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Welcome to ICERM
The Institute for Computational and Experimental Research in Mathematics
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Welcome to ICERM
The Institute for Computational and Experimental Research in Mathematics
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Welcome to ICERM
The Institute for Computational and Experimental Research in Mathematics
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Welcome to ICERM
The Institute for Computational and Experimental Research in Mathematics
This Week at ICERM
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July 14, 2024
There are no events currently scheduled for July 14th.
July 15, 2024
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8:30 - 9:00 am EDTArrive, sign in, get settled- 11th Floor Collaborative Space
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9:00 - 9:30 am EDTCheck in with TAsCheck in with TAs - 10th Floor Collaborative Space
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9:30 - 10:00 am EDTGroup WorkAssigned Group Meeting Space
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10:00 - 10:30 am EDTQuestion of the Day/Group Presentations- 10th Floor Collaborative Space
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10:30 - 11:00 am EDTBreakCoffee Break
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11:00 - 11:30 am EDTGroup Presentations10th Floor Collaborative Space
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11:30 am - 12:00 pm EDTResearch MeetingMeeting - 10th Floor Collaborative Space
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12:00 - 1:30 pm EDTLunch/Free Time
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1:30 - 3:00 pm EDTGroup WorkAssigned Group Meeting Space
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3:00 - 3:30 pm EDTCoffee Break10th Floor Classroom
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3:30 - 4:30 pm EDTGroup Work/ Writing TimeGroup Work - Assigned Group Meeting Space
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6:30 - 8:30 pm EDTGame Night- 10th Floor Collaborative Space
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9:20 - 9:30 am EDTWelcome11th Floor Lecture Hall
- Brendan Hassett, ICERM/Brown University
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9:30 - 10:15 am EDTShortest word problem in braid theory11th Floor Lecture Hall
- Speaker
- Keiko Kawamuro, University of Iowa
- Session Chair
- Nancy Scherich, Elon University
Abstract
Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
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10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
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11:00 - 11:45 am EDTSymmetries of link homology11th Floor Lecture Hall
- Speaker
- Joshua Sussan, CUNY
- Session Chair
- Nancy Scherich, Elon University
Abstract
We construct an action of sl(2) on equivariant Khovanov-Rozansky link homology.
This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner. -
12:00 - 2:00 pm EDTLunch/Free Time
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2:00 - 2:45 pm EDTBi-ordering link complements via braids11th Floor Lecture Hall
- Speaker
- Hannah Turner, Stockton University
- Session Chair
- Caitlin Leverson, Bard College
Abstract
Any link (or knot) group – the fundamental group of a link complement – is left-orderable. However, not many link groups are bi-orderable – that is, admit an order invariant under both left and right multiplication. It is not well understood which link groups are bi-orderable, nor is there is a conjectured topological characterization of links with bi-orderable link groups. I will discuss joint work in progress with Jonathan Johnson and Nancy Scherich to study this problem for braided links – braid closures together with their braid axis. Inspired by Kin-Rolfsen, we focus on braided link groups because algebraic properties of the braid group can be employed in this setting. In particular, I will discuss our implementation of an algorithm which, given a braided link group which is not bi-orderable, will return a definitive "no" and a proof in finite time. Using our program, we give a new infinite family of non-bi-orderable braided links.
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3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
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3:30 - 4:15 pm EDTUniform twisted homological stability of braid groups and moments of quadratic L-functions11th Floor Lecture Hall
- Speaker
- Peter Patzt, University of Oklahoma
- Session Chair
- Caitlin Leverson, Bard College
Abstract
A conjecture of Conrey-Farmer-Keating-Rubinstein-Snaith aims to describe the asymptotics of moments of quadratic L-functions. In joint work with Miller, Petersen, and Randal-Williams and in combination with a paper by Bergström–Diaconu–Petersen–Westerland, we proved a version of this conjecture for function fields. Using the Grothendieck-Lefschetz trace formula, Bergström–Diaconu–Petersen–Westerland showed a connection between the conjecture and the twisted homology of the braid groups. In our paper, we showed what was needed to make this connection. Homological stability says that the k-dimensional homology groups are all isomorphic for a large enough number of strands of the braid groups. This is even known for twisted coefficients pulled back from polynomial representations of the symplectic groups. We proved that the starting point of stability is independent of which irreducible polynomial representation of the symplectic groups one uses. In the talk, I will explain the connections between number theory, the braid groups, the symplectic groups, and homological stability.
