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for="abstracts-0">Show abstracts</label></li><li><input id="abstracts-1" name="abstracts" type="radio" value="hide"> <label for="abstracts-1">Hide abstracts</label></li></ul> </div> <div class="box field is-grouped is-grouped-multiline level-item"> <div class="control"> <span class="select is-small"> <select id="size" name="size"><option value="25">25</option><option selected value="50">50</option><option value="100">100</option><option value="200">200</option></select> </span> <label for="size">results per page</label>. </div> <div class="control"> <label for="order">Sort results by</label> <span class="select is-small"> <select id="order" name="order"><option selected value="-announced_date_first">Announcement date (newest first)</option><option value="announced_date_first">Announcement date (oldest first)</option><option value="-submitted_date">Submission date (newest first)</option><option value="submitted_date">Submission date (oldest first)</option><option value="">Relevance</option></select> </span> </div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2408.03111">arXiv:2408.03111</a> <span> [<a href="https://arxiv.org/pdf/2408.03111">pdf</a>, <a href="https://arxiv.org/ps/2408.03111">ps</a>, <a href="https://arxiv.org/format/2408.03111">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Root data in character varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Zamora%2C+A">Alfonso Zamora</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2408.03111v1-abstract-short" style="display: inline;"> Given $G$ an algebraic reductive group over an algebraically closed field of characteristic zero and $螕$ a finitely generated group, we provide a stratification of the $G$-character variety of $螕$ in terms of conjugacy classes of parabolic subgroups of $G$. Each stratum has the structure of a pseudo-quotient, which is a relaxed GIT notion capturing the topology of the quotient and, therefore, beha… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.03111v1-abstract-full').style.display = 'inline'; document.getElementById('2408.03111v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2408.03111v1-abstract-full" style="display: none;"> Given $G$ an algebraic reductive group over an algebraically closed field of characteristic zero and $螕$ a finitely generated group, we provide a stratification of the $G$-character variety of $螕$ in terms of conjugacy classes of parabolic subgroups of $G$. Each stratum has the structure of a pseudo-quotient, which is a relaxed GIT notion capturing the topology of the quotient and, therefore, behaving well for motivic computations of invariants of the character varieties. These stratifications are constructed by analyzing the root datum of $G$ to encode parabolic classes. Finally, detailed and explicit motivic formulae are provided for cases with Dynkin diagram of types $A$, $B$, $C$ and $D$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2408.03111v1-abstract-full').style.display = 'none'; document.getElementById('2408.03111v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 August, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages. Comments are welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14M35 (Primary); 14L24; 14D20; 14C15; 17B22 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.12286">arXiv:2402.12286</a> <span> [<a href="https://arxiv.org/pdf/2402.12286">pdf</a>, <a href="https://arxiv.org/format/2402.12286">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> </div> <p class="title is-5 mathjax"> Character varieties of torus links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+J">Javier Mart铆nez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.12286v1-abstract-short" style="display: inline;"> In this paper, we study the geometry of the moduli space of representations of the fundamental group of the complement of a torus link into an algebraic group G, an algebraic variety known as the G-character variety of the torus link. These torus links are a family of links in the 3-dimensional sphere formed by stacking several copies of torus knots. We develop an intrinsic stratification of the v… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12286v1-abstract-full').style.display = 'inline'; document.getElementById('2402.12286v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.12286v1-abstract-full" style="display: none;"> In this paper, we study the geometry of the moduli space of representations of the fundamental group of the complement of a torus link into an algebraic group G, an algebraic variety known as the G-character variety of the torus link. These torus links are a family of links in the 3-dimensional sphere formed by stacking several copies of torus knots. We develop an intrinsic stratification of the variety that allows us to relate its geometry with the one of the underlying torus knot. Using this information, we explicitly compute the E-polynomial associated to the Hodge structure of these varieties for $G=SL_2(\mathbb{C})$ and $SL_3(\mathbb{C})$, for an arbitrary torus link, showing an unexpected relation with the number of strands of the link. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12286v1-abstract-full').style.display = 'none'; document.getElementById('2402.12286v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57K31; 14D20; 14C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2309.15331">arXiv:2309.15331</a> <span> [<a href="https://arxiv.org/pdf/2309.15331">pdf</a>, <a href="https://arxiv.org/ps/2309.15331">ps</a>, <a href="https://arxiv.org/format/2309.15331">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Arithmetic-Geometric Correspondence of Character Stacks via Topological Quantum Field Theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Hablicsek%2C+M">M谩rton Hablicsek</a>, <a href="/search/math?searchtype=author&query=Vogel%2C+J">Jesse Vogel</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2309.15331v1-abstract-short" style="display: inline;"> In this paper, we introduce Topological Quantum Field Theories (TQFTs) generalizing the