Date/Time : 08 Aug 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Tirasan Khandhawit, Mahidol University

Seiberg-Witten invariants and monopole Floer homology have been important tools in the study of 3-manifolds and 4-manifolds. In this talk, I will give an overview of the stable homotopy refinement of these invariants. In particular, I will describe Bauer-Furuta invariants of 4-manifolds and Seiberg-Witten-Floer spectra of 3-manifolds. I will also discuss some applications of these refinements.

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Date/Time : 07 Aug 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yongnam Lee, KAIST, Korea

In this talk, I will discuss the positivity of the tangent bundle of smooth projective surfaces. The first part of the talk will be focused on the classification of non-ruled surfaces with pseudo-effective tangent bundles, which is a joint work with Jia Jia and Goulei Zhong. The second part will be focused on weak del Pezzo surfaces with big tangent bundles. The bigness of the tangent bundle is expected to be determined through the configuration of (−2)-curves. Our approach is to produce irreducible effective divisors on the projectivized tangent bundle, called the total dual VMRT, associated to a conic fibration on weak del Pezzo surfaces.

This part is a joint work with Hosung Kim and Jeong-Seop Kim.

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Date/Time : 07 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Jing, University of Toronto

We expose an organizing framework to study higher dimensional infinitary combinatorics based on \v{C}ech cohomology, originating from works by Barry Mitchell, Barbara Osofsky and others. Key combinatorial notions include n-coherence and n-triviality for sequences of functions. We will use some recent vanishing and non-vanishing results to demonstrate “aleph_n is incompact for (n+1)-dimensional combinatorics” and “aleph_{omega+1} can be compact for n-dimensional combinatorics for all n”. Time permitting, we will also discuss the possibility of generalizing classical 2-dimensional properties like being special or being Suslin to higher dimensions. The talk will be purely combinatorial. Joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.

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Date/Time : 01 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Armin Schikorra, University of Pittsburgh

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.

I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione.

Joint work with Dorian Martino

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Date/Time : 31 Jul 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : George Barmpalias, Chinese Academy of Sciences

I will discuss current challenges and progress in algorithmic randomness, focusing on Chaitin’s halting probability, almost everywhere domination and measures of relative randomness. I will offer conjectures, partial results and benchmark problems toward solving the main questions.

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Date/Time : 31 Jul 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jan M. Swart, The Czech Academy of Sciences

In a classical game two players, Alice and Bob, take turns to play n moves each. Alice starts. For each move each player has two options, 1 and 2. The outcome is determined by the exact sequences of moves played by each player. Prior to the game, a winner is assigned to each of the 2^{2n} possible outcomes in an i.i.d. fashion, where p is the probability that Bob is the winner for a given outcome. Then it is known that there exists a value 0<p_c<1 such that the probability that Bob has a winning strategy for large n tends to one if p>p_c and to zero if p<p_c. We study a modification of this game for which the outcome is determined by the exact sequence of moves played by Alice as before, but in the case of Bob all that matters is how often he has played move 1. We show that also in this case, there exists a sharp threshold p’_c that determines which player has with large probability a winning strategy in the limit as n tends to infinity.

Joint work with Anja Sturm and Natalia Cardona-Tob\’on (G\”ottingen).

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Date/Time : 30 Jul 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Min XU, Carnegie Mellon University

Cryo-electron tomography (cryo-ET) 3D imaging technology captures a 3D image of a single cell’s subcellular structures at sub-molecular resolution and in a near-native state. It provides unprecedented opportunities for systematically studying the native spatial organization of subcellular structures, especially macromolecules. However, cryo-ET has high structural complexity and imaging limits, such as high structural diversity and crowding, low signal-to-noise ratio, and missing values. These have made the automated systematic analysis of such images extremely difficult. I will present our recent efforts to develop methods for automated analysis of cryo-ET data.

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Date/Time : 24 Jul 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Shidong Jiang, Flatiron Institute

In this talk, we will first present an overview of fast integral equation methods (FIEMs). Fast algorithms, such as the fast multipole methods and their descendants, have made profound impacts on many areas of scientific computing. Boundary integral equations of the second kind are the natural formulation for boundary value problems of elliptic partial differential equations due to the reduction of dimensionality by one, automatic satisfaction of conditions at infinity for exterior and scattering problems, and well conditioning. The computational bottleneck that arises because the resulting linear system is dense due to the nonlocal and long-range nature of integral operators is largely removed by fast algorithms. FIEMs have been highly successful in solving a large class of problems in fluid mechanics, elasticity, acoustics, and electromagnetics.

Second, we introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. The DMK (dual-space multilevel kernel-splitting) framework uses a hierarchy of grids, computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables without relying on the FFT. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per grid point, even in a fully adaptive context.

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Date/Time : 24 Jul 2024 15:00
Venue : S17 #04-04 (SR3)
Speaker : Chen Qile, Boston College

Gauged Linear Sigma Models introduced by physicists in the 90’s is a way of counting curves in critical loci of a holomorphic function called superpotentials. In this talk, I will report on a joint project in progress with Felix Janda and Yongbin Ruan on computing Gauged Linear Sigma Models via logarithmic compactifications. Our method will lead to quantum lefschetz formulas in all genera, relating Gromov-Witten invariants of complete intersections with enumerative geometry of their ambient spaces. Furthermore the correction terms in these formulas are virtual counts of “generalized special linear series”. In case of Fano complete intersections, we show that the vanishing of these correction terms are controlled by invariants from birational geometry.

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