seminars:arit
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| **TOPICS**: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, Arithmetic Topology, etc. | **TOPICS**: Arithmetic in the broadest sense that includes Number Theory (Elementary Arithmetic, Algebraic, Analytic, Combinatorial, etc.), Algebraic Geometry, Representation Theory, Lie Groups and Lie Algebras, Diophantine Geometry, Geometry of Numbers, Tropical Geometry, Arithmetic Dynamics, Arithmetic Topology, etc. | ||
| - | **PLACE and TIME**: This semester the seminar meets primarily on Tuesdays at 4:00 pm, with possible special lectures at other days and times. The in-house talks will be in-person, while visitors outside of Binghamton area will be by Zoom: [[https://binghamton.zoom.us/j/98485937832|Zoom link]]\\ | + | **PLACE and TIME**: This semester the seminar meets primarily on Tuesdays at 4:00 pm, with possible special lectures at other days and times. The in-house talks will be in-person, while visitors outside of Binghamton area will be in-person or by Zoom: [[https://binghamton.zoom.us/j/98485937832|Zoom link]]\\ |
| **ORGANIZERS**: \\ **Regular Faculy:** [[:people:borisov:|Alexander Borisov]], [[:people:mazur:|Marcin Mazur]], [[:people:adrian:|Adrian Vasiu]]. \\ **Post-Docs:** [[:people:hdang2::|Huy Dang]] | **ORGANIZERS**: \\ **Regular Faculy:** [[:people:borisov:|Alexander Borisov]], [[:people:mazur:|Marcin Mazur]], [[:people:adrian:|Adrian Vasiu]]. \\ **Post-Docs:** [[:people:hdang2::|Huy Dang]] | ||
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| * **October 21** \\ **//Speaker//**: Alexander Borisov \\ **//Title//**: A structure sheaf for Kirch topology on $\mathbb N$ \\ **//Abstract//**: Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. One can hope that in the classical imperfect analogy between arithmetic and geometry this can serve as an arithmetic analog of the usual topology on $\mathbb C$. However, the usual topology on $\mathbb C$ comes with a structure sheaf of complex-analytic functions. As far as I know, no analog for Kirch topology has been proposed before me. I believe that I have stumbled upon just such a thing, more by accident than by a conscious effort: locally LIP functions. These are functions from Kirch-open sets to $\mathbb Z$ such that for every point in the domain there is a Kirch-open neighborhood on which the function is "locally integer polynomial" (LIP): its interpolation polynomial on every finite set has integer coefficients. I will explain why this seems to be a natural object, what I know about it and what I hope to achieve. Some of the material of this talk will be based on my recently published paper: https://math.colgate.edu/~integers/z41/z41.pdf \\ | * **October 21** \\ **//Speaker//**: Alexander Borisov \\ **//Title//**: A structure sheaf for Kirch topology on $\mathbb N$ \\ **//Abstract//**: Kirch topology on $\mathbb N$ goes back to a 1969 paper of Kirch. It can be defined by a basis of open sets that consists of all infinite arithmetic progressions $a+d\mathbb N_0$, such that $gcd(a,d)=1$ and $d$ is square-free. It is Hausdorff, connected, and locally connected. One can hope that in the classical imperfect analogy between arithmetic and geometry this can serve as an arithmetic analog of the usual topology on $\mathbb C$. However, the usual topology on $\mathbb C$ comes with a structure sheaf of complex-analytic functions. As far as I know, no analog for Kirch topology has been proposed before me. I believe that I have stumbled upon just such a thing, more by accident than by a conscious effort: locally LIP functions. These are functions from Kirch-open sets to $\mathbb Z$ such that for every point in the domain there is a Kirch-open neighborhood on which the function is "locally integer polynomial" (LIP): its interpolation polynomial on every finite set has integer coefficients. I will explain why this seems to be a natural object, what I know about it and what I hope to achieve. Some of the material of this talk will be based on my recently published paper: https://math.colgate.edu/~integers/z41/z41.pdf \\ | ||
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| - | * **October 28** \\ **//Speaker//**: TBA \\ **//Title//**: TBA \\ **//Abstract//**: TBA \\ | ||