arithmetic computations done by Hausel and Rodr铆guez-Villegas and the geometric construction done by Logares, Mu帽oz, and Newstead to study cohomological invariants of $G$-representation varieties and $G$-character stacks. We show that these TQFTs are related via a natural transformation that we call the 'arithm… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.15331v1-abstract-full').style.display = 'inline'; document.getElementById('2309.15331v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2309.15331v1-abstract-full" style="display: none;"> In this paper, we introduce Topological Quantum Field Theories (TQFTs) generalizing the arithmetic computations done by Hausel and Rodr铆guez-Villegas and the geometric construction done by Logares, Mu帽oz, and Newstead to study cohomological invariants of $G$-representation varieties and $G$-character stacks. We show that these TQFTs are related via a natural transformation that we call the 'arithmetic-geometric correspondence' generalizing the classical formula of Frobenius on the irreducible characters of a finite group. We use this correspondence to extract some information on the character table of finite groups using the geometric TQFT, and vice versa, we greatly simplify the geometric calculations in the case of upper triangular matrices by lifting its irreducible characters to the geometric setting. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2309.15331v1-abstract-full').style.display = 'none'; document.getElementById('2309.15331v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">41 pages. Comments are welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 57R56; Secondary: 14M35; 14D23; 20C05 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2303.06218">arXiv:2303.06218</a> <span> [<a href="https://arxiv.org/pdf/2303.06218">pdf</a>, <a href="https://arxiv.org/format/2303.06218">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Stratification of $\mathrm{SU}(r)$-character varieties of twisted Hopf links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+J">Javier Mart铆nez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2303.06218v1-abstract-short" style="display: inline;"> We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $螕_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.06218v1-abstract-full').style.display = 'inline'; document.getElementById('2303.06218v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2303.06218v1-abstract-full" style="display: none;"> We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $螕_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Florentino and Lawton. In the case $r = 3$, we also describe different strata of the $\mathrm{SU}(3)$-character variety according to the semi-simple type of the representation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.06218v1-abstract-full').style.display = 'none'; document.getElementById('2303.06218v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Dedicated to Peter E. Newstead, on his $80^{\text{th}}$ birthday. 19 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57K31; 14D20; 14M35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2209.01842">arXiv:2209.01842</a> <span> [<a href="https://arxiv.org/pdf/2209.01842">pdf</a>, <a href="https://arxiv.org/format/2209.01842">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Machine Learning">cs.LG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Dynamical Systems">math.DS</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Machine Learning">stat.ML</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.3390/math9040325">10.3390/math9040325 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Dynamics of Fourier Modes in Torus Generative Adversarial Networks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mozo%2C+A">Alberto Mozo</a>, <a href="/search/math?searchtype=author&query=Talavera%2C+E">Edgar Talavera</a>, <a href="/search/math?searchtype=author&query=G%C3%B3mez-Canaval%2C+S">Sandra G贸mez-Canaval</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2209.01842v1-abstract-short" style="display: inline;"> Generative Adversarial Networks (GANs) are powerful Machine Learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable and typically it is necessary to implement several accessory heuristics to the networks to reach an acceptable convergence of the model. In this paper, we int… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.01842v1-abstract-full').style.display = 'inline'; document.getElementById('2209.01842v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2209.01842v1-abstract-full" style="display: none;"> Generative Adversarial Networks (GANs) are powerful Machine Learning models capable of generating fully synthetic samples of a desired phenomenon with a high resolution. Despite their success, the training process of a GAN is highly unstable and typically it is necessary to implement several accessory heuristics to the networks to reach an acceptable convergence of the model. In this paper, we introduce a novel method to analyze the convergence and stability in the training of Generative Adversarial Networks. For this purpose, we propose to decompose the objective function of the adversary min-max game defining a periodic GAN into its Fourier series. By studying the dynamics of the truncated Fourier series for the continuous Alternating Gradient Descend algorithm, we are able to approximate the real flow and to identify the main features of the convergence of the GAN. This approach is confirmed empirically by studying the training flow in a $2$-parametric GAN aiming to generate an unknown exponential distribution. As byproduct, we show that convergent orbits in GANs are small perturbations of periodic orbits so the Nash equillibria are spiral attractors. This theoretically justifies the slow and unstable training observed in GANs. