Home » Articles posted by Math Department Added on May 09, 2024 by Math DepartmentPlease join us in congratulating Professor Duong H. Phong on his recent election to the National Academy of Sciences!
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Print this pageAdded on May 03, 2024 by Math DepartmentWelcome to the 2024 cohort of graduate students! A list of our new PhD students can be found here.
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Print this pageAdded on April 24, 2024 by Math DepartmentIt is with deep sadness and heavy hearts that we inform you of the death of our colleague and friend, Lars Tyge Nielsen.
Nielsen joined the Columbia University Department of Mathematics in 2012 as the Director of the Mathematics of Finance MA Program. During his time in the department, he was deeply committed to teaching and investing in students’ academic growth and professional development. Through the years, Nielsen attracted an impressive lineup of practitioners to teach and interact with students in the program and constantly worked on improving the curriculum to align with the changing world of mathematical finance. Nielsen was a great colleague, always kind and positive, treating everyone with respect. In his free time, Nielsen loved reading about world history and traveling to Denmark, where he was born and raised.
A funeral service will be held on Wednesday, May 1st at 3:00pm at Holy Trinity Lutheran Church on 3 West 65th Street, New York, NY, 10023. All are welcome. There will be a reception afterwards at the church, refreshments and finger food. The family would like to get an estimate of the number of participants. For this reason, if you are planning to attend, please fill out the form: https://forms.gle/qfJzv8YUT9uqqu5s9.
Print this pageAdded on April 15, 2024 by Math Department
Abstract:
A matrix is called rigid if one must change many of its entries before it becomes a lowrank matrix. Leslie Valiant introduced the notion in 1977 as a tool to prove lower bounds on the number of arithmetic operations needed to compute linear transformations like the discrete Fourier transform. Since then, many connections have been demonstrated between matrix rigidity and other topics in the theory of computation.
Unfortunately, proving that matrices of interest are rigid has shown to be a major challenge. In fact, it remains an open problem to prove that any explicit family of matrices is rigid. By contrast, it is known that a random matrix is rigid with high probability.
In this talk, I’ll give a brief overview of matrix rigidity and its uses in computer science. I’ll then discuss some recent results, including proofs that matrices like Hadamard and Fourier matrices, which were previously conjectured to be rigid, are in fact not rigid. The talk will not assume the audience has a background in computer science.
Time and location: tbd, 4:305:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
Print this page Added on April 11, 2024 by Math Department
Title: Symmetries of manifolds
Abstract: Whenever one studies a mathematical object one ought also to study its symmetries. Manifolds are the central objects of study in topology and geometry, and their groups of symmetries come in many flavours (isometries, diffeomorphisms, homeomorphisms, …). I will discuss some classical and recent results about the spaces of all symmetries of certain simple manifolds, and report on an emerging conjectural picture.
Time and location: Wed. Apr. 17, 4:305:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
Print this pageAdded on February 22, 2024 by Math DepartmentSpecial time, location: Tue. Feb. 27, 4:105:25pm, 407 Math
Title:
Universally counting curves in Calabi–Yau threefolds
Abstract:
Statements such as “there is a unique line between any pair of distinct points in the plane” and “there are 27 lines on any cubic surface” have given rise to the modern theory of enumerative geometry.
To define such “curve counts” in a general setting usually involves choosing a particularly nice compactification of the space of smooth embedded curves (one which admits a natural “virtual fundamental class”). I will propose a new perspective on enumerative invariants which is based instead on a certain “Grothendieck group of 1cycles” and the “universal” curve enumeration invariant taking values in this group. It turns out that if we restrict to complex threefolds with nef anticanonical bundle, this group has a very simple structure: it is generated by “local curves”. This generation result implies some new cases of the MNOP conjecture relating Gromov–Witten and Donaldson–Pandharipande–Thomas invariants of complex threefolds.
Print this pageAdded on February 20, 2024 by Math DepartmentCongratulations to Elena Giorgi, an Assistant Professor in the Department of Mathematics, who was recently awarded a prestigious Sloan Fellowship!
Sloan Fellowship are awarded to “the most promising scientific researchers working today. Their achievements and potential place them among the next generation of scientific leaders in the U.S. and Canada..” For more information on the fellowship please visit: https://sloan.org/fellowships
The full list of 2024 recipients, including two other Columbia faculty, is available at https://sloan.org/fellowships/2024Fellows, and an official press release from Columbia can be read here https://news.columbia.edu/news/threecolumbiafacultymembersnamedsloanresearchfellows.
Print this pageAdded on February 16, 2024 by Math Department
Title: Curvature blow up at big bang singularities
Abstract: Singularities have been accepted as a natural feature in general relativity since the appearance of the singularity theorems of Hawking and Penrose. But these theorems do not say much concerning the nature of singularities. Do the gravitational fields become unbounded? Can the spacetime be extended through the singularity? Recently, many results demonstrating the stability of spatially homogeneous solutions with big bang singularities have appeared. As a consequence, there is an open set of initial data yielding big bang singularities with curvature blow up. However, the purpose of the talk is to illustrate that it is possible to go beyond stability results. In fact, I will present a new result (joint work with Hans Oude Groeniger and Oliver Petersen) in which we identify a general condition on initial data ensuring big bang formation. The solutions need, in this case, not be close to symmetric background solutions. Moreover, the result reproduces previous results in the Einsteinscalar field and Einsteinvacuum settings. Finally, the result is in the Einsteinnonlinear scalar field setting, and therefore yields future and past global nonlinear stability of large classes of spatially locally homogeneous solutions.
Time and location: Wed. Feb. 28, 4:305:30pm, Math 520. Tea will be served in the Math lounge at 4pm.
Print this pageAdded on February 07, 2024 by Math DepartmentSpeaker: Blaine Lawson (Stony Brook)
2/15 – First Lecture: On compact minimal surfaces in S^3
Abstract: I will present a construction of compact minimal surfaces in the Euclidean threesphere. In the orientable case, surfaces of every genus can be minimally embedded. In the nonorientable case, every surface but the real projective plane can be minimally immersed. For the projective plane, no such immersion exists.
These surfaces have some charm, and they also relate to the general topic of singularities of minimal threefolds in four dimensional spaces. This work was inspired many years ago by Professor Eugenio Calabi.
2/16 – Second Lecture: Nonlinear PDE’s and Potential Theories
 How does one deal with a partial differential equation when there is no natural operator?
 Given a differential operator, are there other operators with the same solutions but different useful properties? In fact can one radically change the operator to something tractable, in a way that enables solving the original equation?
 Does the space of subsolutions (or, equivalently, the space of supersolutions) give rise to a potential theory where interesting and relevant theorems can be proved?
These questions arose in my work with Reese Harvey. We discovered that while calibrated manifolds do not usually have analogues of the holomorphic functions that exist in the Kähler case, they do have analogues of plurisubharmonic functions. This started a long investigation. I will discuss various highlights of that work, and also some responses to these questions.
Thursday, February 15, 2024 @ 4:30 PM (407 MATH)
Friday, February 16 , 2024 @ 2:00 PM (312 MATH)
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Print this pageAdded on January 31, 2024 by Math DepartmentSpecial Colloquium
Speaker: Toby Gee (Imperial College London)
Title: Modularity of genus 2 curves
Abstract: I will give an accessible introduction to some problems in the Langlands program. In particular, I will discuss work in progress with George Boxer, Frank Calegari, and Vincent Pilloni in which we prove the modularity of a positive proportion of curves of genus 2.
Date and Time: Tuesday, February 6, 4:30 PM
Location: 407 Mathematics
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