on the Delaunay decompositions of flat surfaces, we give a quantitative generalization of the latter statement. We prove that when the curvature gap is nonzero, there is an explicit upper bound on the lengths of simple geodesic trajectories in the flat sphere. If time allows, we will give extensions of this result to affine structures on punctured spheres and applications to dynamics of germs of biholomorphism tangent to the identity in complex dimension two. This is a work in collaboration with Xavier Buff, Arnaud Cheritat and Kai Fu.

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Date/Time : 28 Oct 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Guillaume Tahar, Guangdong Technion – Israel Institute of Technology

The geodesic flow on the tangent bundle of a Riemann surface endowed with a complex affine structure (or, equivalently, a meromorphic connection) is a dynamical system that exhibits remarkable similarities with the real integral curves of polynomial vector fields in two variables and with the dynamics of germs of biholomorphisms tangent to the identity in complex dimension two. We will present the geometric interpretation of the singularities of the connection, including, time permitting, irregular singularities, and show that many geometric properties of an affine surface can be read from the topology of the bifurcation locus in its tangent bundle. We will also discuss applications to irrational triangular billiards and the classification of geodesic flows up to real affine and holomorphic conjugacy. This is joint work with Xavier Buff and Arnaud Chéritat.

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Date/Time : 21 Oct 2026 16:30
Venue : S16 #04-35
Speaker : Sun Lukang, National University of Singapore

In this talk, I will report on a nonlocal continuity equation on $\mathbb{R}^d$ in which a probability density is driven by attraction toward a background measure $\omega$ and by self-repulsion, with power-law kernels $|x|^{1+a}$ and $|x|^{1+r}$, $a,r\in[0,1)$. I will show global Lagrangian well-posedness, and uniform-in-time bounds on the support when attraction dominates ($a>r$, or $a=r$ with $\omega(\mathbb{R}^d)>1$). For $0\le r\le a<1$, I will characterize zero-flux stationary states through a fractional free-boundary problem, give explicit examples in dimensions one to three that agree with particle simulations, and show that solutions with bounded energy and moments converge along a sequence of times to such a state, and converge narrowly when their $\Omega$-limit set contains a single stationary state. This is joint work with Massimo Fornasier and Hui Huang.

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Date/Time : 20 Oct 2026 14:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Marko Hans Weber, National University of Singapore

We study how social comparisons shape asset prices in an incomplete‐market economy with uninsurable income risk. Agents are heterogeneous in preferences, endowments, and in the weights they place on the consumption of others. Using a mean-field game framework with a directed network described by a finite-rank kernel, we derive in closed form the equilibrium interest rate, stock price, and individual consumption and portfolio policies. A social multiplier matrix captures how network interactions reallocate risk across agents. When the comparison network is not too strongly self-reinforcing, social comparisons reduce the market price of hedgeable risk by raising the effective risk tolerance of the economy, as consumption and social reference benchmarks co-move with common shocks. The network structure governing social comparisons determines whether the precautionary‐savings motive induced by unhedgeable risk is amplified or mitigated. If all agents exert the same influence on others and are equally sensitive to the rest of the system, as in a doubly stochastic comparison network, the specific network topology does not affect equilibrium prices.

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Date/Time : 14 Oct 2026 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Bertram Düring, University of Warwick

Many nonlinear diffusion equations can be interpreted as gradient flows whose dynamics are driven by internal energies and given external potentials, examples include the heat equation and the porous medium equation. When solving these equations numerically, schemes that respect the equations’ special structure are of particular interest. In this talk we present a Lagrangian scheme for nonlinear diffusion equations. For discretisation of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. We present numerical experiments for the porous medium equation in two space dimensions.

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Date/Time : 14 Oct 2026 15:15
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Stefan Junk, Gakushuin University

We consider the directed polymer model in dimensions d≥3, which describes random paths affected by an i.i.d. space-time random environment. In the weak disorder (high temperature) phase, the associated martingale at inverse temperature β is Lp-bounded for 1 ≤ p < p*, where p* = p*(β) > 1 is the critical moment exponent. We show that p*(β) is a (locally) Lipschitz-continuous function of β in the interior of the weak disorder phase. Joint work with Hubert Lacoin (IMPA).

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Date/Time : 14 Oct 2026 17:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Jia Zekun, National University of Singapore

Post’s program asks for a structural thinness property of the complement of an r.e. set that forces Turing incompleteness. The program seemed to collapse when Yates built a complete maximal set. In the 1970s Degtev and Marchenkov revived it by relativizing Post’s notions to an r.e. equivalence relation η: nonrecursive r.e. semirecursive η-maximal sets exist, and every such set is Turing incomplete. The existence of maximal sets is strong. It is equivalent to IΣ₂ over PA⁻ + BΣ₂ (Chong–Yang), and for admissible α it holds exactly when the S₃-projectum of α is ω (Lerman). Downey conjectured that η-maximal sets are easier to obtain. In my previous talk, I showed that PA⁻ + IΣ₁ proves they exist. This talk extends that result in two directions. In each, η-maximal sets turn out to be no harder to build than Friedberg–Muchnik pairs, up to the best current results.

In reverse recursion theory, PA⁻ + IΣ₀ + BΣ₁ + EXP, the theory in which Chong and Mourad proved the Friedberg–Muchnik theorem, proves the existence of a nonrecursive r.e. semirecursive η-maximal set. Moreover, we will outline how a Post-style solution to Post’s problem can be carried out in this theory by bypassing the classical detour through η-hyperhypersimplicity.

In ordinal recursion theory, we show that every limit ordinal β admits a β-r.e. equivalence relation η and a non-β-recursive β-r.e. η-maximal set A. When the Σ₁-cofinality of β is at least its Σ₁-projectum (in particular, when β is admissible), both A and the graph of η can be made regular, tame and strongly β-r.e. This is the same hypothesis under which Friedman obtained regular, strongly β-r.e. Friedberg–Muchnik pairs under <wβ. (When the Σ₁-cofinality of β is less than its Σ₁-projectum, the Friedberg–Muchnik theorem with respect to ≤wβ is generally false and the same theorem with respect to ≤β remains open.

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