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4:30 - 6:00 pm EDTReception11th Floor Collaborative Space
July 16, 2024
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8:30 - 9:00 am EDTArrive, sign in, get settled- 11th Floor Collaborative Space
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9:00 - 10:00 am EDTCheck in with TAs- 10th Floor Collaborative Space
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10:00 - 10:30 am EDTGroup WorkAssigned Group Meeting Space
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10:30 - 11:00 am EDTBreakCoffee Break
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11:00 - 11:15 am EDTQuestion of the Day- 10th Floor Collaborative Space
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11:15 am - 12:00 pm EDTGroup WorkAssigned Group Meeting Space
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12:00 - 1:30 pm EDTLunch/Free Time
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1:30 - 3:00 pm EDTGroup WorkAssigned Group Meeting Space
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3:00 - 3:30 pm EDTCoffee Break10th Floor Collaborative Space
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3:30 - 4:30 pm EDTGroup Work / Writing TimeGroup Work - Assigned Group Meeting Space
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9:30 - 10:15 am EDTSkein Lasagna Modules and Categorified Projectors11th Floor Lecture Hall
- Speaker
- Melissa Zhang, University of California, Davis
- Session Chair
- Miriam Kuzbary, Amherst College
Abstract
In 2018, Morrison, Walker, and Wedrich’s skein lasagna modules are 4-manifold invariants defined using Khovanov-Rozansky homology similarly to how skein modules for 3-manifolds are defined. In 2020, Manolescu and Neithalath developed a formula for computing this invariant for 2-handlebodies by defining an isomorphic object called cabled Khovanov-Rozansky homology; this is computed as a colimit of cables of the attaching link in the Kirby diagram of the 4-manifold.
In joint work with Ian Sullivan, we lift the Manolescu-Neithalath construction to the level of Bar-Natan's tangles and cobordisms, and trade colimits of vector spaces for a homotopy colimit in Bar-Natan's category. As an application, we give a proof that the skein lasagna module of S2xS2 is trivial, confirming a conjecture of Manolescu. Our local techniques also allow for computations of the skein lasagna invariant for other 4-manifolds whose Kirby diagram contains a 0-framed unknot component. Our methods also allow us to relate the Rozansky-Willis invariant of links in S2xS1 to skein lasagna modules. -
10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
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11:00 - 11:45 am EDTCorrection terms of branched double covers and symmetries of immersed curves11th Floor Lecture Hall
- Speaker
- Biji Wong, Duke University
- Session Chair
- Miriam Kuzbary, Amherst College
Abstract
In this talk, we'll discuss recent work to use the immersed curves description of bordered Floer theory to study the d-invariants of branched double covers Sigma_2(L) of links L in the 3-sphere. We'll show that when L is a 2-component plumbing link and Sigma_2(L) is an L-space, then the spin d-invariants of Sigma_2(L) are determined by the signatures of L. This project is joint with J. Hanselman and M. Marengon.
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12:00 - 2:00 pm EDTProblem Session in small groups over catered lunchLunch/Free Time - 11th Floor Lecture Hall
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2:00 - 2:45 pm EDTDetecting corks11th Floor Lecture Hall
- Speaker
- Abhishek Mallick, Rutgers University - New Brusnwick
- Session Chair
- Nicolas Petit, Loyola University Chicago
Abstract
Corks are fundamental to the study of exotic smooth structures on 4-manifolds. In this talk, I will describe how to detect corks and their usefulness. This is joint work with many collaborators over several projects.
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3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
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3:30 - 4:15 pm EDTA topological model for the HOMFLY-PT polynomial11th Floor Lecture Hall
- Speaker
- Christine Ruey Shan Lee, Texas State University
- Session Chair
- Nicolas Petit, Loyola University Chicago
Abstract
A topological model for a knot invariant is a realization of the invariant as graded intersection pairings on coverings of configuration spaces. In this talk I will describe a topological model for the HOMFLY-PT polynomial. I plan to discuss the motivation from previous work by Lawrence and Bigelow giving topological models for the Jones and SL_n polynomials, and the construction, joint with Cristina Anghel, which uses a state sum formulation of the HOMFLY-PT polynomial to construct an intersection pairing on the configuration space of a Heegaard surface of the link.