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2209.01842v1-abstract-full').style.display = 'none'; document.getElementById('2209.01842v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 September, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">27 pages, 8 figures, 1 table. Minor typos corrected from the published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> Primary: 37N99; Secondary: 68T07 <span class="has-text-black-bis has-text-weight-semibold">ACM Class:</span> G.1.7; I.2.6 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2207.09170">arXiv:2207.09170</a> <span> [<a href="https://arxiv.org/pdf/2207.09170">pdf</a>, <a href="https://arxiv.org/format/2207.09170">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Geometry of $\mathrm{SU}(3)$-character varieties of torus knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+J">Javier Mart铆nez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2207.09170v1-abstract-short" style="display: inline;"> We describe the geometry of the character variety of representations of the knot group $螕_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation, and irreducible representations, the latter of two ty… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.09170v1-abstract-full').style.display = 'inline'; document.getElementById('2207.09170v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2207.09170v1-abstract-full" style="display: none;"> We describe the geometry of the character variety of representations of the knot group $螕_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation, and irreducible representations, the latter of two types depending on whether the matrices have distinct eigenvalues, or one of the matrices has one eigenvalue of multiplicity $2$. We describe how the closure of each stratum meets lower strata, and use this to compute the compactly supported Euler characteristic, and to prove that the inclusion of the character variety for $\mathrm{SU}(3)$ into the character variety for $\mathrm{SL}(3,\mathbb{C})$ is a homotopy equivalence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.09170v1-abstract-full').style.display = 'none'; document.getElementById('2207.09170v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 July, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57K31; 14D20; 14M35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2206.00709">arXiv:2206.00709</a> <span> [<a href="https://arxiv.org/pdf/2206.00709">pdf</a>, <a href="https://arxiv.org/format/2206.00709">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.geomphys.2023.104849">10.1016/j.geomphys.2023.104849 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantization of algebraic invariants through Topological Quantum Field Theories </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2206.00709v3-abstract-short" style="display: inline;"> In this paper we investigate the problem of constructing Topological Quantum Field Theories (TQFTs) to quantize algebraic invariants. We exhibit necessary conditions for quantizability based on Euler characteristics. In the case of surfaces, also provide a partial answer in terms of sufficient conditions by means of almost-TQFTs and almost-Frobenius algebras for wide TQFTs. As an application, we s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.00709v3-abstract-full').style.display = 'inline'; document.getElementById('2206.00709v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2206.00709v3-abstract-full" style="display: none;"> In this paper we investigate the problem of constructing Topological Quantum Field Theories (TQFTs) to quantize algebraic invariants. We exhibit necessary conditions for quantizability based on Euler characteristics. In the case of surfaces, also provide a partial answer in terms of sufficient conditions by means of almost-TQFTs and almost-Frobenius algebras for wide TQFTs. As an application, we show that the Poincar茅 polynomial of $G$-representation varieties is not a quantizable invariant by means of a monoidal TQFTs for any algebraic group $G$ of positive dimension. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2206.00709v3-abstract-full').style.display = 'none'; document.getElementById('2206.00709v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 March, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 1 June, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Introduction section improved. Typos corrected. 28 pages, 7 figures, 1 table. Comments are welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R56 (Primary); 18M05; 57K16; 14D21 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.07090">arXiv:2202.07090</a> <span> [<a href="https://arxiv.org/pdf/2202.07090">pdf</a>, <a href="https://arxiv.org/format/2202.07090">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00009-023-02300-w">10.1007/s00009-023-02300-w <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Representation varieties of twisted Hopf links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.07090v2-abstract-short" style="display: inline;"> We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representati… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.07090v2-abstract-full').style.display = 'inline'; document.getElementById('2202.07090v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.07090v2-abstract-full" style="display: none;"> We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.07090v2-abstract-full').style.display = 'none'; document.getElementById('2202.07090v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, 3 figures; v2. Corrected mistake in the E-polynomial in main theorem</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57K31; 14D20; 14C30 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mediterranean Journal of Mathematics volume 20, 2023 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2201.08699">arXiv:2201.08699</a> <span> [<a href="https://arxiv.org/pdf/2201.08699">pdf</a>, <a href="https://arxiv.org/ps/2201.08699">ps</a>, <a href="https://arxiv.org/format/2201.08699">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Virtual Classes of Character Stacks </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Hablicsek%2C+M">M谩rton Hablicsek</a>, <a href="/search/math?searchtype=author&query=Vogel%2C+J">Jesse Vogel</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2201.08699v3-abstract-short" style="display: inline;"> In this paper, we extend the Topological Quantum Field Theory developed by Gonz谩lez-Prieto, Logares, and Mu帽oz for computing virtual classes of $G$-representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks. To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.08699v3-abstract-full').style.display = 'inline'; document.getElementById('2201.08699v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2201.08699v3-abstract-full" style="display: none;"> In this paper, we extend the Topological Quantum Field Theory developed by Gonz谩lez-Prieto, Logares, and Mu帽oz for computing virtual classes of $G$-representation varieties of closed orientable surfaces in the Grothendieck ring of varieties to the setting of the character stacks. To this aim, we define a suitable Grothendieck ring of representable stacks, over which this Topological Quantum Field Theory is defined. In this way, we compute the virtual class of the character stack over $BG$, that is, a motivic decomposition of the representation variety with respect to the natural adjoint action. We apply this framework in two cases providing explicit expressions for the virtual classes of the character stacks of closed orientable surfaces of arbitrary genus. First, in the case of the affine linear group of rank $1$, the virtual class of the character stack fully remembers the natural adjoint action, in particular, the virtual class of the character variety can be straightforwardly derived. Second, we consider the non-connected group $\mathbb{G}_m \rtimes \mathbb{Z}/2\mathbb{Z}$, and we show how our theory allows us to compute motivic information of the character stacks where the classical na茂ve point-counting method fails. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.08699v3-abstract-full').style.display = 'none'; document.getElementById('2201.08699v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 January, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">38 pages. Comments are welcome!</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R56; 14D23; 14M35 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.08945">arXiv:2105.08945</a> <span> [<a href="https://arxiv.org/pdf/2105.08945">pdf</a>, <a href="https://arxiv.org/format/2105.08945">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> The point counting problem in representation varieties of torus knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.08945v2-abstract-short" style="display: inline;"> We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.08945v2-abstract-full').style.display = 'inline'; document.getElementById('2105.08945v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.08945v2-abstract-full" style="display: none;"> We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interplay between these two results in sight of Katz theorem that relates the point count polynomial with the E-polynomial. In particular, we shall show that several point count polynomials exist for these representation varieties, depending on the arithmetic between m,n and the characteristic of the field, whereas only one of them agrees with the actual E-polynomial. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.08945v2-abstract-full').style.display = 'none'; document.getElementById('2105.08945v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages, 4 figures. arXiv admin note: text overlap with arXiv:2104.13651</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14G15; 14D20; 20G15; 14C30 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2104.13651">arXiv:2104.13651</a> <span> [<a href="https://arxiv.org/pdf/2104.13651">pdf</a>, <a href="https://arxiv.org/ps/2104.13651">ps</a>, <a href="https://arxiv.org/format/2104.13651">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> </div> <p class="title is-5 mathjax"> Motive of the representation varieties of torus knots for low rank affine groups </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2104.13651v1-abstract-short" style="display: inline;"> We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C]. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2104.13651v1-abstract-full" style="display: none;"> We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.13651v1-abstract-full').style.display = 'none'; document.getElementById('2104.13651v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 April, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">13 pages, no figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C30; 57R56; 14L24; 14D21 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2011.04603">arXiv:2011.04603</a> <span> [<a href="https://arxiv.org/pdf/2011.04603">pdf</a>, <a href="https://arxiv.org/format/2011.04603">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> On character varieties of singular manifolds </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2011.04603v1-abstract-short" style="display: inline;"> In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.04603v1-abstract-full').style.display = 'inline'; document.getElementById('2011.04603v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2011.04603v1-abstract-full" style="display: none;"> In this paper, we construct a lax monoidal Topological Quantum Field Theory that computes virtual classes, in the Grothendieck ring of algebraic varieties, of $G$-representation varieties over manifolds with conic singularities, which we will call nodefolds. This construction is valid for any algebraic group $G$, in any dimension and also in the parabolic setting. In particular, this TQFT allow us to compute the virtual classes of representation varieties over complex singular planar curves. In addition, in the case $G = \mathrm{SL}_{2}(k)$, the virtual class of the associated character variety over a nodal closed orientable surface is computed both in the non-parabolic and in the parabolic scenarios. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.04603v1-abstract-full').style.display = 'none'; document.getElementById('2011.04603v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 9 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">30 pages, 4 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R56 (Primary); 19E08; 32S50; 14D21; 20G05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2006.01810">arXiv:2006.01810</a> <span> [<a href="https://arxiv.org/pdf/2006.01810">pdf</a>, <a href="https://arxiv.org/ps/2006.01810">ps</a>, <a href="https://arxiv.org/format/2006.01810">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.jalgebra.2022.06.008">10.1016/j.jalgebra.2022.06.008 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Motive of the $SL_4$-character variety of torus knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+A">Angel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2006.01810v3-abstract-short" style="display: inline;"> In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$ of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.01810v3-abstract-full').style.display = 'inline'; document.getElementById('2006.01810v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2006.01810v3-abstract-full" style="display: none;"> In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$ of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.01810v3-abstract-full').style.display = 'none'; document.getElementById('2006.01810v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 June, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">37 pages. Note: download source for the output of all the strata as separate file</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14D20; 57M25; 57M27 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Algebra Volume 610, 2022 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2005.01841">arXiv:2005.01841</a> <span> [<a href="https://arxiv.org/pdf/2005.01841">pdf</a>, <a href="https://arxiv.org/format/2005.01841">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/978-3-030-84721-0_18">10.1007/978-3-030-84721-0_18 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Representation variety for the rank one affine group </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+A">Angel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2005.01841v3-abstract-short" style="display: inline;"> The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which pe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.01841v3-abstract-full').style.display = 'inline'; document.getElementById('2005.01841v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2005.01841v3-abstract-full" style="display: none;"> The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.01841v3-abstract-full').style.display = 'none'; document.getElementById('2005.01841v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages, 3 figures. References added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R56; 14C30; 14D07; 14D21 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications, vol 179, 2021 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.05222">arXiv:1906.05222</a> <span> [<a href="https://arxiv.org/pdf/1906.05222">pdf</a>, <a href="https://arxiv.org/format/1906.05222">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.aim.2020.107148">10.1016/j.aim.2020.107148 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Virtual classes of parabolic $\mathrm{SL}_2(\mathbb{C})$-character varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.05222v2-abstract-short" style="display: inline;"> In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generating a recursive pattern. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.05222v2-abstract-full" style="display: none;"> In this paper, we compute the virtual classes in the Grothendieck ring of algebraic varieties of $\mathrm{SL}_2(\mathbb{C})$-character varieties over compact orientable surfaces with parabolic points of semi-simple type. When the parabolic punctures are chosen to be semi-simple non-generic, we show that a new interaction phenomenon appears generating a recursive pattern. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05222v2-abstract-full').style.display = 'none'; document.getElementById('1906.05222v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">33 pages, 1 figure. Minor corrections added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C30 (Primary) 57R56; 14L24; 14D21 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Mathematics, Vol. 368, 2020, 107148 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.11575">arXiv:1812.11575</a> <span> [<a href="https://arxiv.org/pdf/1812.11575">pdf</a>, <a href="https://arxiv.org/format/1812.11575">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Topological Quantum Field Theories for character varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.11575v1-abstract-short" style="display: inline;"> This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro-geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we develop a formalism that allows us to construct TQFTs… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.11575v1-abstract-full').style.display = 'inline'; document.getElementById('1812.11575v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.11575v1-abstract-full" style="display: none;"> This PhD Thesis is devoted to the study of Hodge structures on a special type of complex algebraic varieties, the so-called character varieties. For this purpose, we propose to use a powerful algebro-geometric tool coming from theoretical physics, known as Topological Quantum Field Theory (TQFT). With this idea in mind, in the present Thesis we develop a formalism that allows us to construct TQFTs from two simpler pieces of data: a field theory (geometric data) and a quantisation (algebraic data). As an application, we construct a TQFT computing Hodge structures on representation varieties and we use it for computing explicity the Deligne-Hodge polynomials of parabolic $\mathrm{SL}_2(\mathbb{C})$-character varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.11575v1-abstract-full').style.display = 'none'; document.getElementById('1812.11575v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">202 pages, 18 figures, PhD thesis (Madrid, 