| * **November 4** \\ **//Speaker//**: Bhargavi Parthasarathy (Syracuse University) \\ **//Title//**: Homomorphisms of maximal Cohen-Macaulay modules over the cone of an elliptic curve \\ **//Abstract//**: Consider the ring $R=k[[x,y,z]]/(f)$ where $f=x^3+y^3+z^3$ with an algebraically closed field $k$ and $char(k)\neq 3$. In a 2002 paper, Laza, Popescu and Pfister used Atiyah’s classification of vector bundles over elliptic curves to obtain a description of the maximal Cohen-Macaulay modules (MCM) over $R$. In particular, the matrix factorizations corresponding to rank one MCMs can be described using points in $V(f)$. If $M, \, N$ are rank one MCMs over $R$, then so is ${\rm Hom}_R(M,N)$. In this talk, I will discuss how the elliptic group law on $f$ can be used to obtain the point in $V(f)$ that describes the matrix factorization corresponding to ${\rm Hom}_R(M,N)$. \\ | * **November 4** \\ **//Speaker//**: Bhargavi Parthasarathy (Syracuse University) \\ **//Title//**: Homomorphisms of maximal Cohen-Macaulay modules over the cone of an elliptic curve \\ **//Abstract//**: Consider the ring $R=k[[x,y,z]]/(f)$ where $f=x^3+y^3+z^3$ with an algebraically closed field $k$ and $char(k)\neq 3$. In a 2002 paper, Laza, Popescu and Pfister used Atiyah’s classification of vector bundles over elliptic curves to obtain a description of the maximal Cohen-Macaulay modules (MCM) over $R$. In particular, the matrix factorizations corresponding to rank one MCMs can be described using points in $V(f)$. If $M, \, N$ are rank one MCMs over $R$, then so is ${\rm Hom}_R(M,N)$. In this talk, I will discuss how the elliptic group law on $f$ can be used to obtain the point in $V(f)$ that describes the matrix factorization corresponding to ${\rm Hom}_R(M,N)$. \\ | ||
| - | * **November 11** \\ **//Speaker//**: TBA \\ **//Title//**: TBA \\ **//Abstract//**: TBA \\ | ||
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| - | * **November 18** \\ **//Speaker//**: TBA \\ **//Title//**: TBA \\ **//Abstract//**: TBA \\ | ||
| - | * **November 25** (Zoom talk only) \\ **//Speaker//**: TBA \\ **//Title//**: TBA \\ **//Abstract//**: TBA \\ | + | * **November 18 (Joint with the Combinatorics Seminar)** \\ **//Speaker//**: Jaeho Shin (Seoul National University) \\ **//Title//**: Biconvex Polytopes and Tropical Linear Spaces \\ **//Abstract//**: Tropical geometry is geometry over exponents of algebraic expressions, using the "logarithmized" operations (min,+) or (max,+). In this setting, one can define tropical convexity and the related notion of biconvex polytopes, which are convex both classically and tropically. There is also a tropical analogue of linear spaces, called tropical linear spaces. Sturmfels conjectured that every biconvex polytope arises as a cell of a tropical linear space. In this talk, I will outline a proof of this conjecture. \\ |
| * **December 2** \\ **//Speaker//**: Bogdan Ion (University of Pittsburgh) \\ **//Title//**: Bernoulli operators, Dirichlet series, and analytic continuation \\ **//Abstract//**: Bernoulli operators are distributions with discrete support associated to Dirichlet series (or rather to the corresponding power series). The most basic case, when the power series has a pole singularity at $z=1$ is analyzed in detail. Its main property is that it naturally acts on the vector space of analytic functions in the plane (with possible isolated singularities) that fall in the image of the Laplace-Mellin transform (for the variable in some half-plane). The action of the Bernoulli operator on the function $t^s$, provides the analytic continuation of the associated Dirichlet series and also detailed information about the location of poles, their resides, and special values. Using examples of arithmetic origin, I will attempt to illustrate what is reasonable to expect when the power series has a non-pole singularity at $z=1$, pointing to an extension of this theory to tempered distributions associated to modular forms.\\ | * **December 2** \\ **//Speaker//**: Bogdan Ion (University of Pittsburgh) \\ **//Title//**: Bernoulli operators, Dirichlet series, and analytic continuation \\ **//Abstract//**: Bernoulli operators are distributions with discrete support associated to Dirichlet series (or rather to the corresponding power series). The most basic case, when the power series has a pole singularity at $z=1$ is analyzed in detail. Its main property is that it naturally acts on the vector space of analytic functions in the plane (with possible isolated singularities) that fall in the image of the Laplace-Mellin transform (for the variable in some half-plane). The action of the Bernoulli operator on the function $t^s$, provides the analytic continuation of the associated Dirichlet series and also detailed information about the location of poles, their resides, and special values. Using examples of arithmetic origin, I will attempt to illustrate what is reasonable to expect when the power series has a non-pole singularity at $z=1$, pointing to an extension of this theory to tempered distributions associated to modular forms.\\ | ||
seminars/arit.1760833656.txt · Last modified: 2025/10/18 20:27 by borisov