July 17, 2024
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8:55 - 9:00 am EDTArrive, sign in, get settled- 11th Floor Collaborative Space
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9:00 - 9:30 am EDTCheck in with TAs- 10th Floor Collaborative Space
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9:30 - 10:30 am EDTGroup WorkAssigned Group Meeting Space
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10:30 - 11:00 am EDTBreakCoffee Break - 10th Floor Collaborative Space
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11:00 - 11:15 am EDTQuestion of the Day- 10th Floor Collaborative Space
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11:15 am - 12:00 pm EDTGroup WorkAssigned Group Meeting Space
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12:00 - 1:30 pm EDTLunch/Free Time
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1:30 - 3:30 pm EDTGroup WorkAssigned Group Meeting Space
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3:00 - 3:30 pm EDTCoffee Break10th Floor Classroom
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3:30 - 4:30 pm EDTGroup Work/ Writing TimeGroup Work - Assigned Group Meeting Space
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9:30 - 10:15 am EDTtba11th Floor Lecture Hall
- Speaker
- Inanc Baykur, University of Massachusetts Amherst
- Session Chair
- Orsola Capovilla-Searle, University of California, Davis
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10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
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11:00 - 11:45 am EDTTBA11th Floor Lecture Hall
- Speaker
- Siddhi Krishna, Columbia University
- Session Chair
- Orsola Capovilla-Searle, University of California, Davis
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12:00 - 12:05 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall
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12:05 - 2:00 pm EDTLunch/Free Time
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2:00 - 2:45 pm EDTNonorientable broken Lefschetz fibrations11th Floor Lecture Hall
- Speaker
- Porter Morgan, University of Massachusetts Amherst
- Session Chair
- Marc Kegel, Humboldt-Universität zu Berlin
Abstract
Broken Lefschetz Fibrations (BLFs) are surjections from a 4-manifold to a sphere with only Lefschetz and indefinite fold singularities. Unlike Lefschetz fibrations, any generic map M^4\to S^2 is homotopic to a BLF. Among BLFs, a particularly nice subset is Simplified Broken Lefschetz Fibrations, which give an explicit handlebody decomposition of their source manifold. In this talk, we’ll review the basic topology of simplified BLFs for closed, smooth, nonorientable 4-manifolds. Then we’ll use simplified BLFs to construct an explicit embedding of nonorientable 4-manifolds into certain 6-manifolds.
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3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
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3:00 - 4:30 pm EDTDesignated collaboration/ work in groups timeGroup Work - 11th Floor Collaborative Space
July 18, 2024
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8:30 - 9:00 am EDTArrive, sign in, get settled- 11th Floor Collaborative Space
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9:00 - 9:30 am EDTCheck in with TAs- 10th Floor Collaborative Space
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9:30 - 10:30 am EDTGroup WorkAssigned Group Meeting Space
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10:30 - 11:00 am EDTBreakCoffee Break
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11:00 - 11:15 am EDTQuestion of the Day- 10th Floor Collaborative Space
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11:15 am - 12:00 pm EDTGroup WorkQuestion of the Day
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12:00 - 1:30 pm EDTLunch/Free Time
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1:30 - 3:00 pm EDTGroup WorkAssigned Group Meeting Space
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3:00 - 3:30 pm EDTCoffee Break10th Floor Collaborative Space
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3:30 - 4:30 pm EDTGroup Work / Writing TimeGroup Work - Assigned Group Meeting Space
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9:30 - 10:15 am EDTCharacterizing and non-characterizing knots by 3-manifolds11th Floor Lecture Hall
- Speaker
- Marc Kegel, Humboldt-Universität zu Berlin
- Session Chair
- Siddhi Krishna, Columbia University
Abstract
From a knot K, we can build 3-manifolds by performing Dehn surgery on that knot. We will discuss some new results explaining in which sense the diffeomorphism types of these 3-manifolds characterize the isotopy class of the knot K. This talk is based on joint work with Abe-Weiss, Baker, Baker-McCoy, Casals-Etnyre, and Piccirillo.