2018)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R56 (Primary); 14C30; 14L24; 14D07; 14D21 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1810.09714">arXiv:1810.09714</a> <span> [<a href="https://arxiv.org/pdf/1810.09714">pdf</a>, <a href="https://arxiv.org/format/1810.09714">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Motivic theory of representation varieties via Topological Quantum Field Theories </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1810.09714v2-abstract-short" style="display: inline;"> In this paper, we use lax monoidal TQFTs as an effective computational method for motivic classes of representation varieties. In particular, we perform the calculation for parabolic $\mathrm{SL}_2(\mathbb{C})$-representation varieties over a closed orientable surface of arbitrary genus and any number of marked points with holonomies of Jordan type. This technique is based on a building method of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.09714v2-abstract-full').style.display = 'inline'; document.getElementById('1810.09714v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1810.09714v2-abstract-full" style="display: none;"> In this paper, we use lax monoidal TQFTs as an effective computational method for motivic classes of representation varieties. In particular, we perform the calculation for parabolic $\mathrm{SL}_2(\mathbb{C})$-representation varieties over a closed orientable surface of arbitrary genus and any number of marked points with holonomies of Jordan type. This technique is based on a building method of lax monoidal TQFTs of physical inspiration that generalizes the construction of Gonz谩lez-Prieto, Logares and Mu帽oz. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1810.09714v2-abstract-full').style.display = 'none'; document.getElementById('1810.09714v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 July, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 October, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages, 5 figures. v2 change of title and major improvements. Extended to computation of motivic classes, removed Section 2 and added Section 3</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14C30 (Primary) 57R56; 14D07; 14D21 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1807.08540">arXiv:1807.08540</a> <span> [<a href="https://arxiv.org/pdf/1807.08540">pdf</a>, <a href="https://arxiv.org/ps/1807.08540">ps</a>, <a href="https://arxiv.org/format/1807.08540">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0219199723500098">10.1142/S0219199723500098 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Pseudo-quotients of algebraic actions and their application to character varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1807.08540v5-abstract-short" style="display: inline;"> In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT q… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.08540v5-abstract-full').style.display = 'inline'; document.getElementById('1807.08540v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1807.08540v5-abstract-full" style="display: none;"> In this paper, we propose a weak version of quotient for the algebraic action of a group on a variety, which we shall call a pseudo-quotient. They arise when we focus on the purely topological properties of good GIT quotients regardless of their algebraic properties. The flexibility granted by their topological nature enables an easier identification in geometric constructions than classical GIT quotients. We obtain several results about the interplay between pseudo-quotients and good quotients. Additionally, we show that in characteristic zero pseudo-quotients are unique up to virtual class in the Grothendieck ring of algebraic varieties. As an application, we compute the virtual class of $\mathrm{SL}_{2}(k)$-character varieties for free groups and surface groups as well as their parabolic counterparts with punctures of Jordan type. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1807.08540v5-abstract-full').style.display = 'none'; document.getElementById('1807.08540v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 July, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">39 pages, no figures. Final version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 14L24 (Primary) 14C25; 20G05 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Communications in Contemporary Mathematics (2023) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.05724">arXiv:1709.05724</a> <span> [<a href="https://arxiv.org/pdf/1709.05724">pdf</a>, <a href="https://arxiv.org/format/1709.05724">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Commutative Algebra">math.AC</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Category Theory">math.CT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.bulsci.2020.102871">10.1016/j.bulsci.2020.102871 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A lax monoidal Topological Quantum Field Theory for representation varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.05724v2-abstract-short" style="display: inline;"> We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoreti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05724v2-abstract-full').style.display = 'inline'; document.getElementById('1709.05724v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.05724v2-abstract-full" style="display: none;"> We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoretic framework allowing its application to other situations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05724v2-abstract-full').style.display = 'none'; document.getElementById('1709.05724v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 November, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, 4 figures. Added references. Added Section 5 with examples</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 57R56 (Primary); 14C30; 14D07; 14D21 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Bulletin des Sciences Math茅matiques, Vol. 161, 2020, 102871 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 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