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10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
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11:00 - 11:45 am EDTContact surgery numbers11th Floor Lecture Hall
- Speaker
- Rima Chatterjee, University of Cologne
- Session Chair
- Marc Kegel, Humboldt-Universität zu Berlin
Abstract
A fundamental result in 3-dimensional contact topology due to Ding-Geiges tells us that any contact 3-manifold can be obtained via doing a surgery on a Legendrian link in the standard contact 3-sphere. So it's natural to ask how simple or complicated a surgery diagram could be for a particular contact manifold? Contact surgery number is a measure of this complexity. In this talk, I will define this notion of complexity and discuss some examples. This is joint work with Marc Kegel.
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12:00 - 2:00 pm EDTLunch/Free Time
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2:00 - 2:45 pm EDTExact Lagragian fillings of Legendrian links and Newton polytopes.11th Floor Lecture Hall
- Speaker
- Orsola Capovilla-Searle, University of California, Davis
- Session Chair
- Biji Wong, Duke University
Abstract
An important problem in contact topology is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Legendrian links can also arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. I will talk about connections between fillings and Newton polytopes.
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3:00 - 3:30 pm EDTCoffee Break11th Floor Collaborative Space
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3:30 - 4:15 pm EDTTBA11th Floor Lecture Hall
- Speaker
- Angela Wu, Louisiana State University
- Session Chair
- Biji Wong, Duke University
July 19, 2024
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8:30 - 9:00 am EDTArrive, sign in, get settled- 11th Floor Collaborative Space
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9:00 - 9:30 am EDTCheck in with TAs- 10th Floor Collaborative Space
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9:30 - 10:30 am EDTGroup Work10th Floor Collaborative Space
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10:30 - 11:00 am EDTBreakCoffee Break
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11:00 - 11:15 am EDTQuestion of the Day- 10th Floor Collaborative Space
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11:15 am - 12:00 pm EDTGroup WorkAssigned Group Meeting Space
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12:00 - 1:30 pm EDTLunch/Free Time
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1:30 - 3:00 pm EDTGroup WorkAssigned Group Meeting Space
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3:00 - 3:30 pm EDTCoffee Break10th Floor Classroom
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3:30 - 4:30 pm EDTGroup Work/ Writing TimeGroup Work - Assigned Group Meeting Space
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6:30 - 8:30 pm EDTMovie Night- 11th Floor Lecture Hall
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9:30 - 10:15 am EDTWork in progress — a gauge-theoretic interpretation of the McKay correspondence11th Floor Lecture Hall
- Speaker
- Jiajun Yan, University of Virginia
- Session Chair
- Jonathan Johnson, Oklahoma State University
Abstract
Let Gamma be a finite subgroup of SU(2). The McKay correspondence states that the McKay quiver of Gamma is isomorphic to the graph of minimal resolution of C^2/Gamma. There are various proofs of the McKay correspondence coming from algebraic geometry and symplectic geometry which we will survey in the talk. Then, we will present an idea / work in progress of a new gauge-theoretic interpretation of the McKay correspondence. To do so, we will review a gauge-theoretic construction of the 4-dimensional hyperkähler ALE spaces for each of which the underlying topological space is the minimal resolution of C^2/Gamma. The main approach is to make use of an S^1-invariant Morse-Bott function arising from the gauge-theoretic construction by identifying its critical points with certain flat connections that induce representations of Gamma via holonomy representation. The idea is partially inspired by the Chern-Simons theory, and some tools from contact geometry also come into play.
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10:30 - 11:00 am EDTCoffee Break11th Floor Collaborative Space
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11:00 - 11:45 am EDTNew quantum invariants from braiding Verma modules11th Floor Lecture Hall
- Speaker
- Sergei Gukov, Caltech
- Session Chair
- Jonathan Johnson, Oklahoma State University
Abstract
In this talk, I will describe recent construction of new link and 3-manifold invariants associated with Verma modules of $U_q (sl_N)$ at generic $q$. The resulting invariants can be combined into a Spin$^c$-decorated TQFT and have a nice property that, for links in general 3-manifolds, they have integer coefficients. In particular, they are expected to admit a categorification and, if time permits, I will outline various ingredients that may go into a construction of 3-manifold homology categorifying $U_q (sl_N)$ invariants at generic $q$.
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12:00 - 2:00 pm EDTLunch/Free Time
July 20, 2024
There are no events currently scheduled for July 20th.
All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC-4).
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